Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces

TL;DR

This paper characterizes the distance from functions in Lipschitz spaces to their subspaces using fractional heat semigroups, providing a new analytical framework for understanding function approximation in these spaces.

Contribution

It introduces a novel fractional semigroup approach to quantify distances in Lipschitz spaces and relates these distances to the size of 'bad' sets via admissible set functions.

Findings

Characterizes distances in Lipschitz spaces using fractional heat semigroups.

Establishes equivalence between critical index and subspace distance.

Applies to various subspaces including Sobolev, Besov, and Triebel–Lizorkin spaces.

Abstract

Let $\Lambda_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{\Lambda_s}: = \inf_{g\in V} \|f-g\|_{\Lambda_s}$ from a function $f\in \Lambda_s$ to a non-dense subspace $V\subset \Lambda_s$ via the fractional semigroup $\{T_{\alpha, t}: =e^{-t (-\Delta)^{\alpha/2}}: t\in (0, \infty)\}$ for any $\alpha\in(0,\infty)$. Given an integer $ r >s/\alpha$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $\Lambda_s$ if and only if there exists a constant $\lambda\in(0,\infty)$ such that \begin{align*} \left|(-\Delta)^{\frac {\alpha r}2} (T_{\alpha, t^\alpha } f)(x) \right|\leq \lambda t^{s -r\alpha }\ \ \text{for any $x\in\mathbb{R}^n$ and $t\in (0, 1]$}.\end{align*} The least such constant is denoted by $\lambda_{ \alpha, r, s}(f)$. For each $f\in \Lambda_s$ and $0<\varepsilon< \lambda_{\alpha,r, s}(f)$, let $$ D_{\alpha, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-\Delta)^{\frac {\alpha r}2} (T_{\alpha, t^\alpha} f)(x) \right|> \varepsilon t^{s -r \alpha }\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $\nu$ on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $\nu$, the \emph{critical index} $ \varepsilon_{\alpha, r, s,\nu}(f):=\inf\{\varepsilon\in(0,\infty):\ \nu(D_{\alpha, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset \Lambda_s$, including intersections of $\Lambda_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $\nu$ depending on $V$ such that $\varepsilon_{\alpha, r, s,\nu}(f)\sim \mathrm{dist}(f, V)_{\Lambda_s}.$

Equations

$$\left|(-\Delta {)}^{\frac{\alpha r}{2}}(T_{\alpha ,t^{\alpha }}f)(x)\right|\le \lambda t^{s-r\alpha }\text{for any x∈Rn and t∈(0,1]}.$$

$$D_{\alpha ,r}(s,f,\epsilon ):=\left\{(x,t)\in {\mathbb{R}}^n\times (0,1]:\left|(-\Delta {)}^{\frac{\alpha r}{2}}(T_{\alpha ,t^{\alpha }}f)(x)\right|>\epsilon t^{s-r\alpha }\right\}$$

$$\Vert f{\Vert }_{\mathrm{BMO}}:=\underset{B}{\sup }\frac{1}{|B|}{\int }_B\left|f(x)-f_B\right|dx<\infty ,$$

$$d{\left(f,L^{\infty }\right)}_{\mathrm{BMO}}:=\underset{g\in L^{\infty }}{\inf }\Vert f-g{\Vert }_{\mathrm{BMO}}\text{for}f\in \mathrm{BMO}$$

$$\underset{B}{\sup }\frac{\mid \{x\in B:\left|f(x)-f_B\right|>\lambda \}|}{|B|}\le e^{-\frac{\lambda }{\delta }}\text{ for large}\lambda ,$$

$$d{\left(f,L^{\infty }\right)}_{\mathrm{BMO}}{\sim }_n\delta (f):=\inf \left\{\delta \in (0,\infty ):\text{eqref}1-1-b\text{holds}\right\}.$$

$$\delta (f){\sim }_n\inf \left\{\sum _{j=1}^n{\Vert g_j\Vert }_{\infty }:g_0,g_1,\dots ,g_n\in L^{\infty }\text{and}f=g_0+\sum _{j=1}^n{R}_j{g}_j\right\},$$

$$d{\left(f,{\mathrm{J}}_s(\mathrm{bmo})\right)}_{{\Lambda }_s}:=$$

$$\Vert f{\Vert }_{\mathrm{bmo}}:=\Vert f{\Vert }_{\mathrm{BMO}}+\underset{k\in {\mathbb{Z}}^n}{\sup }{\left[{\int }_{k+[0,1{)}^n}|f(x){|}^{2}dx\right]}^{\frac{1}{2}}<\infty .$$

$${\Delta }_h(f)(x):=f\left(x+\frac{h}{2}\right)-f\left(x-\frac{h}{2}\right),{\Delta }_h^{i+1}:={\Delta }_h^i({\Delta }_h),i\in \mathbb{N}.$$

$$\Vert f{\Vert }_{{\Lambda }_s}:=\Vert f{\Vert }_{L^{\infty }}+\underset{y\in (0,1]}{\sup }\underset{x\in {\mathbb{R}}^n}{\sup }\frac{{\Delta }_{r}f(x,y)}{y^s}<\infty ,$$

$${\Delta }_{k}f(x,y)=\underset{h\in {\mathbb{R}}^n,|h|=y}{\sup }|{\Delta }_h^{k}f(x)|,x\in {\mathbb{R}}^n,y\in (0,\infty ),k\in \mathbb{N}.$$

$${\Delta }_{r}f(x,y)\le L{y}^s\text{for all}x\in {\mathbb{R}}^n\text{and}y\in (0,1].$$

$$\underset{I\in \mathcal{D}}{\sup }\frac{1}{|I|}{\int }_0^{\ell (I)}\left|\left\{x\in I:{\Delta }_{r}f(x,y)>\epsilon y^s\right\}\right|\frac{dy}{y}<\infty .$$

$$S_r(s,f,\epsilon ):$$

$$d{\gamma }_{r,s,f,\epsilon }(x,y):=\frac{1_{S_r(s,f,\epsilon )}(x,y)1_{[0,1]}(y)dxdy}{y},(x,y)\in {\mathbb{R}}_{+}^{n+1}.$$

$$d(f,{\mathrm{J}}_s(\mathrm{bmo}){)}_{{\Lambda }_s}{\sim }_{s,n,r}{\epsilon }_{r,s}(f):=\inf \left\{\epsilon \in (0,\infty ):\text{eqref1-1-2 holds}\right\}.$$

$$f\in {\overline{{\mathrm{J}}_s(\mathrm{bmo})}}^{{\Lambda }_s}⟺f\in {\Lambda }_s\text{and}{\epsilon }_{r,s}(f)=0.$$

$${\overline{E}}^{{\Lambda }_s}=\left\{f\in {\Lambda }_s:d(f,E{)}_{{\Lambda }_s}=0\right\}\text{with}d{\left(f,E\right)}_{{\Lambda }_s}:=\underset{g\in E}{\inf }\Vert f-g{\Vert }_{{\Lambda }_s}.$$

$$d{\left(f,E\right)}_{{\Lambda }_s}:=d{\left(f,E\cap {\Lambda }_s\right)}_{{\Lambda }_s}\text{and}{\overline{E}}^{{\Lambda }_s}:={\overline{E\cap {\Lambda }_s}}^{{\Lambda }_s}.$$

$$d(f,W^{1,p}{)}_{{\Lambda }_s}{\sim }_{s,p,n}\inf \left\{\epsilon \in (0,\infty ):{\Vert {\int }_{\left\{y\in (0,1]:y^{-1}{\Delta }_{2}f(\cdot ,y)>\epsilon \right\}}\frac{dy}{y}\Vert }_{L^{\frac{p}{2}}}<\infty \right\}.$$

$$d(f,B_{p,q}^s{)}_{{\Lambda }_s}{\sim }_{r,s,p,q,n}\inf \left\{\epsilon \in (0,\infty ):B_{p,q,\epsilon }^s(f)<\infty \right\},$$

$$B_{p,q,\epsilon }^s(f):={\Vert {\left\{{\int }_{2^{-j-1}}^{2^{-j}}\left|\left\{x\in {\mathbb{R}}^n:y^{-s}{\Delta }_{r}f(x,y)>\epsilon \right\}\right|\frac{dy}{y}\right\}}_{j=0}^{\infty }\Vert }_{{\ell }^{\frac{q}{p}}({\mathbb{Z}}_{+})}.$$

$$d(f,F_{p,q}^s{)}_{{\Lambda }_s}{\sim }_{r,s,p,q,n}\inf \left\{\epsilon \in (0,\infty ):F_{p,q,\epsilon }^s(f)<\infty \right\},$$

$$F_{p,q,\epsilon }^s(f):={\Vert {\int }_{\left\{y\in [0,1]:y^{-s}{\Delta }_{r}f(x,y)>\epsilon \right\}}\frac{dy}{y}\Vert }_{L^{\frac{p}{q}}}.$$

$$T_{\alpha ,t}:=e^{-t(-\Delta {)}^{\alpha /2}},t\in (0,\infty ),\alpha \in (0,\infty ),$$

$$\mathcal{F}\left((-\Delta {)}^{\alpha /2}f\right)(\xi ):=|2\pi \xi {|}^{\alpha }\mathcal{F}f(\xi ).$$

$$\mathcal{F}f(\xi )\equiv \hat{f}(\xi ):={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\cdot \xi }dx.$$

$$\mathcal{F}(T_{\alpha ,t}f)(\xi )=e^{-t|2\pi \xi {|}^{\alpha }}\mathcal{F}f(\xi ),\xi \in {\mathbb{R}}^n.$$

$${\partial }_{t}u(x,t)+(-\Delta {)}^{\alpha /2}u(x,t)=0,x\in {\mathbb{R}}^n,t\in (0,\infty ),$$

Full text

1. 1 Introduction

2. 2 Summary of main results

3. 3 Framework and general results

4. 4 Preliminary results

5. 5 Kernel estimates of fractional heat semigroups

6. 6 Proof of Theorem 3.5 (heat semigroup characterization of $d(f,\!\Lambda_{\!X}^{\!s})_{\!\Lambda_{s}}$ under Assumption I) based on Propositions 6.1, 6.2, and 6.3

7. 7 Proof of Proposition 6.1

8. 8 Proof of Proposition 6.2

9. 9 Proof of Proposition 6.3

10. 10 Proof of Theorem 3.12 (heat semigroup characterization of $d(f,\Lambda_{X}^{s})_{\Lambda_{s}}$ under Assumption II)

11. 11 Applications to specific spaces

12. 11.1 Besov spaces

13. 11.2 Triebel–Lizorkin spaces

14. 11.3 Besov-type spaces

15. 11.4 Triebel–Lizorkin-type spaces

1 Introduction

Let $\mathop{\mathrm{\,BMO\,}}$ denote the space of all locally integrable functions $f$ on $\mathbb{R}^{n}$ with bounded mean oscillation

[TABLE]

where the supremum is taken over all Euclidean balls $B$ in $\mathbb{R}^{n}$ and $f_{B}:=\frac{1}{|B|}\int_{B}f(x)\,dx$. As is well known, $L^{\infty}\subsetneqq\mathrm{BMO}$. In 1978, Garnett and Jones [15] characterized the distance

[TABLE]

by the infimum of the constant $\delta\in(0,\infty)$ in the John–Nirenberg inequality (see [27]),

[TABLE]

proving that

[TABLE]

Here and throughout the article, $|E|$ denotes the Lebesgue measure of $E\subset\mathbb{R}^{n}$, the symbol $A\sim_{p,q,r}D$ means that there exist positive constants $C_{1},C_{2}$ depending only on the subscripts $p,q,r,\ldots$ such that $C_{1}A\leq D\leq C_{2}A$. This remarkable result has several important applications. First, it shows that, for any $f\in\mathop{\mathrm{\,BMO\,}}$,

[TABLE]

which complements the Fefferman–Stein characterization that $f\in\mathop{\mathrm{\,BMO\,}}$ if and only if there exist $g_{0},g_{1},\ldots,g_{n}\in L^{\infty}$ such that $f=g_{0}+\sum_{j=1}^{n}R_{j}g_{j}$ ( [15, pp.  374–375]). Here $R_{j}$ is the $j$-th Riesz transform. Second, it can be used to generalize the classical Helson–Szegö theorem to higher dimensions, involving weights $w$ such that the Riesz transforms are bounded on $L^{2}(wdx)$ ([15, Corollary 1.2]). Third, using it, Jones [28] derived sharp estimates for solutions to the corona problem for $H^{\infty}$ of the upper half-plane or unit disk, and in [29] he established the factorization of $A_{p}$ weights. Lastly, Bourgain [2] applied it to study embedding $L^{1}$ in $L^{1}/H^{1}$, proving that $L^{1}$ is isomorphic to a subspace of $L^{1}/H^{1}$. For more details, we refer to [9] and the references therein.

In [39], Saksman and Soler i Gibert studied a similar problem for distances in the nonhomogeneous Lipschitz space $\Lambda_{s}=(\Lambda_{s},\|\cdot\|_{\Lambda_{s}})$ of smooth order $s\in(0,\infty)$ on $\mathbb{R}^{n}$, seeking to determine the distance (up to a positive constant multiple) in $\Lambda_{s}$ for all $f\in\Lambda_{s}$ and $s\in(0,1]$:

[TABLE]

Here, $\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}})$ is the image of the space $\mathop{\mathrm{\,bmo\,}}$ under the Bessel potential $\mathrm{J}_{s}:=(I-\Delta)^{-\frac{s}{2}}$, and $\mathop{\mathrm{bmo}}$ is a nonhomogeneous version of the space $\mathop{\mathrm{\,BMO\,}}$ consisting of all functions $f\in\mathop{\mathrm{\,BMO\,}}$ with

[TABLE]

To state the results in [39], we begin by introducing some necessary symbols. For any $k\in\mathbb{N}$ and $h\in\mathbb{R}^{n}$, the $k$-th order symmetric difference operator $\Delta_{h}^{k}$ is defined recursively as follows: for any $f:\mathbb{R}^{n}\to\mathbb{R}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

Given $s\in(0,\infty)$, the Lipschitz space $\Lambda_{s}$ on $\mathbb{R}^{n}$ is the Banach space of all continuous functions $f:\mathbb{R}^{n}\to\mathbb{R}$ such that

[TABLE]

where $r=\lfloor s\rfloor+1$, $\lfloor s\rfloor$ denotes the largest integer not greater than $s$, and

[TABLE]

It is well known that replacing $r=\lfloor s\rfloor+1$ with any positive integer $r>s$ in the definition of $\|\cdot\|_{\Lambda_{s}}$ results in an equivalent norm. Below we will fix $s\in(0,\infty)$ and an integer $r>s$. Note that by the definition, $f\in L^{\infty}$ belongs to the space $\Lambda_{s}$ if and only if there exists a constant $L\in(0,\infty)$ such that

[TABLE]

Given $f\in\Lambda_{s}$, we define $\varepsilon_{r,s}(f)$ to be the infimum of all $\varepsilon\in(0,\infty)$ such that

[TABLE]

Here and throughout the article, $\mathcal{D}$ denotes the collection of all dyadic cubes $I$ in $\mathbb{R}^{n}$ of edge length $\ell(I)\leq 1$. For any $f\in\Lambda_{s}$ and $\varepsilon\in(0,\infty)$, we also define the set

[TABLE]

and the measure

[TABLE]

It is easily seen that (1.3) holds if and only if $\gamma_{r,s,f,\varepsilon}$ is a Carleson measure on $\mathbb{R}^{n+1}_{+}$. This means that $\varepsilon_{r,s}(f)$ is the critical index such that $\gamma_{r,s,f,\varepsilon}$ is a Carleson measure whenever $\varepsilon\in(\varepsilon_{r,s}(f),\infty)$. Remarkably, Saksman and Soler i Gibert [39] proved that, for any $s\in(0,1]$, the critical index $\varepsilon_{r,s}(f)$ characterizes the distance $d(f,\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}}))_{\Lambda_{s}}$ for any $f\in\Lambda_{s}$, showing the following theorem.

Theorem A**.**

([39]) If $s\in(0,1)$ and $r=1$ or if $s\in(0,1]$ and $r=2$, then, for any $f\in\Lambda_{s}$,

[TABLE]

Several remarks are in order. First, we point out that Theorem A was previously established by Nicolau and Soler i Gibert in [36] for $n=s=1$, however with a proof that can not be extended to higher-dimensional cases. Second, Theorem A yields the following characterization of the closure of $\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}})$ under the topology of $\Lambda_{s}$:

[TABLE]

Here and throughout the article, we denote the closure of a set $E\subset\Lambda^{s}$ under the topology of $\Lambda_{s}$ by $\overline{E}^{\Lambda_{s}}$. Clearly, for any $E\subset\Lambda_{s}$,

[TABLE]

In the case when $E$ does not lie entirely in the space $\Lambda_{s}$, we will still define with a slight abuse of the symbol

[TABLE]

In a recent article [11], we significantly extended Theorem A by using higher-order finite differences to characterize the distances $d(f,V)_{\Lambda_{s}}$ from $f\in\Lambda_{s}$ to various non-dense subspaces $V\cap\Lambda_{s}$ of $\Lambda_{s}$ for the full range of $s\in(0,\infty)$. The spaces $V$ considered in [11] include the space $\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}})$ for all $s\in(0,\infty)$, the Besov spaces $B_{p,q}^{s}$ and the Triebel–Lizorkin spaces $F_{p,q}^{s}$ for all $s\in(0,\infty)$ and $p,q\in(0,\infty]$, the Besov-type spaces, and the Triebel–Lizorkin-type spaces.

For reader’s convenience, below we summarize the main results of [11] for the aforementioned specific well-known function spaces. Assume that $s\in(0,\infty)$ and $r>s$ is a given integer. Let $p\in(0,\infty)$ and $q\in(0,\infty]$. Let $\mathcal{C}_{0}$ denote the space of all continuous functions $f:\mathbb{R}^{n}\to\mathbb{R}$ such that $\lim_{|x|\to\infty}f(x)=0$. The following results were proved in [11]:

• •

For the space $\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}})$ and $f\in\Lambda_{s}$, the conclusion of Theorem A holds for the full range of $s\in(0,\infty)$.

• •

For the Sobolev space $W^{1,p}$ and $f\in\mathcal{C}_{0}\cap\Lambda_{1}$,

[TABLE]

• •

For the Besov space $B_{p,q}^{s}$ and $f\in\mathcal{C}_{0}\cap\Lambda_{s}$,

[TABLE]

where

[TABLE]

• •

For the Triebel–Lizorkin space $F_{p,q}^{s}$ and $f\in\mathcal{C}_{0}\cap\Lambda_{s}$,

[TABLE]

where

[TABLE]

This article is highly motivated by the work in [39] and serves as a continuation of the article [11]. Our main purpose is to characterize the distance $\mathrm{dist}(f,V)_{\Lambda_{s}}:=\inf_{g\in V}\|f-g\|_{\Lambda_{s}}$ from $f\in\Lambda_{s}$ to a non-dense subspace $V\subset\Lambda_{s}$ in terms of the fractional heat semigroup

[TABLE]

where $\Delta=\sum_{j=1}^{n}\partial_{j}^{2}$ with $\partial_{j}:=\frac{\partial}{\partial x_{j}}$ for any $j\in\{1,\ldots,n\}$ denotes the Laplacian on $\mathbb{R}^{n}$. To proceed, let us recall briefly some symbols and definitions.

For any $\alpha\in(0,\infty)$, let $(-\Delta)^{\alpha/2}$ denote the fractional power of the Laplacian $-\Delta$ on $\mathbb{R}^{n}$ defined, in a distributional sense, by setting, for any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Here and throughout the article, $\mathcal{F}$ denotes the usual Fourier transform on $\mathbb{R}^{n}$ defined by setting, for any $f\in L^{1}$ and $\xi\in\mathbb{R}^{n}$,

[TABLE]

Both the Fourier transform $\mathcal{F}$ and its inverse $\mathcal{F}^{-1}$ extend to the space $\mathcal{S}^{\prime}$ of tempered distributions in the standard way. Equivalently, in terms of the Fourier transform, the fractional heat semigroup $T_{\alpha,t}$ in (1.4) satisfies

[TABLE]

Clearly, if $\alpha=1$, then $T_{\alpha,t}$ is the Poisson semigroup on $\mathbb{R}^{n}$, while, for $\alpha=2$, $T_{\alpha,2}$ is the usual heat semigroup on $\mathbb{R}^{n}$.

The fractional heat semigroup $T_{\alpha,t}$ can be used to solve the fractional heat equation,

[TABLE]

which describes diffusion processes that are non-local and exhibit anomalous behavior such as long-range interactions. Clearly, the solution of this equation is $u(x,t)=T_{\alpha,t}u_{0}(x)$, where $u_{0}(x):=u(x,0)$ is the initial condition. The fractional heat semigroup has also been used extensively to solve other fractional partial differential equations that appear in probability, financial mathematics, elasticity, and biology. For details, we refer to Section 1.1 of [47].

The fractional heat semigroup plays a significant role in characterizing various function spaces, including Besov, Triebel–Lizorkin, and Sobolev spaces; see [8, 37, 38] and the references therein. For example, given $s\in(0,\infty)$ and any integer $r>s/\alpha$, a function $f\in\mathcal{C}\cap L^{\infty}$ belongs to the space $\Lambda_{s}$ if and only if there exists a constant $\lambda\in(0,\infty)$ such that the estimate

[TABLE]

holds for all $x\in\mathbb{R}^{n}$ and $0<t\leq 1$, where $\mathcal{C}$ denotes the space of all continuous functions on $\mathbb{R}^{n}$. In this case, we also have

[TABLE]

Our aim in this article is to establish results analogous to those in [11] with $y^{-s}\Delta_{r}f(x,y)$ replaced by $y^{r-\frac{s}{\alpha}}|(-\Delta)^{\frac{\alpha r}{2}}(T_{\alpha,y}f)(x)|$, within the unified framework introduced in [11]. The main results will be described in the next section.

The remainder of this article is organized as follows. In Section 2, we summarize the main results of this article along with necessary background, using minimal symbols. In particular, we formulate our main results more precisely for several classical function spaces, including Sobolev, Besov, Triebel–Lizorkin, and Besov-type spaces.

In Section 3, we describe in detail a unified general framework within which our main results are established. This framework was first introduced in [11]. Theorems 3.5 and 3.12 provide precise formulations of our main results in this setting, where a function subspace $V\subset\Lambda_{s}$ of smooth order $s\in(0,\infty)$ is referred to as the Daubechies $s$-Lipschitz $X$-based space. This space is defined via Daubechies wavelet expansions and a given quasi-normed lattice $X$ of function sequences.

Sections 4–10 are devoted to the proofs of our main results within this general framework. Given the length and complexity of the arguments, the proofs are divided across several sections. A key technical tool is the family of higher-order ball average operators. Specifically, Section 4 collects several useful results from our earlier article [11], which will be used repeatedly in the subsequent analysis.

In Section 5, we establish pointwise kernel estimates for the fractional heat semigroup $T_{\alpha,t}$ with $\alpha,t\in(0,\infty)$, and use them to derive uniform norm estimates for the derivatives $\partial_{t}^{i}\partial_{x}^{\beta}[T_{\alpha,t}f(x)]$, where $f\in\Lambda_{s}$, $\beta\in\mathbb{Z}_{+}^{n}$ and $i\in\mathbb{Z}_{+}$. In particular, we characterize the space $\Lambda_{s}$ in terms of the uniform norms of these derivatives. These results play a crucial role in the proofs of the main theorems in later sections.

Section 6 outlines the main steps of the proof of Theorem 3.5, our first main result. This proof relies on three major technical propositions, which are also formulated in Section 6, but are proved in subsequent sections. A complete proof of Theorem 3.5, conditional on these propositions, is also given here.

The proofs of the three key propositions are carried out in Sections 7–9. These arguments are quite technical, but constitute the core of the proof of Theorem 3.5.

Section 10 presents the proof of Theorem 3.12, which is deduced from the aforementioned three propositions established in Sections 7–9.

Finally, in Section 11, the last section, we illustrate how our general results apply to various classical function spaces. In each case, it remains to verify that the space in question satisfies the assumptions required by our general framework, a task largely accomplished in our earlier work [11].

We conclude this section with some conventions on symbols. Throughout the article, since we work in $\mathbb{R}^{n}$, we omit the underlying space $\mathbb{R}^{n}$ in all the related symbols whenever there is no risk of ambiguity. Let $\mathbb{N}:=\{1,2,\ldots\}$ and $\mathbb{Z}_{+}:=\mathbb{N}\cup\{0\}$. We use $\mathbf{0}$ to denote the origin of $\mathbb{R}^{n}$. All functions on $\mathbb{R}^{n}$ and subsets of $\mathbb{R}^{n}$ are assumed to be Lebesgue measurable. Given a set $E\subset\mathbb{R}^{n}$, we denote its Lebesgue measure by $|E|$ and its characteristic function by $\mathbf{1}_{E}$. For any measurable set $E\subset\mathbb{R}^{n}$ with $|E|<\infty$ and any locally integrable function $f\in L^{1}_{\mathrm{loc}}$, we define the average of $f$ over $E$ by

[TABLE]

We denote the upper half-space $\mathbb{R}^{n}\times(0,\infty)$ by $\mathbb{R}_{+}^{n+1}$. For any $x\in\mathbb{R}^{n}$ and $r\in(0,\infty)$, let $B(x,r):=\{y\in\mathbb{R}^{n}:|x-y|<r\}$. We write $\mathbb{B}$ for the collection of all Euclidean balls $B(x,r)$ with $x\in\mathbb{R}^{n}$ and $r\in(0,\infty)$. For any $\gamma\in(0,\infty)$ and any ball $B:=B(x_{B},r_{B})$, we write $\gamma B:=B(x_{B},\gamma r_{B})$. We use the letter $C$ to denote a general positive constant, which may vary from line to line but may depend on parameters indicated by subscripts. The symbol $f\lesssim g$ means that $f\leq Cg$ for some constant $C\in(0,\infty)$. If both $f\lesssim g$ and $g\lesssim f$ hold, we write $f\sim g$. If $f\leq Cg$ and $g=h$ or $g\leq h$, we write $f\lesssim g=h$ or $f\lesssim g\leq h$, respectively. For a finite set $A$, we denote its cardinality by $\sharp A$. Finally, in all proofs we consistently retain the symbols introduced in the original theorem (or related statement).

2 Summary of main results

Our goal is to characterize the distance $\mathrm{dist}(f,V)_{\Lambda_{s}}:=\inf_{g\in V}\|f-g\|_{\Lambda_{s}}$ from a function $f\in\Lambda_{s}$ to a non-dense subspace $V\subset\Lambda_{s}$ in terms of the fractional heat semigroup $T_{\alpha,t}=e^{-t(-\Delta)^{\alpha/2}}$, $t\in(0,\infty)$. Let us describe our problem more precisely as follows. Let $f\in\mathcal{C}\cap L^{\infty}$. For convenience, we let

[TABLE]

Let $r\in\mathbb{N}$ be such that $\alpha r>s$. By definition,

[TABLE]

here and thereafter, $\partial_{n+1}:=\partial_{t}$. Thus, by (1.7), $f\in\Lambda_{s}$ if and only if there exists a constant $\lambda\in(0,\infty)$ such that the estimate

[TABLE]

holds for all $x\in\mathbb{R}^{n}$ and $t\in(0,1]$. We denote by $\lambda_{\alpha,r,s}(f)$ the least such constant. For any $0<\varepsilon<\lambda_{\alpha,r,s}(f)$, define

[TABLE]

to be the set of all points $(x,t)\in\mathbb{R}^{n}\times(0,1]$ at which (2.1) with $\varepsilon$ in place of $\lambda$ fails. To quantify the size of this set, we introduce a class of extended nonnegative valued admissible set functions $\nu$ on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n}\times[0,1])$ such that $\nu(\emptyset)=0$ and $\nu(E_{1})\leq\nu(E_{2})$ whenever $E_{1},E_{2}\in\mathcal{B}(\mathbb{R}^{n}\times[0,1])$ and $E_{1}\subset E_{2}$. Given each admissible function $\nu$, we define

[TABLE]

That is, $\varepsilon_{\alpha,r,s,\nu}(f)$ is the critical index for which $\nu(D_{\alpha,r}(s,f,\varepsilon))<\infty$ for any $\varepsilon>\varepsilon_{\alpha,r,s,\nu}(f)$. Our main question is: what conditions should be imposed on a subspace $V$ of $\Lambda_{s}$ to guarantee the existence of an admissible function $\nu$ such that, for any $f\in\Lambda_{s}$, the distance $\mathrm{dist}(f,V)_{\Lambda_{s}}$ can be characterized by the critical index $\varepsilon_{\alpha,r,s,\nu}(f)$?

We will give a partial answer to this question in a unified framework, showing that, under certain conditions on a given subspace $V\subset\Lambda_{s}$, there always exists an admissible function $\nu$ which can be expressed explicitly such that $\varepsilon_{\alpha,r,s,\nu}(f)\sim\mathrm{dist}(f,\Lambda_{s})_{\Lambda_{s}}$. Our general result applies to a broad class of subspaces $V\subset\Lambda_{s}$, including intersections of $\Lambda_{s}$ with Sobolev, Besov, Triebel–Lizorkin, and Besov-type spaces. Our results for these spaces will be in full analogy with the corresponding results proved in [11] , where we need to replace $y^{-s}\Delta_{r}f(x,y)$ by $y^{\alpha r-s}|\partial_{n+1}^{r}u_{\alpha}(x,y^{\alpha})|$. More specifically, the following results for these specific function spaces can be deduced directly from our general result: Let $r\in\mathbb{N}$ be such that $\alpha r>s+3$, and let $u_{\alpha}(x,y):=T_{\alpha,y}f(x)$ for any $f\in\Lambda_{s}$ and $(x,y)\in\mathbb{R}_{+}^{n+1}$.

• •

For the space $\mathrm{J}_{s}(\mathop{\mathrm{\,bmo\,}})$ and any $f\in\Lambda_{s}$,

[TABLE]

where

[TABLE]

• •

For the Besov space $B_{p,q}^{s}$ with $p\in(0,\infty)$ and $q\in(0,\infty]$ and for any $f\in\mathcal{C}_{0}\cap\Lambda_{s}$,

[TABLE]

where

[TABLE]

• •

For the Triebel–Lizorkin spaces $F_{p,q}^{s}$ with $p\in(0,\infty)$ and $q\in(0,\infty]$ and for any $f\in\mathcal{C}_{0}\cap\Lambda_{s}$,

[TABLE]

where

[TABLE]

For $\alpha=1$, the operators $T_{\alpha,t}$ reduce to the Poisson semigroup. In this case, the estimate (2.3) for $0<s\leq 1$ and $r=2$ was previously established in [39, Theorem 4]. By applying the fractional heat equation (1.6) twice, one can show that the function $u(x,t)=T_{1,t}f(x)$ satisfies the Laplace equation:

[TABLE]

This equation plays a crucial role in the proof for $\alpha=1$ in [39], where harmonic properties of the Poisson semigroup are utilized.

For general fractional heat semigroups, however, the situation is more subtle, and the proofs are technically more involved. Indeed, when $\alpha\neq 1,2$, the fractional heat equation (1.6) no longer corresponds to a local differential operator and, in particular, the Laplace equation is unavailable. To address this issue, we develop a new approach based on higher-order ball average operators and certain combinatorial identities. This method not only simplifies the analysis but also enables us to establish the desired result for the full range $s\in(0,\infty)$ and $\alpha\in(0,\infty)$.

As a direct application of our main results, we can describe the closures $\overline{V\cap\Lambda_{s}}^{\Lambda_{s}}$ of various classical function spaces $V$ with respect to the topology of $\Lambda_{s}$. For instance, in the case of the Besov space $B^{s}_{p,q}$ with $s\in(0,\infty)$, $p\in(0,\infty]$ and $q\in(0,\infty]$, our characterizations can be stated as follows. Let $r\in\mathbb{N}$ be such that $\alpha r>s+3$, and let $u_{\alpha}(x,y):=T_{\alpha,y}f(x)$ for any $f\in\Lambda_{s}$ and $(x,y)\in\mathbb{R}_{+}^{n+1}$.

• •

If $p\in(0,\infty)$, then $f\in\overline{B^{s}_{p,q}\cap\Lambda_{s}}^{\Lambda_{s}}$ if and only if $B_{\alpha,r,p,q,\varepsilon}^{s}(f)<\infty$ for any $\varepsilon\in(0,\infty)$ and

[TABLE]

where $B_{\alpha,r,p,q,\varepsilon}^{s}(f)$ is defined in (2.4) and $\varphi$ is the father wavelet in a Daubechies wavelet system of regularity $L>\max\{s,n(\max\{\frac{1}{p},1\}-1)-s\}$.

• •

If $p=\infty$, then $f\in\overline{B^{s}_{\infty,q}}^{\Lambda_{s}}$ if and only if

[TABLE]

where, for any $j\in\mathbb{Z},$

[TABLE]

This article establishes a more general result within a unified framework introduced in [11]. In this setting, a function space $V$ of smooth order $s\in(0,\infty)$, referred to as the Daubechies $s$-Lipschitz $X$-based space, is defined via Daubechies wavelet expansions and a given quasi-normed lattice $X$ of function sequences (see Definitions 3.1 and 3.2). This framework includes all the classical function spaces mentioned earlier, making the previously stated results become direct corollaries of the general theorem. The details of this general framework will be presented in the next section.

Finally, we point out that various characterizations of function spaces will be used throughout this article. For wavelet characterizations of Besov and Triebel–Lizorkin spaces, we refer to [34, 52, 53, 54]; for their difference characterizations, see [50, 51, 55]. Wavelet characterizations of Besov-type and Triebel–Lizorkin-type spaces can be found in [62, 33, 45, 24, 48, 14], while their difference characterizations are treated in [40, 62, 45, 26, 25]. For further extensions of these spaces, see also [3, 4, 5, 6, 7, 21, 19, 17, 20, 18]. In addition to the classical function spaces mentioned above, our main result applies to a broader class of spaces that admit wavelet-type characterizations.

3 Framework and general results

Before proceeding, let us briefly review the Daubechies wavelet system on $\mathbb{R}^{n}$ and related facts. We denote by $\mathcal{C}_{\mathrm{c}}^{r}$ the set of all $r$ times continuously differentiable real-valued functions on $\mathbb{R}^{n}$ with compact support. For any $\gamma:=(\gamma_{1},\ldots,\gamma_{n})\in\mathbb{Z}_{+}^{n}$ and $x:=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$, let $x^{\gamma}:=x_{1}^{\gamma_{1}}\cdots x_{n}^{\gamma_{n}}$. According to [34, Sections 3.8 and 3.9], for any integer $L\in(1,\infty)$ there exist functions $\varphi,\psi_{l}\in C_{\mathrm{c}}^{L}$ with $l\in\{1,\ldots,2^{n}-1\}=:\mathcal{N}$ such that, for any $\gamma\in\mathbb{Z}_{+}^{n}$ with $|\gamma|\leq L$,

[TABLE]

and the union of both

[TABLE]

and

[TABLE]

forms an orthonormal basis of $L^{2}$. The system $\Phi\cup\Psi$ is called the Daubechies wavelet system of regularity $L$ on $\mathbb{R}^{n}$. We index this system as usual via dyadic cubes as follows. For any $j\in\mathbb{Z}_{+}$, let $\mathcal{D}_{j}$ denote the class of all dyadic cubes in $\mathcal{D}$ with edge length $2^{-j}$. Let $\Omega:=\Omega_{0}\cup\Omega_{1}$, where $\Omega_{0}:=\{0\}\times\mathcal{D}_{0}$ and $\Omega_{1}:=\mathcal{N}\times\mathcal{D}$. For any $\omega=(0,I)\in\Omega_{0}$ with $I=k+[0,1)^{n}\in\mathcal{D}_{0}$, let

[TABLE]

while, for any $\omega=(l_{\omega},I_{\omega})\in\Omega_{1}$ with $l_{\omega}\in\mathcal{N}$ and $I_{\omega}=2^{-j}(k+[0,1)^{n})\in\mathcal{D}_{j}$, let

[TABLE]

For any $\omega\in\Omega$ and $f\in L^{1}_{\mathrm{loc}}$ (the set of all locally integrable functions on $\mathbb{R}^{n}$), let

[TABLE]

Throughout this article, we will fix a Daubechies wavelet system $\Phi\cup\Psi$ with large regularity $L$.

The definition of the function space $V$ in our framework also requires the concept of quasi-normed lattice of function sequences, which we now introduce. For any given $i,j\in\mathbb{Z}_{+}$, let $\delta_{i,j}:=1$ or [math] depending on whether $i=j$ or $i\neq j$. We denote by $\mathscr{M}$ the set of all measurable functions $f:\mathbb{R}^{n}\to\mathbb{R}\cup\{\pm\infty\}$ that are finite almost everywhere on $\mathbb{R}^{n}$ and by $\mathscr{M}_{\mathbb{Z}_{+}}$ the set of all sequences $\{f_{j}\}_{j\in\mathbb{Z}_{+}}$ of functions in $\mathscr{M}$. We use boldface letters $\mathbf{G},\mathbf{F},\ldots$ to denote elements in $\mathscr{M}_{\mathbb{Z}_{+}}$. For any $\gamma\in\mathbb{C}$ and $\mathbf{F}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}},\mathbf{G}:=\{g_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$, define $|\mathbf{F}|:=\{|f_{j}|\}_{j\in\mathbb{Z}_{+}},\ \gamma\mathbf{F}:=\{\gamma f_{j}\}_{j\in\mathbb{Z}_{+}}$, and $\mathbf{F}\pm\mathbf{G}:=\{f_{j}\pm g_{j}\}_{j\in\mathbb{Z}_{+}}.$ Also, we write $\mathbf{F}\leq\mathbf{G}$ if, for any $j\in\mathbb{Z}_{+}$, $f_{j}\leq g_{j}$ almost everywhere on $\mathbb{R}^{n}$. We define the left shift $\textnormal{Sh}_{L}$ and the right shift $\textnormal{Sh}_{R}$ on the space $\mathscr{M}_{\mathbb{Z}_{+}}$ as follows: for any $\mathbf{F}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$,

[TABLE]

Definition 3.1**.**

Let $\|\cdot\|_{X}:\mathscr{M}_{\mathbb{Z}_{+}}\to[0,\infty]$ be an extended-valued quasi-norm defined on the entire space $\mathscr{M}_{\mathbb{Z}_{+}}$ satisfying that $\|\mathbf{F}\|_{X}\leq\|\mathbf{G}\|_{X}$ whenever $\mathbf{F},\mathbf{G}\in\mathscr{M}_{\mathbb{Z}_{+}}$ and $|\mathbf{F}|\leq|\mathbf{G}|$. Let $X$ denote the space of all function sequences $\mathbf{F}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that $\|\mathbf{F}\|_{X}<\infty$. We call $X=(X,\|\cdot\|_{X})$ a quasi-normed lattice of function sequences if it satisfies the following two conditions:

• (i)

there exists a positive constant $C$ such that, for any $\mathbf{F}\in X$,

[TABLE]

• (ii)

for any bounded function $f:\mathbb{R}^{n}\to\mathbb{R}$ with compact support, $\{f,0,0,\ldots\}\in X$.

The function space with smooth order $s\in(0,\infty)$ in our general framework is called the Daubechies $s$-Lipschitz $X$-based space. It is defined below through the Daubechies wavelet expansion and a given quasi-normed lattice $X$ of function sequences.

Definition 3.2**.**

Let $X$ be a quasi-normed lattice of function sequences. Let $s\in(0,\infty)$. Assume that the regularity $L$ of the Daubechies wavelet system is strictly bigger than $s$. Then the Daubechies $s$-Lipschitz $X$-based space $\Lambda_{X}^{s}$ is defined to be the set of all functions $f\in\Lambda_{s}$ such that

[TABLE]

Remark 3.3**.**

In the above definition, the space $\Lambda_{X}^{s}$ formally depends on a particular choice of the wavelet systems. However, most function spaces in analysis that admit wavelet characterizations can also be equivalently defined through alternative tools such as finite differences and semigroup methods. In such cases, any Daubechies wavelet system with sufficient regularity may be used in their wavelet characterization. Therefore, for the validity of our main results for these function spaces, we may always fix a wavelet system with sufficiently large regularity $L$.

Many classical function spaces, including those mentioned function spaces above, can be viewed as the Daubechies s-Lipschitz $X$-based spaces, associated with a suitable quasi-normed lattice $X$ of function sequences, according to the usual wavelet characterizations of these spaces. For example, the wavelet characterizations of Besov spaces $B^{s}_{p,q}$ and the Triebel–Lizorkin spaces $F^{s}_{p,q}$ for any $p,q,s\in(0,\infty)$ (see [55, Proposition 1.11 and Corollary 2]) yield the following equivalences (assuming the wavelet system is sufficiently regular):

[TABLE]

and

[TABLE]

This means that both $B^{s}_{p,q}$ and $F^{s}_{p,q}$ can be regarded as Daubechies s-Lipschitz $X$-based spaces, with the quasi-norm on the underlying function sequence lattice $X$ given, respectively, by

[TABLE]

and

[TABLE]

We are now in a position to formulate the main results in the general setting. For simplicity, we adopt the following symbols for the remainder of this section:

• •

$X$ is a quasi-normed lattice of function sequences, and $\Lambda_{X}^{s}$ is the Daubechies $s$-Lipschitz $X$-based space for a fixed $s\in(0,\infty)$;

• •

$L$ denotes the regularity of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$;

• •

$T_{\alpha,t}$ is the fractional heat semigroup of order $\alpha\in(0,\infty)$ defined in (1.5), $r$ is an integer such that $\alpha r>s+3$, and $u_{\alpha}(x,t):=T_{\alpha,t}f(x)$ for any $f\in\Lambda_{s}$ and $(x,t)\in\mathbb{R}^{n+1}_{+}$.

For any $f\in\Lambda_{s}$ and $\varepsilon\in(0,\infty)$, recall that the sets $D_{\alpha,r}(s,f,\varepsilon)$ and $D_{\alpha,r,j}(s,f,\varepsilon)$, $j\in\mathbb{Z}_{+}$, are defined in (2.2) and (2.5), respectively. We also define

[TABLE]

These assumptions and symbols are understood and will not be repeated in the following statements.

For simplicity, we refer to $\alpha,s,r,L,n$ and the positive constant $C$ in Definition 3.1 as framework parameters. The framework parameters also include the parameter $\theta$ and the implicit positive constant $C$ in Assumption I below if assumed as well as the general positive constant $C$ in Definition 3.10 below if Assumption II below is assumed. Most implicit constants in this article depend on these framework parameters.

The formulation of our first main result requires the following assumption for the given quasi-normed lattice $X$ of function sequences.

Assumption I**.**

(Doubling condition of $X$) There exist positive constants $\theta$ and $C$ such that, for any sequence $\{B_{k,j}\}_{k\in\mathbb{N},j\in\mathbb{Z}_{+}}\subset\mathbb{B}$ of Euclidean balls in $\mathbb{R}^{n}$, one has

[TABLE]

Remark 3.4**.**

From Assumption I (Doubling condition of $X$), it follows that, for any $\beta\in(1,\infty)$ and any sequence $\{B_{k,j}\}_{k\in\mathbb{N},j\in\mathbb{Z}_{+}}$ in $\mathbb{B}$ of Euclidean balls in $\mathbb{R}^{n}$, one has

[TABLE]

Let us also define the following two quantities for any $f\in\Lambda_{s}$ under Assumption I:

[TABLE]

and

[TABLE]

where the set $V_{0}(s,f,\varepsilon)$ is defined in (3.2).

Theorem 3.5**.**

Assume that $L>\alpha r+1$ and $X$ satisfies Assumption I for some constant $\theta\in(0,\infty)$. Then, for any $f\in\Lambda_{s}$,

[TABLE]

where the positive equivalence constants depend only on the framework parameters.

Remark 3.6**.**

According to [11, Corollary 4.1], if the quasi-norm $\|\cdot\|_{X}$ satisfies the additional condition

[TABLE]

then the quantity $\varepsilon_{X}^{0}f$ defined in (3.3) admits the following equivalent expression in terms of the father wavelet $\varphi$ of the Daubechies wavelet system given in (3.1):

[TABLE]

As a result, under this additional assumption (3.5) on $\|\cdot\|_{X}$ and under the assumptions of Theorem 3.5, we have

[TABLE]

As a corollary of Theorem 3.5, we immediately obtain the following characterization of the closure $\overline{\Lambda_{X}^{s}}^{\Lambda_{s}}$ of $\Lambda_{X}^{s}$ under Assumption I.

Corollary 3.7**.**

Assume that $L>\alpha r+1$ and $X$ satisfies Assumption I for some constant $\theta\in(0,\infty)$. Then a function $f\in\Lambda_{s}$ belongs to $\overline{\Lambda_{X}^{s}}^{\Lambda_{s}}$ if and only if, for any $\varepsilon\in(0,\infty)$,

[TABLE]

One limitation of Theorem 3.5 is that Assumption I does not hold for certain function spaces associated with endpoint parameters, such as the Triebel–Lizorkin space $F^{s}_{\infty,q}$ and the Besov space $B^{s}_{\infty,q}$. See Section 11 for a detailed discussion. As a result, Theorem 3.5 is not applicable to these spaces. However, such endpoint spaces typically satisfy a different weaker condition, namely, Assumption II below. Under that assumption, we will establish an alternative result, Theorem 3.12, which extends our framework to include function spaces with endpoint parameters.

The formulation of Assumption II is more involved, requiring introducing some additional symbols and definitions. We denote by $\mathscr{P}({\mathbb{R}}^{n+1}_{+})$ the collection of all measurable subsets of ${\mathbb{R}}^{n+1}_{+}$. Let

[TABLE]

be the collection of all sequences of measurable subsets of ${\mathbb{R}}^{n+1}_{+}$. For any $\{A_{j}\}_{j\in\mathbb{Z}_{+}},\{B_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})$, we write $\{A_{j}\}_{j\in\mathbb{Z}_{+}}\subset\{B_{j}\}_{j\in\mathbb{Z}_{+}}$ if $A_{j}\subset B_{j}$ for any $j\in\mathbb{Z}_{+}$.

Definition 3.8**.**

The left shift $\textnormal{Sh}_{L}$ and the right shift $\textnormal{Sh}_{R}$ on $\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})$ are defined, respectively, as follows: for any $\mathbf{F}:=\{A_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})$,

[TABLE]

We also use the Poincaré hyperbolic metric $\rho:\mathbb{R}_{+}^{n+1}\times\mathbb{R}_{+}^{n+1}\to\mathbb{R}$ in the upper-half space $\mathbb{R}_{+}^{n+1}$, which is defined by setting, for any $\mathbf{x}:=(x,x_{n+1}),\mathbf{y}:=(y,y_{n+1})\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

Here, we write a vector in $\mathbb{R}_{+}^{n+1}$ in the form $\mathbf{x}:=(x,x_{n+1})$ or $\mathbf{y}:=(y,y_{n+1})$, where $x,y\in\mathbb{R}^{n}$ and $x_{n+1},y_{n+1}\in(0,\infty)$. Properties of the metric $\rho$ can be found in Appendix of [11].

For any $\mathbf{x}\in\mathbb{R}^{n+1}_{+}$ and $t\in(0,\infty)$, define

[TABLE]

Given $R\in(0,\infty)$, we define the hyperbolic $R$-neighborhood $A_{R}$ of a subset $A\subset\mathbb{R}_{+}^{n+1}$ by setting

[TABLE]

where $\rho\left(\textbf{x},A\right):=\inf_{\textbf{y}\in A}\rho\left(\textbf{x},\textbf{y}\right)$.

Definition 3.9**.**

(Property I) We say that a set $A\subset\mathbb{R}_{+}^{n+1}$ has Property I with constants $\delta,\delta^{\prime}\in(0,\infty)$ if

[TABLE]

In general, we say a set $A\subset\mathbb{R}_{+}^{n+1}$ has Property I if (3.6) holds for some constants $\delta,\delta^{\prime}\in(0,\frac{1}{10})$.

Observe that Property I describes some geometric properties of the boundary of sets.

Definition 3.10**.**

An extended non-negative valued set function $\nu:\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})\rightarrow[0,\infty]$ is called a Carleson-type measure if it satisfies the following conditions for some constant $C\in(0,\infty)$ and all $\{A_{j}\}_{j\in\mathbb{Z}_{+}},\{B_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})$:

• (i)

$\nu(\{A_{j}\}_{j\in\mathbb{Z}_{+}})\leq\nu(\{B_{j}\}_{j\in\mathbb{Z}_{+}})$ whenever $\{A_{j}\}_{j\in\mathbb{Z}_{+}}\subset\{B_{j}\}_{j\in\mathbb{Z}_{+}}$;

• (ii)

$\nu(\{A_{j}\cup B_{j}\}_{j\in\mathbb{Z}_{+}})\leq C\left[\nu(\{A_{j}\}_{j\in\mathbb{Z}_{+}})+\nu(\{B_{j}\}_{j\in\mathbb{Z}_{+}})\right]$;

• (iii)

$\nu(\textnormal{Sh}_{L}\{A_{j}\}_{j\in\mathbb{Z}_{+}})+\nu(\textnormal{Sh}_{R}\{A_{j}\}_{j\in\mathbb{Z}_{+}})\leq C\nu(\{A_{j}\}_{j\in\mathbb{Z}_{+}})$;

• (iv)

if every set in the sequence $\{A_{j}\}_{j\in\mathbb{Z}_{+}}$ has Property I with uniform constants $\delta,\delta^{\prime}\in(0,\frac{1}{10})$ (independent of $j$), then, for any $R\in(1,\infty)$,

[TABLE]

Given a function $\nu:\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})\rightarrow[0,\infty]$, we also use a slight abuse of symbol that, for any $A\in\mathscr{P}({\mathbb{R}}^{n+1}_{+})$, $\nu(A):=\nu\left(\left\{A,\emptyset,\emptyset,\ldots\right\}\right).$ For each dyadic cube $I\in\mathcal{D}$, we define $T(I):=I\times(\frac{\ell(I)}{2},\ell(I)]$.

In our second result, we assume the quasi-normed lattice $X$ satisfies the following assumption.

Assumption II**.**

(Carleson-type measure condition of $X$) There exists a Carleson-type measure $\nu:\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})\rightarrow[0,\infty]$ such that, for any collection $\mathcal{A}$ of dyadic cubes in $\mathcal{D}$,

[TABLE]

where it is agreed that $\sum_{I\in\emptyset}\mathbf{1}_{I}=0$.

Remark 3.11**.**

• (i)

It was pointed out in [11, Remark 2.8] that, if $X$ satisfies Assumption I, then $X$ also satisfies Assumption II if we choose, for any $\{A_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})$,

[TABLE]

• (ii)

It was shown in [11, Lemmas 5.7 and 5.15] that Assumption II holds for $\Lambda_{X}^{s}:=F^{s}_{\infty,q}$ with $q\in(0,\infty]$ and for $\Lambda_{X}^{s}:=B^{s}_{\infty,q}$ with $q\in(0,\infty]$, where $s\in(0,\infty)$.

Theorem 3.12**.**

Assume that $L>\alpha r+1$ and $X$ satisfies Assumption II for some Carleson-type measure $\nu:\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})\rightarrow[0,\infty]$. Then, for any $f\in\Lambda_{s}$,

[TABLE]

where

[TABLE]

[TABLE]

and the positive equivalence constants are independent of $f$.

As a direct corollary of Theorem 3.12, we immediately obtain the following characterization of the closure $\overline{\Lambda_{X}^{s}}^{\Lambda_{s}}$ of $\Lambda_{X}^{s}$ under Assumption II.

Corollary 3.13**.**

Assume that $L>\alpha r+1$ and $X$ satisfies Assumption II for some Carleson-type measure $\nu:\mathscr{P}_{\mathbb{Z}_{+}}(\mathbb{R}^{n+1}_{+})\rightarrow[0,\infty]$. Then a function $f\in\Lambda_{s}$ belongs to $\overline{\Lambda_{X}^{s}}^{\Lambda_{s}}$ if and only if, for any $\varepsilon\in(0,\infty)$,

[TABLE]

The proofs of Theorems 3.5 and 3.12 are quite long, so we divide them into several sections (Sections 4–10) and steps. A key technical tool is the family of higher-order ball average operators. Specifically, Section 4 collects several useful results from our earlier article [11], which will be used repeatedly in the subsequent analysis. In Section 5, we establish pointwise kernel estimates for the fractional heat semigroup $T_{\alpha,t}$, with $\alpha,t\in(0,\infty)$, and use them to derive uniform norm estimates for the derivatives $\partial_{t}^{i}\partial_{x}^{\beta}[T_{\alpha,t}f(x)]$, where $f\in\Lambda_{s}$, $\beta\in\mathbb{Z}_{+}^{n}$, and $i\in\mathbb{Z}_{+}$. In particular, we characterize the space $\Lambda_{s}$ in terms of the uniform norms of these derivatives. These results play a crucial role in the proofs of the main theorems in later sections. Section 6 outlines the main steps of the proof of Theorem 3.5, our first main result. This proof relies on three major technical propositions, which are also formulated in Section 6, but are proved in subsequent sections. A complete proof of Theorem 3.5, conditional on these propositions, is also given here. The proofs of the three key propositions are carried out in Sections 7–9. These arguments are quite technical, but constitute the core of the proof of Theorem 3.5. Section 10 presents the proof of Theorem 3.12, which is deduced from the aforementioned three propositions established in Sections 7–9.

4 Preliminary results

In proving the main results, we will use several useful estimates established in [11], which we present in this section.

We first recall some facts related to the Poincaré hyperbolic metric in the upper-half space $\mathbb{R}_{+}^{n+1}$.

Lemma 4.1**.**

[11, (A.4) and (A.5)]*

If $\mathbf{x},\mathbf{z}\in\mathbb{R}^{n+1}_{+}$ and $\rho(\mathbf{x},\mathbf{z})\leq 1/2$, then*

[TABLE]

and

[TABLE]

Lemma 4.2**.**

[11, Lemma A.11]* There exists a positive constant $C$ such that, for any $R\in(1,\infty)$ and $(x,y)\in\mathbb{R}_{+}^{n+1}$,*

[TABLE]

where $C\in(0,\infty)$ is a constant depending only on $n$.

Lemma 4.3**.**

[11, Lemma A.10]**

• (i)

If $\{A^{\alpha}\}_{\alpha\in\mathcal{A}}$ is a collection of subsets of $\mathbb{R}^{n+1}_{+}$ having Property I with constants $\delta,\delta^{\prime}\in(0,\frac{1}{10})$ (i.e., (3.6) is satisfied for any set in the collection), then the union $\bigcup_{\alpha\in\mathcal{A}}A^{\alpha}$ also has Property I with constants depending only on $\delta,\delta^{\prime}$.

• (ii)

For any $E\subset\mathbb{R}^{n+1}_{+}$ and any constant $a\in(0,\infty)$, the set $E_{a}:=\{\mathbf{x}\in\mathbb{R}_{+}^{n+1}:\ d(\mathbf{x},E)<a\}$ has Property I with constants depending only on $a$.

Finally, the hyperbolic length $L(\gamma)$ of a piecewise $C^{1}$-curve

[TABLE]

is given by setting

[TABLE]

It is well known (see [11, Theorem A.4]) that, for any two distinct points $\mathbf{p},\mathbf{q}\in\mathbb{R}_{+}^{n+1}$, there exists a unique $C^{1}$-curve $\gamma(\mathbf{p},\mathbf{q})$ connecting $\mathbf{p}$ and $\mathbf{q}$ such that

[TABLE]

with the infimum being taken over all piecewise $C^{1}$-curves $\gamma$ in $\mathbb{R}_{+}^{n+1}$ connecting $\mathbf{p},\mathbf{q}\in\mathbb{R}_{+}^{n+1}$. The curve $\gamma(\mathbf{p},\mathbf{q})$ is called the geodesic connecting $\mathbf{p}$ and $\mathbf{q}$.

Lemma 4.4**.**

Let $f\in C^{1}(\mathbb{R}_{+}^{n+1})$ and $\mathbf{p},\mathbf{q}\in\mathbb{R}_{+}^{n+1}$. Then

[TABLE]

where $\mathbf{x}=(x,x_{n+1})\in\mathbb{R}_{+}^{n+1}$ and $\gamma(\mathbf{p},\mathbf{q})$ is the geodesic connecting $\mathbf{p}$ and $\mathbf{q}$.

Proof.

Let $\gamma:\ [0,1]\to\mathbb{R}_{+}^{n+1}$ be a parametric representation of the geodesic $\gamma(\mathbf{p},\mathbf{q})$ such that $\gamma(0)=\mathbf{p}$ and $\gamma(1)=\mathbf{q}$. Then we have

[TABLE]

This finishes the proof of Lemma 4.4. ∎

We now state several key estimates established in [11]. Let $X$ be a quasi-normed lattice of function sequences, and let $s\in(0,\infty)$. Denote by $\Lambda_{X}^{s}$ the Daubechies $s$-Lipschitz $X$-based space. Let $r_{1}\in\mathbb{N}$ be such that $r_{1}>s$, and let $L$ denote the regularity of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$. We retain this symbol in the statements of the next four lemmas. All positive constants of equivalence in the estimates below are independent of $f\in\Lambda_{s}$.

The following lemmas, established in [11], play a crucial role in the proofs of the main results of this article.

Lemma 4.5**.**

[11, Lemma 4.13]* Assume that $X$ satisfies Assumption I for some $\theta\in(0,\infty)$. Let $\delta\in(0,1/4)$ and $R>\delta$. Then, for any sequence $\{A_{j}\}_{j\in\mathbb{Z}_{+}}$ of measurable subsets of $\mathbb{R}^{n+1}_{+}$,*

[TABLE]

where $C\in(0,\infty)$ is a constant depending only on $R$, $\delta$ and the framework parameters.

Lemma 4.6**.**

[11, Theorem 2.4]* Assume that $L>\max\{s,r_{1}-1\}$ and $X$ satisfies Assumption I for some $\theta\in(0,\infty)$. Then, for any $f\in\Lambda_{s}$,*

[TABLE]

where the positive equivalence constants depend only on the framework parameters, the quantity $\varepsilon_{X}^{0}f$ is given in (3.3),

[TABLE]

and, for any $j\in\mathbb{Z}$,

[TABLE]

Lemma 4.7**.**

[11, Theorem 4.4]* Assume that $f\in\Lambda_{s}$ and $0<\varepsilon_{1}<\varepsilon$. Let $S_{r_{1},j}(s,f,\cdot)$ be defined in (4.3). Then there exists a positive constant $\delta$, depending only on $f$, $\varepsilon$, $\varepsilon_{1}$ and the framework parameters, such that, for any $j\in\mathbb{Z}$,*

[TABLE]

Lemma 4.8**.**

[11, Theorem 2.5]* Assume that $L>\max\{s,r_{1}-1\}$ and $X$ satisfies Assumption II for some Carleson-type measure $\nu$. Then, for any $f\in\Lambda_{s}$,*

[TABLE]

where the positive equivalence constants depend only on the framework parameters, $\varepsilon_{X,\nu}^{0}$ is defined in (3.7), and

[TABLE]

5 Kernel estimates of fractional heat

semigroups

In this section, we establish pointwise kernel estimates for the fractional heat semigroup $T_{\alpha,t}$, with $t\in(0,\infty)$ and $\alpha\in(0,\infty)$, and use them to derive uniform norm estimates for the derivatives $\partial_{t}^{i}\partial_{x}^{\beta}[T_{\alpha,t}f(x)]$, where $f\in\Lambda_{s}$, $\beta\in\mathbb{Z}_{+}^{n}$ and $i\in\mathbb{Z}_{+}$. In particular, we characterize the space $\Lambda_{s}$ in terms of the uniform norms of these derivatives. These results play a crucial role in the proofs of the main theorems presented in subsequent sections. Finally, we note that some of these results are likely known at least in special cases such as $\alpha=1$ or $\alpha=2$; however, due to the lack of precise references, we provide self-contained proofs for completeness.

We begin with estimates for fractional heat kernels. For any $\alpha\in(0,\infty)$, let $K_{\alpha}$ be the Fourier transform of the function $e^{-|2\pi\cdot|^{\alpha}}$; that is,

[TABLE]

In particular, when $\alpha=1$, $K_{\alpha}$ is the classical Poisson kernel, $K_{1}(x)=C_{n}(1+|x|^{2})^{-\frac{n+1}{2}}$ for any $x\in\mathbb{R}^{n}$, while, for $\alpha=2$, $K_{\alpha}$ coincides with the heat kernel, $K_{2}(x)=C_{n}e^{-|x|^{2}/4}.$ For general $\alpha\in(0,\infty)$, the function $K_{\alpha}$ satisfies the following estimates.

Theorem 5.1**.**

For any given $\alpha\in(0,\infty)$, $K_{\alpha}$ is a $C^{\infty}$-function on $\mathbb{R}^{n}$ satisfying that there exists a positive constant $C_{k,\alpha,n}$, depending only on $k,\alpha$ and $n$, such that, for any $k\in\mathbb{Z}_{+}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

In particular, Theorem 5.1 implies that $K_{\alpha}$ is a radial and integrable function on $\mathbb{R}^{n}$. Since

[TABLE]

it follows that

[TABLE]

where

[TABLE]

As a direct consequence of Theorem 5.1, we obtain the following result.

Corollary 5.2**.**

Let $\alpha\in(0,\infty)$ and $k,j\in\mathbb{Z}_{+}$. Then there exists a positive constant $C$ such that, for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$,

[TABLE]

For later applications, it will be more convenient to express the estimate (5.3) in the form

[TABLE]

where

[TABLE]

The estimates in Theorem 5.1 for the fractional heat kernels are likely known, particularly for $\alpha\in(0,2]$, but since we were unable to locate precise references, we will include a self-contained proof for completeness. When $\alpha$ is a positive even integer, the function $e^{-|\cdot|^{\alpha}}$ is a Schwartz function on $\mathbb{R}^{n}$, and therefore the estimate (5.1) is not sharp. For example, in the case $\alpha=2$, the kernel $K_{2}(x)=c_{n}e^{-|x|^{2}}$ is the classical heat kernel, which decays exponentially at infinity. Nevertheless, the estimates given in the theorem are sufficient for our purposes.

To prove Theorem 5.1, we begin by recalling some known facts about homogeneous distributions. Let $\mathcal{S}$ denote the space of all Schwartz functions on $\mathbb{R}^{n}$. For any $\gamma:=(\gamma_{1},\ldots,\gamma_{n})\in\mathbb{Z}_{+}^{n}$ and $x:=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$, let $|\gamma|:=\gamma_{1}+\cdots+\gamma_{n}$, $x^{\gamma}:=x_{1}^{\gamma_{1}}\cdots x_{n}^{\gamma_{n}}$, and $\partial^{\gamma}:=(\frac{\partial}{\partial x_{1}})^{\gamma_{1}}\cdots(\frac{\partial}{\partial x_{n}})^{\gamma_{n}}$.

Definition 5.3**.**

[16, pp. 128, (2.4.7)] Let $z\in\mathbb{C}$ be such that $\text{Re}\,z>-n-m-1$ for a positive integer $m\in\mathbb{N}$. The homogeneous tempered distribution $u_{z}:\mathcal{S}\to\mathbb{C}$ is defined for any $f\in\mathcal{S}$ by setting

[TABLE]

where $a_{n,z}=\frac{\pi^{\frac{z+n}{2}}}{\Gamma\left(\frac{z+n}{2}\right)},$

[TABLE]

It is well known that the above definition is independent of the selection of the integer $m$ and, moreover, if $\mathrm{Re}z>-n$, then the homogeneous distribution $u_{z}$ coincides with the locally integrable function $a_{n,z}|\cdot|^{z}$ on $\mathbb{R}^{n}$ (see [16, p. 127]). Furthermore, a straightforward calculation shows that (5.3) can be equivalently expressed, in terms of a $C^{\infty}$-function $\eta$ on $\mathbb{R}^{n}$ such that $\eta(x)=1$ for $|x|\leq 1$ and $\eta(x)=0$ for $|x|\geq 2$, as follows:

[TABLE]

where

[TABLE]

We now recall the following well-known result on the distributional Fourier transform of homogeneous distributions.

Lemma 5.4**.**

[16, Theorem 2.4.6]* For any $z\in\mathbb{C}$, $\widehat{u_{z}}=u_{-n-z},$ where $\widehat{u_{z}}$ denotes the distributional Fourier transform of $u_{z}$.*

For later use, we define the modulation and the translation operators for any $f\in L_{\mathrm{loc}}^{1}$, $\xi\in\mathbb{R}^{n}$, and $x\in\mathbb{R}^{n}$, respectively, by setting

[TABLE]

The following lemma will play an important role in the proof of Theorem 5.1.

Lemma 5.5**.**

Assume that $\gamma\in(0,\infty)$, $\beta\in\mathbb{Z}_{+}^{n}$, and $g\in\mathcal{S}$. Let

[TABLE]

Then there exists a positive constant $C_{n,\gamma,\beta,g}$, depending only on $n,\gamma,\beta$ and $g$, such that, for any $x\in\mathbb{R}^{n}$,

[TABLE]

Proof.

For convenience, we let $g_{\beta}(x):=x^{\beta}g(x)$ for any $x\in\mathbb{R}^{n}$ and let $G=\mathcal{F}^{-1}g$. Then both $G$ and $g_{\beta}$ are Schwartz functions, and $\mathcal{F}^{-1}g_{\beta}=c_{\beta}\partial^{\beta}G$. Since the homogeneous distribution $u_{\gamma}$ coincides with the locally integrable function $c_{n,\gamma}|\cdot|^{\gamma}$, we have

[TABLE]

where the last step uses Lemma 5.4.

Next, let $m\in\mathbb{N}$ be such that $m\geq\gamma+1$, and let $\eta$ be a $C^{\infty}$-function on $\mathbb{R}^{n}$ such that $\eta(x)=1$ for $|x|\leq 1$ and $\eta(x)=0$ for $|x|\geq 2$. Using (5) with $z=-n-\gamma$ and taking into account the fact that $G\in\mathcal{S}$, we conclude that

[TABLE]

However, a straightforward calculation shows that

[TABLE]

Thus, putting the above estimates together, we obtain the desired estimate for $|H_{\gamma,\beta}(x)|$, which hence completes the proof of Lemma 5.5. ∎

Proof of Theorem 5.1.

Let $G_{\alpha}(x):=e^{-|2\pi x|^{\alpha}}$ for any $x\in\mathbb{R}^{n}$. Without loss of generality, we may assume that $\alpha$ is not an even integer because otherwise $G_{\alpha}$ is a Schwartz function and the stated estimate holds trivially. Since $K_{\alpha}=\widehat{G_{\alpha}}$ and the function $(1+|\cdot|)^{i}G_{\alpha}$ is integrable over $\mathbb{R}^{n}$ for any integer $i\in\mathbb{N}$, it follows that $K_{\alpha}$ is a $C^{\infty}$-function on $\mathbb{R}^{n}$.

To prove the estimate (5.1), let $\beta\in\mathbb{Z}_{+}^{n}$ be such that $|\beta|=k$, and let $\eta$ be a $C^{\infty}$-function on $\mathbb{R}^{n}$ such that $\eta(x)=1$ for $|x|\leq 1$ and $\eta(x)=0$ for $|x|\geq 2$. We then write

[TABLE]

where $I_{\alpha,\beta}(x):=c\int_{\mathbb{R}^{n}}\eta(y)e^{-|2\pi y|^{\alpha}}y^{\beta}e^{-2\pi ix\cdot y}\,dy,$

[TABLE]

and $c$ is a constant independent of $x$.

For the integral $J_{\alpha,\beta}$, we write $J_{\alpha,\beta}(x)=c\widehat{F_{\alpha,\beta}}(x)$, where $F_{\alpha,\beta}(y):=[1-\eta(y)]e^{-|2\pi y|^{\alpha}}y^{\beta}$ for any $y\in\mathbb{R}^{n}$. Since $F_{\alpha,\beta}$ is a Schwartz function on $\mathbb{R}^{n}$, we deduce that

[TABLE]

To estimate the integral $I_{\alpha,\beta}(x)$, let $N,\ell\in\mathbb{N}$ be such that $n+\alpha+k\leq N<n+k+\alpha+1$ and $N+2\leq\ell\alpha$. Write

[TABLE]

where

[TABLE]

and

[TABLE]

Since $\ell\alpha+k\geq N+2$, it follows from Taylor’s theorem that $D_{\ell}\in C_{c}^{N}$, which implies

[TABLE]

It remains to estimate the integral $I_{\alpha,\beta,j}(x)$ for $0\leq j\leq\ell$. For $j=0$, since $y^{\beta}\eta(y)$ is a Schwartz function, it follows that

[TABLE]

For $1\leq j\leq\ell$, we use Lemma 5.5 to obtain

[TABLE]

Substituting the above estimates into (5.10) yields

[TABLE]

Finally, combining (5.9) and (5.11) with (5.8), we conclude the desired estimate (5.1). This finishes the proof of Theorem 5.1. ∎

Next, we apply the estimates from Theorem 5.1 to characterize the space $\Lambda_{s}$ through the derivatives of the fractional heat semigroup. To this end, let us first review some semigroup properties of $T_{\alpha,t}$ with $\alpha\in(0,\infty)$ and $t\in[0,\infty)$. Let $Y$ denote the Banach space of all bounded uniformly continuous functions on $\mathbb{R}^{n}$, equipped with the uniform norm $\|\cdot\|_{\infty}$. For simplicity, we fix $\alpha\in(0,\infty)$ and let $\mathcal{A}:=(-\Delta)^{\alpha/2}$. For each integer $r\in\mathbb{N}$, define

[TABLE]

Clearly, ${\mathcal{D}}(\mathcal{A}^{r})$ contains all bounded continuous functions on $\mathbb{R}^{n}$ whose distributional Fourier transforms have compact support. In particular, this implies that ${\mathcal{D}}(\mathcal{A}^{r})$ is a dense subset of $Y$.

By Corollary 5.2 and the definition (1.5), it is easy to verify that the family $\{T_{\alpha,t}\}_{t\in(0,\infty)}$ consists of uniformly bounded operators on $L^{\infty}$ satisfying the following conditions:

1. (i)

$T_{\alpha,0}=I$ and $\displaystyle\lim_{t\rightarrow 0+}\|T_{\alpha,t}f-f\|_{\infty}=0$ for any $f\in Y$, here and hereafter, we denote $\|\cdot\|_{L^{\infty}}$ simply by $\|\cdot\|_{\infty}$; 2. (ii)

$T_{\alpha,t_{1}+t_{2}}=T_{\alpha,t_{1}}T_{\alpha,t_{2}}$ for all $t_{1},t_{2}\in[0,\infty)$; 3. (iii)

for any $f\in\mathcal{D}(\mathcal{A})$,

[TABLE] 4. (iv)

for any $t\in(0,\infty)$ and $f\in Y$, we have $T_{\alpha,t}f\in{\mathcal{D}}(\mathcal{A})$ and $\frac{\partial}{\partial t}T_{\alpha,t}f=\mathcal{A}T_{\alpha,t}f$ ; moreover, there exists a constant $C\in(1,\infty)$ such that, for any $t\in(0,\infty)$ and $f\in Y$,

[TABLE] 5. (v)

for any $t\in(0,\infty)$ and $f\in\mathcal{D}(\mathcal{A})$, we have

[TABLE]

In particular, this means that $\{T_{\alpha,t}\}_{t\in(0,\infty)}$ is a holomorphic semigroup on the space $Y$ with the infinitesimal generator $\mathcal{A}=(-\Delta)^{\alpha/2}$ (see [13, Section 5]). It follows by [13, Theorem 5.1] that, for any $r\in\mathbb{N}$ and $f\in Y$,

[TABLE]

where the positive constants of equivalence are independent of $f$ and $t$, and $K(f,\mathcal{A}^{r},t)_{\infty}$ is the $K$-functional defined by

[TABLE]

Furthermore, from the proof of [13, Theorem 5.1], it follows that there exists a positive integer $m$, depending only on $\alpha$ and $n$, such that, for any $r\in\mathbb{N}$ and $f\in Y$,

[TABLE]

The space $\Lambda_{s}$ can be characterized in terms of the fractional heat semigroup as follows.

Theorem 5.6**.**

Let $s,\alpha\in(0,\infty)$ and $r\in\mathbb{N}$ be such that $\alpha r>s$, and let $f\in L^{\infty}\cap\mathcal{C}$. Then $f$ belongs to the space $\Lambda_{s}$ if and only if

[TABLE]

in which case,

[TABLE]

Proof.

As is well known (see [1, Theorem 6.7.4]),

[TABLE]

in which case, we have

[TABLE]

If $f\in\Lambda_{s}$, then, using (5.12) and (5.13), we obtain

[TABLE]

where the last step used (5.17).

Conversely, assume that (5.14) holds; that is, there exists a constant $A\in(0,\infty)$ such that

[TABLE]

Using (5.12) and (5.17), we find that

[TABLE]

Let $u_{\alpha}(x,t):=T_{\alpha,t}f(x)$ for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. Then, for any $t\geq 0$, we have

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

which, using (5.18), implies

[TABLE]

Substituting this last estimate with $t^{\alpha}$ in place of $t$ into (5.19) yields the desired estimate:

[TABLE]

This finishes the proof of Theorem 5.6. ∎

Let $f\in\Lambda_{s}$. Define $u_{\alpha}(x,t)=T_{\alpha,t}f(x)$ for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. By (5.15), if $r\in\mathbb{N}$ and $r>\frac{s}{\alpha}$, then

[TABLE]

The following theorem extends the inequality (5.20).

Theorem 5.7**.**

Let $i\in\mathbb{Z}_{+}$ and $\beta\in\mathbb{Z}_{+}^{n}$ with $|\beta|+i\alpha>s$. Let $f\in\Lambda_{s}$ and $u_{\alpha}(x,t):=T_{\alpha,t}f(x)$ for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. Then there exists a positive constant $C_{\alpha,\beta,n,i}$, depending only on $\alpha,\beta,n$ and $i$, such that, for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$,

[TABLE]

Proof.

We divide the proof into two steps.

Step 1. We prove that (5.21) holds for all $\beta\in\mathbb{Z}_{+}^{n}$ and $i\in\mathbb{Z}_{+}$ with $i>\frac{s}{\alpha}$.

Let $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. By (5.2), we have, for any $t_{1},t_{2}\in(0,\infty)$,

[TABLE]

This implies

[TABLE]

Setting $t_{1}=t_{2}=\frac{t}{2}$ yields

[TABLE]

By (5.20) applied to $r=i>\frac{s}{\alpha}$, we then conclude that

[TABLE]

which, using (5.3), is estimated by

[TABLE]

This proves (5.21) for all $\beta\in\mathbb{Z}_{+}^{n}$ and $i\in\mathbb{Z}_{+}$ with $i>\frac{s}{\alpha}$.

Step 2. Prove that, if $j\in\mathbb{Z}_{+}$, $\beta\in\mathbb{Z}_{+}^{n}$, and $|\beta|+j\alpha>s$, then, for any $(x,t)\in\mathbb{R}_{+}^{n+1}$, we have

[TABLE]

We claim that the present theorem follows from (5.22) and (5.21) by using the case already established in Step 1. Indeed, once (5.22) is proven, then, for any $j\in\mathbb{Z}_{+}$ and $\beta\in\mathbb{Z}_{+}^{n}$ with $|\beta|+j\alpha>s$, we have

[TABLE]

This means that, if (5.21) holds for some $i=j+1$ and $\beta\in\mathbb{Z}_{+}^{n}$ with $|\beta|+(j+1)\alpha>s$, then it also holds for $i=j$ and the same $\beta\in\mathbb{Z}_{+}^{n}$ satisfying $|\beta|+j\alpha>s$. Since (5.21) has already been verified for all $i>\frac{s}{\alpha}$ and $\beta\in\mathbb{Z}_{+}^{n}$ in Step 1, this implies that (5.21) holds for all $i\in\mathbb{Z}_{+}$ and $\beta\in\mathbb{Z}_{+}^{n}$ such that $|\beta|+j\alpha>s$.

To show (5.22), we first observe from (5.2) that

[TABLE]

Thus, using (5.3), we obtain

[TABLE]

It follows that

[TABLE]

where the last step used the assumption that $j+\frac{|\beta|}{\alpha}>\frac{s}{\alpha}$. This yields the desired estimate (5.22), which completes the proof of Theorem 5.7. ∎

6 Proof of Theorem 3.5 (heat semigroup

characterization of $d(f,\!\Lambda_{\!X}^{\!s})_{\!\Lambda_{s}}$ under Assumption I) based on Propositions 6.1, 6.2, and 6.3

The proof of Theorem 3.5 is quite long. We outline it in this section. Let $s,\alpha\in(0,\infty)$ and $r\in\mathbb{N}$ be such that $\alpha r>s+3$. Let $L$ denote the regularity of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$. Recall that we refer to $\alpha,s,r,L,\theta$ and $n$ as framework parameters. Throughout this section, the letters $C$ and $c$ denote sufficiently large and sufficiently small constants, respectively, both depending only on the framework parameters. The exact values of these constants are not important.

We decompose the proof of Theorem 3.5 into several steps. In the first step, we establish the following proposition, which provides an estimate for the neighborhood of the set $D_{\alpha,r,j}(s,f,\varepsilon)$ under the hyperbolic metric.

Proposition 6.1**.**

Let $f\in\Lambda_{s}$ and $\varepsilon,\varepsilon_{1}\in(0,\infty)$ with $\varepsilon>\varepsilon_{1}$. Then there exists a positive constant $\delta$, depending only on $\|f\|_{\Lambda_{s}}$, $\varepsilon$, $\varepsilon_{1}$, and the framework parameters, such that, for any $j\in\mathbb{\mathbb{missing}}{Z}$,

[TABLE]

where, for any $j\in\mathbb{Z}$,

[TABLE]

The proof of this proposition will be given in Section 7.

Next, in the second step, we use finite differences to establish certain lower estimates of the neighborhood of the set $D_{\alpha,r,j}(s,f,\varepsilon)$. For any integer $r_{1}>s$, $\varepsilon\in(0,\infty)$, $j\in\mathbb{Z}$, and $f\in\Lambda_{s}$, define

[TABLE]

Proposition 6.2**.**

Let $r_{1}\in\mathbb{N}$ be such that $r_{1}>\alpha r+1$. Then, given any $f\in\Lambda_{s}$ and $\varepsilon\in(0,\infty)$, there exists a constant $R\in(2,\infty)$, depending only on $\|f\|_{\Lambda_{s}}$, $\varepsilon$, $r_{1}$ and the framework parameters, such that, for any $j\in\mathbb{Z}_{+}$,

[TABLE]

where $m$ denotes the positive integer such that $2^{m-1}<R\leq 2^{m}$.

The proof of this proposition will be given in Section 9.

Finally, in the last step, we use finite differences to establish certain upper estimates of the neighborhood of the set $D_{\alpha,r,j}(s,f,\varepsilon)$.

Proposition 6.3**.**

Assume that $\alpha r>s+3$. Let $\ell\in\mathbb{N}$ be such that $\frac{s}{2}<\ell<\frac{\alpha r-1}{2}$. Then, given any $\varepsilon\in(0,\infty)$ and $f\in\Lambda_{s}$, there exists a positive constant $R\in(1,\infty)$, depending only on $\|f\|_{\Lambda_{s}}$, $\varepsilon$, and the framework parameters, such that, for any $j\in\mathbb{Z}_{+}$,

[TABLE]

The proof of Proposition 6.3 will be given in Section 9.

For the moment, we take Propositions 6.1-6.3 for granted and proceed with the proof of Theorem 3.5.

Proof of Theorem 3.5 assuming

Propositions 6.1-6.3.

First, we prove the upper bound,

[TABLE]

where the constant $C$ depends only on the framework parameters. Let $r_{1}\in\mathbb{N}$ be such that $r_{1}>s$. By Lemma 4.6, it is enough to show that there exists a constant $c\in(0,1)$, depending only on the framework parameters, such that, for any $\varepsilon\in(0,\infty)$,

[TABLE]

Here and throughout the proof, the implicit constant in $\lesssim$ may depend on $f$, the framework parameters, and other parameters involved the proof, such as $\varepsilon$, $r_{1}$, $R$, $m$ and $\delta$.

Indeed, once (6.2) is proven, then, by (3.4) and (4.2), we conclude that

[TABLE]

which, using Lemma 4.6, will imply the upper estimate (6.1).

To show (6.2), we use Proposition 6.2 to obtain

[TABLE]

which, using Lemma 4.5 and the boundedness of the left shift, is estimated by

[TABLE]

Applying Proposition 6.1 to the pair of parameters $(c\varepsilon,2^{-1}c\varepsilon)$ and choosing the constant $\delta\in(0,1)$ small enough yield

[TABLE]

where the last step used the boundedness of the left shift. This shows the estimate (6.2) and hence the upper bound (6.1).

Next, we prove the lower estimate:

[TABLE]

Let $\ell\in\mathbb{N}$ be such that $\frac{s}{2}<\ell<\frac{\alpha r-1}{2}$. By Lemma 4.6 applied to $r=2\ell$, it is sufficient to show that

[TABLE]

Indeed, (6.4) implies

[TABLE]

and hence the lower estimate (6.3), according to Lemma 4.6.

To show (6.4), we use Proposition 6.3 and the boundedness of left shift to obtain

[TABLE]

which, using Lemma 4.5 again, is estimated by

[TABLE]

Then (6.4) follows from Lemma 4.7 with $\varepsilon_{1}=\frac{1}{2}c\varepsilon$ and the boundedness of the right shift. This finishes the proof of Theorem 3.5. ∎

It remains to prove Propositions 6.1–6.3, which will given in the next three sections (Section 7–9), respectively.

7 Proof of Proposition 6.1

This section is devoted to the proof of Proposition 6.1. Let $s,\alpha\in(0,\infty)$, $r\in\mathbb{N}$, and $\alpha r>s$. We will fix these framework parameters throughout this section.

The following lemma plays a crucial role in the proof of Proposition 6.1.

Lemma 7.1**.**

Let $f\in\Lambda_{s}$ and let $u_{\alpha}(x,t):=T_{\alpha,t}f(x)$ for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. Then there exists a positive constant $C_{n,r,s,\alpha}$, depending only on $r,s,\alpha$ and $n$, such that, for any $\mathbf{x}=(x,x_{n+1}),\mathbf{z}=(z,z_{n+1})\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

Proof.

Let $F(\mathbf{x}):=x_{n+1}^{\alpha r-s}\partial_{n+1}^{r}u_{\alpha}(x,x_{n+1}^{\alpha})$ for any $\mathbf{x}=(x,x_{n+1})\in\mathbb{R}_{+}^{n+1}$. Then, by Lemma 4.4, we have, for any $\mathbf{x}=(x,x_{n+1})$, $\mathbf{z}=(z,z_{n+1})\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

where $\mathbf{w}=(w,w_{n+1})\in\mathbb{R}^{n}\times(0,\infty)$. However, a straightforward calculation shows that, for any $\mathbf{w}:=(w,w_{n+1})\in\mathbb{R}^{n}\times(0,\infty)$,

[TABLE]

which, using Theorem 5.7, is bounded above by $C\|f\|_{\Lambda_{s}}$. This combined with (7.2) yields the desired estimate (7.1), and hence finishes the proof of the Lemma 7.1. ∎

Proof of Proposition 6.1.

Define, for any $\lambda\in(0,\infty)$,

[TABLE]

Assume that $\varepsilon_{1}\in(0,\infty)$ and $\varepsilon>\varepsilon_{1}$. Let $\delta\in(0,1/10)$ be a constant such that

[TABLE]

where $C_{n,r,s,\alpha}$ is the same constant as in (7.1). By (7.1), for any $\mathbf{x}=(x,x_{n+1})$, $\mathbf{z}=(z,z_{n+1})\in\mathbb{R}_{+}^{n+1}$ with $\rho(\mathbf{x},\mathbf{z})<\delta$, we have

[TABLE]

implying

[TABLE]

This means that $B_{\rho}(\mathbf{x},\delta)\subset D_{\alpha,r}(s,f,\varepsilon_{1})$ for any $\mathbf{x}\in D_{\alpha,r}(s,f,\varepsilon)$. Thus,

[TABLE]

To conclude the proof of Proposition 6.1, let $\mathbf{z}:=(z,z_{n+1})\in(D_{\alpha,r,j}(s,f,\varepsilon))_{\delta}$. Then there exists $\mathbf{x}=(x,x_{n+1})\in D_{\alpha,r,j}(s,f,\varepsilon)$ such that $\rho(\mathbf{x},\mathbf{z})<\delta$. Furthermore, (7.3) implies that $\mathbf{z}\in D_{\alpha,r}(s,f,\varepsilon_{1})$. However, using (4.1), we obtain

[TABLE]

It follows that

[TABLE]

Since this holds for any $\mathbf{z}\in(D_{\alpha,r,j}(s,f,\varepsilon))_{\delta}$, we infer that

[TABLE]

This finishes the proof of Proposition 6.1. ∎

8 Proof of Proposition 6.2

This section is devoted to the proof of Proposition 6.2. Throughout this section, we fix the framework parameters $\alpha,s\in(0,\infty)$ and $r\in\mathbb{N}$ such that $\alpha r>s$. For each given function $f\in\Lambda_{s}$, we define $u_{\alpha,f}(x,t):=T_{\alpha,t}f(x)$ for any $x\in\mathbb{R}^{n}$ and $t\in(0,\infty)$. To simplify symbol, we will use $u$ to denote $u_{\alpha,f}$ whenever no confusion is possible. This symbol will be used consistently throughout this section.

Proposition 6.2 follows directly from the lemma below, which provides an estimate of finite differences in terms of the derivatives $\partial_{n+1}^{r}u$.

Lemma 8.1**.**

Let $r_{1}\in\mathbb{N}$ be such that $r_{1}>\alpha r+1$. Then there exists a constant $\delta\in(0,1)$, depending only on $\alpha$ and $s$, such that, for any $f\in\Lambda_{s}$, $(x,y)\in\mathbb{R}_{+}^{n+1}$, and $R>2(r_{1}+2)^{\frac{\alpha+1}{\alpha}}$,

[TABLE]

where the constant $C$ depends only on $r_{1}$ and the framework parameters.

For the moment, we take Lemma 8.1 for granted and proceed with the proof of Proposition 6.2.

Proof of Proposition 6.2 (assuming

Lemma 8.1).

Let $(x,y)\in S_{r_{1},j}(s,f,\varepsilon)$. By (4.3), we have $2^{-j-1}<y\leq 2^{-j}$ and $y^{-s}\Delta_{r_{1}}f(x,y)>\varepsilon$. Furthermore, by Lemma 8.1, for any $R_{1}>2(r_{1}+2)^{\frac{\alpha+1}{\alpha}}$,

[TABLE]

where $C\in(0,\infty)$ is independent of $f$, $R_{1}$, and $(x,y)\in\mathbb{R}_{+}^{n+1}$. Now, choosing the constant $R_{1}$ sufficiently large so that $CR_{1}^{-\delta}\|f\|_{\Lambda_{s}}\leq\frac{\varepsilon}{2}$ yields

[TABLE]

Thus, there must exist $(x^{\prime},t)\in\mathbb{R}_{+}^{n+1}$ such that

[TABLE]

and

[TABLE]

In particular, by Lemma 4.2, this implies

[TABLE]

with $R=CR_{1}$ for some constant $C>1$ depending only on the framework parameters.

Next, let $m$ be a positive integer such that $2^{m-1}<R\leq 2^{m}$. Then $2^{-m-j-1}<R^{-1}y<t\leq 2^{-1}y\leq 2^{-1-j}.$ This together with (8.1) implies

[TABLE]

Finally, combining (8.3) with (8.2), we obtain $(x,y)\in\bigcup_{i=j+1}^{j+m}(D_{\alpha,r,i}(s,f,c\varepsilon))_{R}.$ Since this holds for any $(x,y)\in S_{r_{1},j}(s,f,\varepsilon)$, we complete the proof of Proposition 6.2. ∎

It remains to prove Lemma 8.1. We need two additional lemmas.

Lemma 8.2**.**

Let $r_{1},r_{2}\in\mathbb{N}$ be such that $r_{1}\geq r_{2}>\frac{s}{\alpha}$, and let $f\in\Lambda_{s}$. Then, for any $R>1$, $\tau\in(0,1)$, and $(x,y)\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

where the constant $C$ depends only on $r_{1},r_{2},\tau$, and the framework parameters.

Proof.

Using (5.2) and the semigroup property of $T_{\alpha,t}$, we obtain, for any $x\in\mathbb{R}^{n}$ and $t_{1},t_{2}\in(0,\infty)$,

[TABLE]

This implies that

[TABLE]

Setting $t_{1}=\tau y$ and $t_{2}=(1-\tau)y$, we obtain

[TABLE]

This, combined with (5.4) and (5.5), yields

[TABLE]

Given any constant $R>1$, we break this last integral into two parts $\int_{|z|\leq Ry^{\frac{1}{\alpha}}}+\int_{|z|>Ry^{\frac{1}{\alpha}}}$. We then deduce

[TABLE]

where

[TABLE]

and

[TABLE]

The term $I_{1}(x,y)$ can be estimated as follows:

[TABLE]

To estimate the term $I_{2}(x,y)$, we use Theorem 5.6. Recalling that

[TABLE]

we deduce

[TABLE]

A combination of the above estimates then yields the estimate (8.4) for all $r_{1}\geq r_{2}>\frac{s}{\alpha}$, which completes the proof of Lemma 8.2. ∎

Lemma 8.3**.**

Let $i\in\mathbb{Z}_{+}$ and $\beta\in\mathbb{Z}_{+}^{n}$ be such that $0\leq i<r$ and $|\beta|>\alpha(r-i)+1$. Let $\tau\in(0,1)$. Then there exists a positive constant $C$, depending only on $\tau$, $\beta$, and the framework parameters, such that, for any $f\in\Lambda_{s}$, $(x,y)\in\mathbb{R}_{+}^{n+1}$, and $R>2$,

[TABLE]

Proof.

Since $\partial_{n+1}^{i}u(x,y^{\alpha})=(-\Delta)^{\frac{\alpha i}{2}}T_{\alpha,y^{\alpha}}f(x),$ it follows that, for any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Let $\tau_{2}\in(0,1)$ be such that $\tau_{2}^{\alpha}=1-\tau^{\alpha}$. We then write

[TABLE]

where $C$ is a constant and $M_{i}(\xi)=C\frac{\xi^{\beta}}{|\xi|^{\alpha r-\alpha i}}e^{-|2\pi\tau_{2}\xi|^{\alpha}}.$ Since $|\beta|>\alpha r-\alpha i+1$, it is easily seen that

[TABLE]

Thus, there exists a function $G_{i}\in L^{1}$ such that $\widehat{G_{i}}=M_{i}$ and $|G_{i}(x)|\lesssim(1+|x|)^{-n-1}$ for any $x\in\mathbb{R}^{n}.$ It then follows from (8.5) that

[TABLE]

Breaking this last integral into two parts $\int_{|z|\leq Ry}+\int_{|z|>Ry}$, we conclude that

[TABLE]

which completes the proof of Lemma 8.3. ∎

Proof of Lemma 8.1 .

Applying Taylor’s theorem to the function $t\mapsto u(x,t)$ at the point $t=y^{\alpha}\in(0,\infty)$, we obtain

[TABLE]

Let $h\in\mathbb{R}^{n}$ be such that $|h|=y$. Taking $r_{1}$-th order symmetric difference with respect to the variable $x$ in the direction of $h\in\mathbb{R}^{n}$ on both sides of (8.6) yields

[TABLE]

where $\Delta_{h}^{r_{1}}$ denotes the symmetric difference of order $r$ given in (1.2) and the symbol $\Delta_{h,x}^{r_{1}}$ means that the difference operator $\Delta_{h}^{r_{1}}$ is acting on the variable $x$. It follows from (1.2) that

[TABLE]

where

[TABLE]

and $\overline{[x,z]}:=\{tz+(1-t)x:\ \ t\in[0,1]\}$ for any $x,z\in\mathbb{R}^{n}$.

For the first term $I_{1}$, we use Lemma 8.3 to obtain, for any $R_{1}>2r_{1}$,

[TABLE]

The second term $I_{2}$ can be estimated as follows: for any constant $R_{1}>2r_{1}$,

[TABLE]

where we used (1.2) in the first step, (5.15) in the second step and Lemma 8.2 in the last step.

Finally, combining (8.8), (8) with (8.7), and setting $\tau=\frac{1}{2}$, we obtain

[TABLE]

where $R=2R_{1}^{1+\frac{1}{\alpha}}$ and $\delta:=\frac{\alpha}{\alpha+1}\min\{1,\frac{s}{\alpha}\}$. This finishes the proof of Lemma 8.1. ∎

9 Proof of Proposition 6.3

This section is devoted to the proof of Proposition 6.3. We fix the framework parameters $\alpha,s\in(0,\infty)$ and $r\in\mathbb{N}$ such that $\alpha r>s+3$. For any given function $f\in\Lambda_{s}$, we will use $u(x,y)$ to denote $u_{\alpha,f}(x,t):=T_{\alpha,t}f(x)$ whenever no confusion arises.

Proposition 6.3 follows directly from the lemma below, which provides an estimate of the quantity $|\partial_{n+1}^{r}u(x,t^{\alpha})|$ in terms of finite differences.

Lemma 9.1**.**

Assume that $\alpha r>s+3$. Let $\ell\in\mathbb{N}$ be such that $\frac{s}{2}<\ell<\frac{\alpha r-1}{2}$. Then, for any $f\in\Lambda_{s}$, $(x,t)\in\mathbb{R}_{+}^{n+1}$, and $R>8\ell$,

[TABLE]

where the positive constant $C$ is independent of $f$, $R$, and $(x,t)$.

For the moment, we take Lemma 9.1 for granted and proceed with the proof of Proposition 6.3.

Proof of Proposition 6.3 (assuming

Lemma 9.1).

Let $(x,t)\in D_{\alpha,r,j}(s,f,\varepsilon)$. Then $2^{-j-1}<t\leq 2^{-j}$ and, by Lemma 9.1,

[TABLE]

where $c_{\ell}=(8\ell)^{-1}$ and $R_{1}$ is a sufficiently large constant such that

[TABLE]

Thus, there exists $(x^{\prime},y)\in\mathbb{R}_{+}^{n+1}$ such that $c_{\ell}t\leq y\leq 2^{-1}t$, $|x-x^{\prime}|\leq R_{1}t$, and

[TABLE]

In particular, by Lemma 4.2, this implies $(x,t)\in B_{\rho}\left((x^{\prime},y),R\right)$ for some constant $R=CR_{1}>R_{1}$.

Let $m=m_{\ell}$ denote the positive integer such that $2^{-m}<(8\ell)^{-1}\leq 2^{-m+1}$. Then

[TABLE]

and hence, by (9.2), we have

[TABLE]

Since $(x,t)\in B_{\rho}\left((x^{\prime},y),R\right)$, it follows that

[TABLE]

From this holds for an arbitrary $(x,t)\in D_{\alpha,r,j}(s,f,\varepsilon)$, we deduce that

[TABLE]

which is as desired, This finishes the proof of Proposition 6.3 because $m\leq 8\ell$. ∎

It remains to prove Lemma 9.1, which is technically more involved. The crucial tool in the proof is the higher-order ball average operators, along with some previously established properties of these operators, which we now recall.

Given $\ell\in\mathbb{N}$ and $t\in(0,\infty)$, the $2\ell$-th order ball average operator $B_{\ell,t}$ is defined by setting, for any $f\in L_{\text{loc}}^{1}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

where

[TABLE]

We will use the following known result on the approximation by ball average operators.

Lemma 9.2**.**

([12, Lemma 5.11] and [10, Theorem 1]) Given any $\ell\in\mathbb{N}$, there exists a constant $C=C_{\ell,n}\in(0,\infty)$, depending only on $\ell$ and $n$, such that, for any $t\in(0,\infty)$, $f\in\mathcal{C}\cap L^{\infty}$, and $x\in\mathbb{R}^{n}$,

[TABLE]

Furthermore, if $\ell>s/2$, then there exists a positive constant $C_{\ell,s,n}$, depending only on $\ell,s$ and $n$, such that, for any $t\in(0,\infty)$ and $f\in\Lambda_{s}$,

[TABLE]

The estimate (9.3) was established in our recent article [12, Lemma 5.11], while (9.4) follows directly from [10, Theorem 1] and (5.16).

We also need some known properties for the Fourier transform of $B_{\ell,t}f$. A straightforward computation (see [10, Lemma 2]) shows that, for any $\xi\in\mathbb{R}^{n}$,

[TABLE]

where

[TABLE]

The following lemma collects some useful estimates of the function $m_{\ell}$, which can be found in [10].

Lemma 9.3**.**

[10, (9), Lemma 3, (16), (23)–(25)]* Let $\ell\in\mathbb{N}$. Then the following statements hold.*

1. (i)

For any $\xi\in\mathbb{R}^{n}$,

[TABLE] 2. (ii)

For any $j\in\mathbb{Z}_{+}$, there exists a positive constant $C=C_{j,\ell}$ such that, for any $\xi\in\mathbb{R}^{n}$,

[TABLE] 3. (iii)

There exists a constant $\gamma\in(0,1)$ such that $0<m_{\ell}(\xi)\leq\gamma<1$ whenever $|\xi|\geq 1$. 4. (iv)

$\frac{1-m_{\ell}(\cdot)}{|\cdot|^{2\ell}}$* extends to a positive $C^{\infty}$-function on $\mathbb{R}^{n}$.*

Proof of Lemma 9.1.

Throughout the proof below, the letter $C$ denotes a general positive constant depending only on $\ell$ and the framework parameters. Note that $\alpha r>2\ell+1$.

First, by definition, we have, for any $(\xi,t)\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

where $\phi_{\alpha,r}(\xi):=c_{n,r}|\xi|^{\alpha r}e^{-|2\pi\xi|^{\alpha}}.$ Here and throughout the proof, the Fourier transform is understood in a distributional sense. For simplicity, we let $F_{\ell,t}:=f-B_{\ell,t}f$. Then, using (9.5), we may write, for any $(\xi,t)\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

where

[TABLE]

Next, we claim that there exists a function $G_{\alpha,r}\in L^{1}$ satisfying that $\widehat{G_{\alpha,r}}=\mu_{\alpha,r}$ and

[TABLE]

Once this claim is proven, we define a family of uniformly bounded convolution operators $\{U_{t}\}_{t\in(0,\infty)}$ on $L^{\infty}$ by setting, for any $g\in L^{\infty}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

By (9.6), we then obtain the following integral representation for $t^{\alpha r}\partial_{n+1}^{r}u(x,t^{\alpha})$:

[TABLE]

To show the claim, we define

[TABLE]

and we use Lemma 9.3(i) and the knowledge $\alpha r>2\ell+1$. Since $\mu_{\alpha,r}$ is a radial, integrable function on $\mathbb{R}^{n}$, $G_{\alpha,r}=\widehat{\mu_{\alpha,r}}$ is a uniformly bounded, continuous, radial function on $\mathbb{R}^{n}$ with distributional Fourier transform $\mu_{\alpha,r}$. Using (9.7), we write, for any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Lemma 9.3(i) implies that, for any given $i\in\mathbb{Z_{+}}$ and any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Notice also that, for any given $i\in\mathbb{Z_{+}}$ and any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Thus, by the Leibniz rule, we obtain, for any given $i\in\mathbb{Z_{+}}$ and any $\xi\in\mathbb{R}^{n}$,

[TABLE]

Since $\alpha r>2\ell+1$, this implies that $|\nabla^{i}\mu_{\alpha,r}(\xi)|\in L^{1}$ for $i=0,\ldots,n+1$. Thus, by (9.10), it follows that, for any $\beta\in\mathbb{Z}_{+}^{n}$ with $|\beta|\leq n+1$,

[TABLE]

which implies the desired estimate (9.8) and hence completes the proof of the claim.

Finally, we prove (9.1). Indeed, using (9.8) and (9.9), we obtain, for any $R>1$ and $(x,t)\in\mathbb{R}_{+}^{n+1}$,

[TABLE]

which, using Lemma 9.2, is estimated by

[TABLE]

This proves (9.1) for $R>8\ell$, which completes the proof of Lemma 9.1. ∎

10 Proof of Theorem 3.12 (heat semigroup

characterization of

$d(f,\Lambda_{X}^{s})_{\Lambda_{s}}$ under Assumption II)

This section is devoted to the proof of Theorem 3.12, which relies on Propositions 6.1-6.3 established in previous sections.

Let $\ell\in\mathbb{N}$ be such that $\frac{s}{2}<\ell<\frac{\alpha r-1}{2}$. Let

[TABLE]

First, we prove the upper bound:

[TABLE]

By Lemma 4.8 applied to $r_{1}=2\ell$, it is enough to show that

[TABLE]

To prove (10.2), we let $\varepsilon\in(0,\infty)$ and use Proposition 6.2 to obtain a constant $R\in(1,\infty)$ such that

[TABLE]

where $c\in(0,1)$ is a sufficiently small constant depending only on the framework parameters. By Lemma 4.3(ii), for any $\delta\in(0,1)$, the sets $\left(D_{\alpha,r,j+1}(s,f,c\varepsilon)\right)_{\delta}$, $j\in\mathbb{Z}$, has Property I with uniform constants. Thus, it follows from Definition 3.10 (iii) that, for any $\delta\in(0,1)$,

[TABLE]

On the other hand, however, by Proposition 6.1 and both (i) and (ii) of Definition 3.10, we can find a constant $\delta\in(0,1)$, depending on $f$ and $\varepsilon$, such that

[TABLE]

where the last step used the boundedness of the right shift. Thus, the following implication holds:

[TABLE]

from which the desired inequality (10.2) follows. This proves the upper estimate (10.1).

Next, we show the lower estimate:

[TABLE]

By Lemma 4.8, it is enough to show that

[TABLE]

Using Proposition 6.3 and both (i) and (ii) of Definition 3.10, we find a constant $R\in(1,\infty)$ such that

[TABLE]

where the constant $c\in(0,1)$ depends only on the framework parameters. Similar to the above proof (10.2), using Lemma 4.3(ii) and Definition 3.10(iii), we conclude that, for any $\delta\in(0,1)$,

[TABLE]

On the other hand, however, by Lemma 4.7 and both (i) and (ii) of Definition 3.10, there exists a constant $\delta\in(0,1)$, depending on $f$ and $\varepsilon$, such that

[TABLE]

Thus, the following implication holds:

[TABLE]

from which (10.4) follows. This proves the upper estimate (10.3) and hence finishes the proof of Theorem 3.12.

11 Applications to specific spaces

In this section, we give some specific examples and show how to apply the main results obtained in the previous sections to these examples.

11.1 Besov spaces

Now, we present the concept of Besov spaces as follows; see, for example, [55, Definition 1.1]. As above, we denote by $\mathcal{S}$ the space of all Schwartz functions on ${\mathbb{R}^{n}}$ and $\mathcal{S}^{\prime}$ its topological dual space (i.e., the space of all tempered distributions on $\mathbb{R}^{n}$). Let $\eta\in\mathcal{S}$ satisfy $0\leq\eta\leq 1,$

[TABLE]

For any $x\in\mathbb{R}^{n}$, let $a(x):=\eta\left(x\right)-\eta(2x)$. Then $\mathop{\mathrm{\,supp\,}}a\subset\left\{x\in\mathbb{R}^{n}:\ 1/2\leq|x|\leq 3/2\right\}$ and, for any $x\in\mathbb{R}^{n}$,

[TABLE]

Define $\{\phi_{k}\}_{k\in\mathbb{N}}$ by setting, for any $x\in\mathbb{R}^{n}$,

[TABLE]

and, for any $k\in\mathbb{N}$,

[TABLE]

Clearly, $\sum_{k\in\mathbb{Z}_{+}}\widehat{\phi}_{k}=1$ on $\mathbb{R}^{n}$.

Definition 11.1**.**

Let $p,q\in(0,\infty]$ and $s\in{\mathbb{R}}$. Then the Besov space $B^{s}_{p,q}$ is defined to be the set of all $f\in\mathcal{S}^{\prime}$ such that

[TABLE]

where the usual modification is made when $q=\infty$.

Let $q,p\in(0,\infty]$. The space $l^{q}(L^{p})_{\mathbb{Z}_{+}}$ is defined to be the set of all $\mathbf{F}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that

[TABLE]

where the usual modification is made when $q=\infty$. From the definition of the space $l^{q}(L^{p})_{\mathbb{Z}_{+}}$, it is easy to deduce that $l^{q}(L^{p})_{\mathbb{Z}_{+}}$ is a quasi-normed lattice of function sequences.

The following lemma is about the wavelet characterization of Besov spaces (see, for example, [55, Proposition 1.11]).

Lemma 11.2**.**

Let $s\in(0,\infty)$ and $p,q\in(0,\infty]$. Assume that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that $L>\max\{s,n(\max\{\frac{1}{p},1\}-1)-s\}$. Then $B^{s}_{p,q}\cap\Lambda_{s}=\Lambda_{X}^{s}$ with equivalent quasi-norm, where $\Lambda_{X}^{s}$ is the Daubechies $s$-Lipschitz $X$-based space with $X:=l^{q}(L^{p})_{\mathbb{Z}_{+}}$.

We still need the following two lemmas, which are precisely, respectively, [11, Lemmas 4.3 and 4.4].

Lemma 11.3**.**

Let $p\in(0,\infty)$ and $q\in(0,\infty]$. Then $l^{q}(L^{p})_{\mathbb{Z}_{+}}$ satisfies Assumption I.

Lemma 11.4**.**

Let $q\in(0,\infty)$. Then $l^{q}(L^{\infty})_{\mathbb{Z}_{+}}$ satisfies Assumption II.

Using Theorems 3.5 and 3.12 and Lemmas 11.2, 11.3, and 11.4, we obtain the following conclusions.

Theorem 11.5**.**

Let $\alpha,s\in(0,\infty)$, $p\in(0,\infty)$, and $q\in(0,\infty]$. Assume that $r\in\mathbb{N}$ with $\alpha r>s+3$ and that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that $L>\max\{s,n(\max\{\frac{1}{p},1\}-1)-s\}$ and $L\geq r-1$. Let $\varphi$ be the same as in (3.1). Then the following statements hold.

• (i)

For any $f\in\Lambda_{s}$,

[TABLE]

with positive equivalence constants independent of $f$.

• (ii)

For any $f\in\Lambda_{s}$ and $q\in(0,\infty)$,

[TABLE]

with positive equivalence constants independent of $f$.

Using Corollaries 3.7 and 3.13 and Lemmas 11.2, 11.3, and 11.4, we obtain the following conclusions.

Theorem 11.6**.**

Under the same assumption as in Theorem 11.5, then the following statements hold.

• (i)

$f\in\overline{B^{s}_{p,q}\cap\Lambda_{s}}^{\Lambda_{s}}$* if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$,*

[TABLE]

• (ii)

$f\in\overline{B^{s}_{\infty,q}}^{\Lambda_{s}}$* if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$,*

[TABLE]

11.2 Triebel–Lizorkin spaces

Next, we recall the definition of Triebel–Lizorkin spaces as follows; see, for example, [55, Definition 1.1]. Let $\{\phi_{j}\}_{j\in\mathbb{Z}_{+}}$ be the same as in (11.1) and (11.2).

Definition 11.7**.**

Let $q\in(0,\infty]$ and $s\in{\mathbb{R}}$.

• (i)

If $p\in(0,\infty)$, then the Triebel–Lizorkin space $F^{s}_{p,q}$ is defined to be the set of all $f\in\mathcal{S}^{\prime}$ such that

[TABLE]

• (ii)

The Triebel–Lizorkin space $F^{s}_{\infty,q}$ is defined to be the set of all $f\in\mathcal{S}^{\prime}$ such that

[TABLE]

where, for any $J\in\mathbb{Z}_{+}$ and $M\in\mathbb{Z}^{n}$, $I_{J,M}:=2^{-J}(M+[0,1)^{n})$.

Let $q,p\in(0,\infty]$. The space $L^{p}(l^{q})_{\mathbb{Z}_{+}}$ is defined to be the set of all $\mathbf{F}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that

[TABLE]

where the usual modification is made when $q=\infty$. The space $F_{\infty,q}(\mathbb{R}^{n},\mathbb{Z}_{+})$ is defined to be the set of all $\mathbf{F}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that

[TABLE]

where the usual modification is made when $q=\infty$.

From these definitions, it is easy to infer that $L^{p}(l^{q})_{\mathbb{Z}_{+}}$ and $F_{\infty,q}(\mathbb{R}^{n},\mathbb{Z}_{+})$ are quasi-normed lattices of function sequences.

The following lemma is about the semigroup characterization of Triebel–Lizorkin spaces (see, for example, [55, Corollary 2]).

Lemma 11.8**.**

Let $s\in(0,\infty)$ and $p,q\in(0,\infty]$. Assume that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that $L>\max\{s,n(\max\{\frac{1}{p},\frac{1}{q},1\}-1)-s\}$. Then $F^{s}_{p,q}\cap\Lambda_{s}=\Lambda_{X}^{s}$, where $\Lambda_{X}^{s}$ is the Daubechies $s$-Lipschitz $X$-based space with $X:=L^{p}(l^{q})_{\mathbb{Z}_{+}}$ when $p\neq\infty$ or with $X:=F_{\infty,q}(\mathbb{R}^{n},\mathbb{Z}_{+})$ when $p=\infty$.

Here, and thereafter, the Hardy–Littlewood maximal operator $\mathcal{M}$ is defined by setting, for any $f\in L_{{\mathop{\mathrm{\,loc\,}}}}^{1}$ and $x\in\mathbb{R}^{n}$,

[TABLE]

where the supremum is taken over all balls $B\subset\mathbb{R}^{n}$ containing $x$.

We need the following lemma, which is precisely [11, Lemma 4.11].

Lemma 11.9**.**

Let $s,p\in(0,\infty)$ and $q\in(0,\infty]$. Then $L^{p}(l^{q})_{\mathbb{Z}_{+}}$ satisfies Assumption I.

For any cube $I\subset\mathbb{R}^{n}$, we always use $\ell(I)$ to denote its edge length and let $\widehat{I}:=I\times[0,\ell(I))$. For any given measurable set $A\subset\mathbb{R}_{+}^{n+1}$, let

[TABLE]

Let $d\mu(x,x_{n+1}):=\frac{dx\,dx_{n+1}}{x_{n+1}}.$ Since we assume that the edge length of any cube in $\mathcal{D}$ is at most $1$, we deduce that, for any $A\subset\mathbb{R}_{+}^{n+1}$, $M(A)=M\left(A\cap(\mathbb{R}^{n}\times(0,1])\right).$ In other words, $M(A)<\infty$ is equivalent to that $\mathbf{1}_{A\cap(\mathbb{R}^{n}\times(0,1])}d\mu$ is a Carleson measure.

We still need the following two lemmas, which are exactly, respectively, [11, Lemmas 4.13 and 5.7].

Lemma 11.10**.**

Let $A\subset\mathbb{R}^{n+1}_{+}$ be a measurable set such that $M(A)<\infty$, where $M$ is the same as in (11.3). Assume that $A$ has Property I with constants $\delta,\delta^{\prime}\in(0,1/10)$. Then, for any $R\in(0,\infty)$, $M(A_{R})<\infty$.

Lemma 11.11**.**

Let $q\in(0,\infty]$. Then $F_{\infty,q}(\mathbb{R}^{n},\mathbb{Z}_{+})$ satisfies Assumption II.

Using Theorems 3.5 and 3.12 and Lemmas 11.8, 11.9, and 11.11, we obtain the following conclusions.

Theorem 11.12**.**

Let $\alpha,s\in(0,\infty)$, $p\in(0,\infty)$, and $q\in(0,\infty]$. Assume that $r\in\mathbb{N}$ with $\alpha r>s+3$ and that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that $L>\max\{s,n(\max\{\frac{1}{p},\frac{1}{q},1\}-1)-s\}$ and $L\geq r-1$. Let $\varphi$ be the same as in (3.1). Then the following statements hold.

• (i)

For any $f\in\Lambda_{s}$,

[TABLE]

with positive equivalence constants independent of $f$.

• (ii)

For any $f\in\Lambda_{s}$,

[TABLE]

with positive equivalence constants independent of $f$.

Remark 11.13**.**

In (i) and (ii) of Theorem 11.12, if $s\in(0,1]$, $\alpha=1$, and $q=2$, then the conclusions are exactly [39, Theorems 4], while Theorem 11.12 in other cases is completely new.

Using Corollaries 3.7 and 3.13 and Lemmas 11.8, 11.9, and 11.11, we obtain the following conclusions.

Theorem 11.14**.**

Under the same assumption as in Theorem 11.12, then the following statements hold.

• (i)

$f\in\overline{F^{s}_{p,q}\cap\Lambda_{s}}^{\Lambda_{s}}$* if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$,*

[TABLE]

• (ii)

$f\in\overline{F^{s}_{\infty,q}}^{\Lambda_{s}}$* if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$, $M(D_{\alpha,r}(s,f,\varepsilon))<\infty.$*

11.3 Besov-type spaces

The Besov-type and Triebel–Lizorkin-type spaces were intensively investigated in [24, 33, 44, 57, 58, 62, 63] and, in particular, the Triebel–Lizorkin-type spaces were introduced in [56, 57, 62] to connect Triebel–Lizorkin spaces and $Q$ spaces. Furthermore, they also have a close relation with Besov–Morrey and Triebel–Lizorkin–Morrey spaces introduced in [30, 49], which are systematically studied in [19, 21, 40, 41, 42, 43, 45, 46]. We refer to [44, 59, 60, 61] for more studies on Besov-type and Triebel–Lizorkin-type spaces and to [17, 18, 20, 22, 23, 31, 32, 35] for more variants and their applications.

For any $p$, $q\in\mathbb{R}$, let $p\vee q:=\max\{p,q\}$ and $p\wedge q:=\min\{p,q\}$; furthermore, let $p_{+}:=\max\{p,0\}$. Let $\mathscr{Q}$ be the set of all dyadic cubes. For any $Q\in\mathscr{Q}$, let $j_{Q}:=-\mathop{\mathrm{\,log\,}}_{2}\ell(Q).$ Now, we recall the concept of Besov-type spaces. Let $s\in\mathbb{R}$, $p,q\in(0,\infty]$, and $\tau\in[0,\infty)$. The space $(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$ is defined to be the set of all $\mathbf{f}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that

[TABLE]

where the usual modification is made when $q=\infty$. Let $\{\phi_{j}\}_{j\in\mathbb{Z}_{+}}$ be the same as in (11.1) and (11.2). The Besov-type space $B^{s,\tau}_{p,q}$ is defined to be the set of all $f\in\mathcal{S}^{\prime}$ such that

[TABLE]

From the definition of the space $(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$, it is easy to deduce that $(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$ is a quasi-normed lattice of function sequences.

The following lemma is about the wavelet characterization of Besov-type spaces (see, for example, [62, Section 4.2] or [33, Theorem 6.3(i)]).

Lemma 11.15**.**

Let $s\in(0,\infty)$, $p,q\in(0,\infty]$, and $\tau\in[0,\infty)$. Assume that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that

[TABLE]

Then $B^{s,\tau}_{p,q}\cap\Lambda_{s}=\Lambda_{X}^{s}$, where $\Lambda_{X}^{s}$ is the Daubechies $s$-Lipschitz $X$-based space with $X:=(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$.

We still need the following lemma, which is precisely [11, Lemma 5.36].

Lemma 11.16**.**

Let $\tau\in[0,1/p)$, $p\in(0,\infty)$, and $q\in(0,\infty]$. Then $(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$ satisfies Assumption I.

Using Theorem 3.5 and Lemmas 11.15 and 11.16, we obtain the following conclusions.

Theorem 11.17**.**

Let $\tau\in[0,1/p)$, $\alpha,s\in(0,\infty)$, $p\in(0,\infty)$, and $q\in(0,\infty]$. Assume that $r\in\mathbb{N}$ with $\alpha r>s+3$ and that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that (11.4) and $L\geq r-1$. Let $\varphi$ be the same as in (3.1). Let

[TABLE]

Then, for any $f\in\Lambda_{s}$,

[TABLE]

with positive equivalence constants independent of $f$.

Remark 11.18**.**

Whether Theorem 11.17 in the case $p=\infty$ holds true remains unclear at present.

Using Corollary 3.7 and Lemmas 11.15 and 11.16, we obtain the following conclusions.

Theorem 11.19**.**

Under the same assumption as in Theorem 11.17, then $f\in\overline{B^{s,\tau}_{p,q}\cap\Lambda_{s}}^{\Lambda_{s}}$ if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$,

[TABLE]

and

[TABLE]

11.4 Triebel–Lizorkin-type spaces

Next, we present the concept of Triebel–Lizorkin spaces as follows; see, for example, [56, 57, 62]. Let $\{\phi_{j}\}_{j\in\mathbb{Z}_{+}}$ be the same as in (11.1) and (11.2). Let $s\in\mathbb{R}$, $p\in(0,\infty)$, $q\in(0,\infty]$, and $\tau\in[0,\infty)$. The space $(L^{p}_{\tau}l^{q})_{\mathbb{Z}_{+}}$ is defined to be the set of all $\mathbf{f}:=\{f_{j}\}_{j\in\mathbb{Z}_{+}}\in\mathscr{M}_{\mathbb{Z}_{+}}$ such that

[TABLE]

where the usual modification is made when $q=\infty$. The Triebel–Lizorkin-type space $F^{s,\tau}_{p,q}$ is defined to be the set of all $f\in\mathcal{S}^{\prime}$ such that

[TABLE]

From these definitions, it is easy to infer that $(L^{p}_{\tau}l^{q})_{\mathbb{Z}_{+}}$ is a quasi-normed lattice of function sequences.

The following lemma is about the wavelet characterization of Triebel–Lizorkin-type spaces (see, for example, [62, Section 4.2] or [33, Theorem 6.3(i)]).

Lemma 11.20**.**

Let $s\in(0,\infty)$, $p\in(0,\infty)$, $q\in(0,\infty]$, and $\tau\in[0,\infty)$. Assume that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that

[TABLE]

Then $F^{s,\tau}_{p,q}\cap\Lambda_{s}=\Lambda_{X}^{s}$, where $\Lambda_{X}^{s}$ is the Daubechies $s$-Lipschitz $X$-based space with $X:=(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$.

We still need the following lemma, which is exactly [11, Lemmas 5.45].

Lemma 11.21**.**

Let $s,p\in(0,\infty)$, $\tau\in[0,1/p)$, and $q\in(0,\infty]$. Then $(l^{q}L^{p}_{\tau})_{\mathbb{Z}_{+}}$ satisfies Assumption I.

Using Theorem 3.5 and Lemmas 11.20 and 11.21, we obtain the following conclusions.

Theorem 11.22**.**

Let $\alpha,s\in(0,\infty)$, $\tau\in[0,1/p)$, $p\in(0,\infty)$, and $q\in(0,\infty]$. Assume that $r\in\mathbb{N}$ with $\alpha r>s+3$ and that the regularity parameter $L\in\mathbb{N}$ of the Daubechies wavelet system $\{\psi_{\omega}\}_{\omega\in\Omega}$ satisfies that (11.20) and $L\geq r-1$. Let $\varphi$ be the same as in (3.1). Let

[TABLE]

Then, for any $f\in\Lambda_{s}$,

[TABLE]

with positive equivalence constants independent of $f$.

Using Corollary 3.7 and Lemmas 11.20 and 11.21, we obtain the following conclusions.

Theorem 11.23**.**

Under the same assumption as in Theorem 11.22, then $f\in\overline{F^{s,\tau}_{p,q}\cap\Lambda_{s}}^{\Lambda_{s}}$ if and only if $f\in\Lambda_{s}$ and, for any $\varepsilon\in(0,\infty)$,

[TABLE]

and

[TABLE]