Intrinsic nature of the Stein-Weiss $H^1$-inequality

TL;DR

This paper investigates the intrinsic properties of the Stein-Weiss $H^1$-inequality, revealing a characterization of certain function spaces through Riesz transforms and duality, and addressing Bourgain-Brezis' decomposition problem.

Contribution

It provides a new characterization of the Riesz transform component in the Stein-Weiss inequality using duality and BMO, linking it to Bourgain-Brezis' problem.

Findings

Characterizes functions in $I_s([ ing{H}^{s,1}_-]^*)$ via Riesz transforms and BMO.

Links the Stein-Weiss inequality to Bourgain-Brezis' decomposition problem.

Establishes a connection between $H^1$-inequality and Riesz transform decompositions.

Abstract

This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$ \|I_s u\|_{L^\frac{n}{n-s}}\lesssim \|u\|_{L^1}+\|\vec{R}u\|_{L^{1}}=\|u\|_{H^1} $$ through the tracing and duality laws based on Riesz's singular integral operator $I_s$. We discover that $f\in I_s\big([\mathring{H}^{s,1}_{-}]^\ast\big)$ if and only if $\exists\ \vec{g}=(g_1,...,g_n)\in \big(L^\infty\big)^n$ such that $f=\vec{R}\cdot\vec{g}=\sum_{j=1}^n R_jg_j$ in $\mathrm{BMO}$ (the John-Nirenberg space introduced in their 1961 {\it Comm. Pure Appl. Math.} paper \cite{JN}) where $\vec{R}=(R_1,...,R_n)$ is the vector-valued Riesz transform - this characterizes the Riesz transform part $\vec{R}\cdot\big(L^\infty\big)^n$ of Fefferman-Stein's decomposition (established in their 1972 {\it Acta Math} paper \cite{FS}) for $\mathrm{BMO}=L^\infty+\vec{R}\cdot\big(L^\infty\big)^n$ and yet indicates that $I_s\big([\mathring{H}^{s,1}_-]^\ast\big)$ is indeed a solution to Bourgain-Brezis' problem under $n\ge 2$: ``What are the function spaces $X, W^{1,n}\subset X\subset \mathrm{BMO}$, such that every $F\in X$ has a decomposition $F=\sum_{j=1}^n R_j Y_j$ where $Y_j\in L^\infty$?" (posed in their 2003 {\it J. Amer. Math. Soc.} paper \cite{BB}).

Equations

$$\Vert I_{s}u{\Vert }_{L^{\frac{n}{n-s}}}\lesssim \Vert u{\Vert }_{L^1}+\Vert \vec{R}u{\Vert }_{L^1}=\Vert u{\Vert }_{H^1}$$

$$\Vert u{\Vert }_{H^p}=\Vert u{\Vert }_{L^p}+\Vert \vec{R}u{\Vert }_{L^p}<\infty ,$$

$$\vec{R}=(R_1,\dots ,R_n)$$

$$\vec{R}u=(R_{1}u,...,R_{n}u)\&R_{j}u(x)=\left(\frac{\Gamma (\frac{n+1}{2})}{{\pi }^{\frac{n+1}{2}}}\right)\text{p.v.}{\int }_{{\mathbb{R}}^n}\frac{x_j-y_j}{|x-y{|}^{n+1}}u(y)dy\text{a.e.}x\in {\mathbb{R}}^n$$

$$\vec{f}=(f_1,\dots ,f_n)$$

$$\Vert \vec{f}{\Vert }_{L^p}=\sum _{j=1}^n\Vert f_j{\Vert }_{L^p}.$$

$$u\in \bigcup _{p\in [1,\frac{n}{s})}{L}^p$$

$$I_{s}u(x)=\left(\frac{\Gamma (\frac{n-s}{2})}{{\pi }^{\frac{n}{2}}{2}^s\Gamma (\frac{s}{2})}\right){\int }_{{\mathbb{R}}^n}|x-y{|}^{s-n}u(y)dy\text{a.e.}x\in {\mathbb{R}}^n.$$

$$0<s<1\&1\le p<\frac{n}{s},$$

$$\Vert I_{s}u{\Vert }_{L^{\frac{pn}{n-sp}}}\lesssim \Vert u{\Vert }_{L^p}+\Vert \vec{R}u{\Vert }_{L^p}\approx \Vert u{\Vert }_{H^p}\forall u\in H^p.$$

$$(-\Delta {)}^{\frac{s}{2}}u(x)=\{\begin{array}{ll}{I}_{-s}u(x)=c_{n,s}{\int }_{{\mathbb{R}}^n}\frac{u(x+y)}{|y{|}^{n+s}}dy& \text{as}s\in (-1,0)\\ u(x)& \text{as}s=0\\ c_{n,s,+}{\int }_{{\mathbb{R}}^n}\frac{u(x+y)-u(x)}{|y{|}^{n+s}}dy& \text{as}s\in (0,1)\end{array}$$

$$\nabla^{s}u(x)=\Bigg{(}\frac{\partial^{s}u}{\partial x_{j}^{s}}\Bigg{)}_{j=1}^{n}=\vec{R}(-\Delta)^{\frac{s}{2}}u(x)=c_{n,s,-}\int_{\mathbb{R}^{n}}\frac{y\big{(}u(x)-u(x-y)\big{)}}{|y|^{n+1+s}}\,dy,$$

$$\begin{cases}c_{n,s}=\frac{\Gamma(\frac{n-s}{2})}{\pi^{\frac{n}{2}}2^{s}\Gamma(\frac{s}{2})}\\ c_{n,s,+}=\frac{s2^{s-1}\Gamma\big{(}\frac{n+s}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}1-\frac{s}{2}\big{)}}\\ c_{n,s,-}=\frac{2^{s}\Gamma\big{(}\frac{n+s+1}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}\frac{1-s}{2}\big{)}}.\end{cases}$$

$$\begin{cases}(-\Delta)^{\frac{s}{2}}u(x)=c_{+}\bigg{(}\frac{d^{s}}{dx^{s}_{+}}+\frac{d^{s}}{dx^{s}_{-}}\bigg{)}u(x)\\ \nabla^{s}u(x)=c_{-}\bigg{(}\frac{d^{s}}{dx^{s}_{+}}-\frac{d^{s}}{dx^{s}_{-}}\bigg{)}u(x)\\ \frac{d^{s}}{dx^{s}_{\pm}}u(x)=\frac{s}{\Gamma(1-s)}\int_{\pm\infty}^{0}\frac{t(u(x+t)-u(x))}{|t|^{2+s}}\,dt.\end{cases}$$

$${\nabla }_{+}^{s}u=(-\Delta {)}^{\frac{s}{2}}u\&{\nabla }_{-}^{s}u={\nabla }^{s}u=\vec{R}(-\Delta {)}^{\frac{s}{2}}u.$$

$$\nabla =({\partial }_{x_1},\dots ,{\partial }_{x_n}).$$

$$\Vert u{\Vert }_{L^{\frac{pn}{n-sp}}}\lesssim \Vert {\nabla }_{+}^{s}u{\Vert }_{L^p}+\Vert {\nabla }_{-}^{s}u{\Vert }_{L^p}\forall u\in I_s(C_c^{\infty }\cap H^p).$$

$$\Vert u{\Vert }_{L^{\frac{pn}{n-sp}}}\lesssim \Vert {\nabla }_{+}^{s}u{\Vert }_{L^p}\forall u\in I_s(C_c^{\infty }\cap H^p).$$

$$\Vert u{\Vert }_{L^{\frac{pn}{n-sp}}}\lesssim \Vert {\nabla }_{-}^{s}u{\Vert }_{L^p}\forall u\in I_s(C_c^{\infty }\cap H^p).$$

$$\|u\|_{L^{\frac{n}{n-s},\infty}}=\sup_{t>0}t\big{|}\big{\{}x\in{{{\mathbb{R}}}^{n}}:\ |u(x)|>t\big{\}}\big{|}^{\frac{n-s}{n}}\lesssim\|\nabla^{s}_{\pm}u\|_{L^{1}}\qquad\forall\ \ u\in I_{s}(C_{c}^{\infty}\cap H^{1}),$$

$$\text{id}=-\sum _{j=1}^n{R}_j^2\&\Vert R_{j}u{\Vert }_{L^{\frac{n}{n-s},\infty }}\lesssim \Vert u{\Vert }_{L^{\frac{n}{n-s},\infty }}\forall (j,u)\in \{1,2,\dots ,n\}\times L^{\frac{n}{n-s},\infty }.$$

$${\rho }_{N,\alpha }(f)=\underset{x\in {\mathbb{R}}^n}{\sup }(1+|x{|}^N)|D^{\alpha }f(x)|<\infty \text{holds for}\{\begin{array}{l}N\in {\mathbb{Z}}_{+}=\mathbb{N}\cup \{0\}\\ \alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{Z}}_{+}^n\\ D^{\alpha }={\partial }_{x_1}^{{\alpha }_1}\cdots {\partial }_{x_n}^{{\alpha }_n}.\end{array}$$

$${\mathcal{S}}_s=\left\{f\in C^{\infty }:{\rho }_{n+s,\alpha }(\varphi )=\underset{x\in {\mathbb{R}}^n}{\sup }(1+|x{|}^{n+s})|D^{\alpha }f(x)|<\infty \forall \alpha \in {\mathbb{Z}}_{+}^n\right\},$$

$$u\in {\mathcal{S}}_s^{\prime }\subset {\mathcal{S}}^{\prime }$$

$$H^{s,1}=\left\{u\in {\mathcal{S}}_s^{\prime }:[u{]}_{H^{s,1}}=\Vert (-\Delta {)}^{\frac{s}{2}}u{\Vert }_{H^1}<\infty \right\}.$$

$$u_1-u_2=\text{constant}\iff [u_1{]}_{H^{s,1}}=[u_2{]}_{H^{s,1}}.$$

$$H_{\pm }^{s,1}=\left\{u\in {\mathcal{S}}_s^{\prime }:[u{]}_{H_{\pm }^{s,1}}=\Vert {\nabla }_{\pm }^{s}u{\Vert }_{L^1}<\infty \right\},$$

$$H^{s,1}=H_{+}^{s,1}\cap H_{-}^{s,1}.$$

$${\mathring{H}}_{\pm }^{s,1}$$

$$H^{s,1}={\mathring{H}}_{+}^{s,1}\cap {\mathring{H}}_{-}^{s,1}$$

Full text

[TABLE]

through the tracing and duality laws based on Riesz’s singular integral operator $I_{s}$. We discover that f\in I_{s}\big{(}[\mathring{H}^{s,1}_{-}]^{\ast}\big{)} if and only if \exists\ \vec{g}=(g_{1},...,g_{n})\in\big{(}L^{\infty}\big{)}^{n} such that $f=\vec{R}\cdot\vec{g}=\sum_{j=1}^{n}R_{j}g_{j}$ in $\mathrm{BMO}$ (the John-Nirenberg space introduced in their 1961 Comm. Pure Appl. Math. paper [14]) where $\vec{R}=(R_{1},...,R_{n})$ is the vector-valued Riesz transform - this characterizes the Riesz transform part \vec{R}\cdot\big{(}L^{\infty}\big{)}^{n} of Fefferman-Stein’s decomposition (established in their 1972 Acta Math paper [9]) for \mathrm{BMO}=L^{\infty}+\vec{R}\cdot\big{(}L^{\infty}\big{)}^{n} and yet indicates that I_{s}\big{(}[\mathring{H}^{s,1}_{-}]^{\ast}\big{)} is indeed a solution to Bourgain-Brezis’ problem under $n\geq 2$: “What are the function spaces $X,W^{1,n}\subset X\subset\mathrm{BMO}$, such that every $F\in X$ has a decomposition $F=\sum_{j=1}^{n}R_{j}Y_{j}$ where $Y_{j}\in L^{\infty}$?” (posed in their 2003 J. Amer. Math. Soc. paper [6]).

LL was supported by the National Natural Science Foundation of China (# 11771446); JX was supported by NSERC of Canada (# 202979463102000).

1. 1 Introduction

2. 1.1 The Stein-Weiss $H^{p}$-inequalities

3. 1.2 Overview of the principal results

4. 2 Dense subspaces of $H^{s,1}\,\&\,H^{s,1}_{\pm}$

5. 2.1 Initial definitions of $\nabla^{s}_{\pm}$

6. 2.2 Dense subspaces of $H^{s,1}\ \&\ H^{s,1}_{+}$

7. 3 Tracing laws for $H^{s,1}\,\&\,H^{s,1}_{\pm}$

8. 3.1 Strong/weak estimates for $\text{Cap}_{X\in\{H^{s,1},H^{s,1}_{\pm}\}}$

9. 3.2 Restrictions/traces of $H^{s,1}\ \&\ H^{s,1}_{\pm}$

10. 4 Duality laws for $H^{s,1}\,\&\,\mathring{H}^{s,1}_{\pm}$

11. 4.1 Adjoint operators of $\nabla^{s}_{\pm}$ via $\{\mathcal{S},{\mathop{\mathrm{missing}}{\,BMO\,}}\}$

12. 4.2 Dualities of $H^{s,1}\ \&\ \mathring{H}^{s,1}_{\pm}$

1. Introduction

1.1. The Stein-Weiss $H^{p}$-inequalities

For $(n,p)\in\mathbb{N}\times[1,\infty)$, denote by $H^{p}$ the real Hardy space on the Euclidean space ${{{\mathbb{R}}}^{n}}$, consisting of all functions $f$ in the Lebesgue space $L^{p}$ with

[TABLE]

where

[TABLE]

is the vector-valued Riesz transform on ${{{\mathbb{R}}}^{n}}$, with

[TABLE]

and $\Gamma$ being the Gamma function. Also, for a vector-valued function

[TABLE]

let

[TABLE]

Note that $H^{p}$ coincides with the classical Lebesgue space $L^{p}$ whenever $p\in(1,\infty)$ and the $(0,1)\ni s$-th order Riesz singular integral operator $I_{s}$ acting on a function

[TABLE]

is defined by

[TABLE]

We refer the reader to Stein’s seminal texts [33, 34] for more about these basic notions. The well-known Stein-Weiss $H^{p}$-inequality (cf. [35]) states that under

[TABLE]

the Riesz-Hardy potential space $I_{s}(H^{p})$ can be continuously embedded into $L^{\frac{pn}{n-sp}}$, that is,

[TABLE]

Let $C_{c}^{\infty}$ be the collection of all infinitely differentiable functions compactly supported in ${{{\mathbb{R}}}^{n}}$. Note that $C_{c}^{\infty}\cap H^{p}$ is dense in $H^{p}$ for any $p\in[1,\infty)$. For any $u\in C_{c}^{\infty}$ let

[TABLE]

and

[TABLE]

where (cf. [7, Definition 1.1, Lemma 1.4] for $c_{n,s,+}$ and §2 below for $c_{n,s,-}$)

[TABLE]

In particular, if $0<s<n=1$ then there are two $s$-dependent constants $c_{\pm}$ to make the following Liouville fractional derivative formulas (cf. [27]):

[TABLE]

Hence it is natural and reasonable to adopt the notations

[TABLE]

The operators $\nabla^{s}_{+}$ and $\nabla^{s}_{-}$ can be viewed as the fractional extensions of the gradient operator

[TABLE]

Accordingly, for any $s\in(0,1)$, the Stein-Weiss inequality (1.1) (cf. [25]) amounts to

[TABLE]

Of course, it is appropriate to mention the following basic facts:

• $\rhd$

If $0<s<1<p<n/s$, then the right-hand-side of (1.2) can be replaced by $\|\nabla^{s}_{\pm}u\|_{L^{p}}$. More precisely, on the one hand, the boundedness of $\vec{R}$ on $L^{p>1}$ and (1.2) give (cf. [26, Lemma 2.4])

[TABLE]

One the other hand, [26, Theorem 1.8] derives

[TABLE]

• $\rhd$

If $0<s<p=1<n$, then the right-hand-side of (1.2) can be replaced by $\|\nabla^{s}_{-}u\|_{L^{1}}$ (cf. [25, Theorem $A^{\prime}$]) but cannot be replaced by $\|\nabla^{s}_{+}u\|_{L^{1}}$ (cf. [33, p.119]).

• $\rhd$

If $0<s<p=1=n$, then the right-hand-side of (1.2) cannot be replaced by either $\|\nabla^{s}_{+}u\|_{L^{1}}$ or $\|\nabla^{s}_{-}u\|_{L^{1}}$. A counterexample is given in [25, Section 3.3].

• $\rhd$

If $0<s<p=1\leq n$, then instead of the strong-type estimates, one has the following weak-type inequality:

[TABLE]

while the case for $\|\nabla^{s}_{+}u\|_{L^{1}}$ is due to the boundedness of $I_{s}$ from $L^{1}$ to $L^{\frac{n}{n-s},\infty}$ (cf. [2] or [33, p. 119]) and for $\|\nabla^{s}_{-}u\|_{L^{1}}$ follows further from [21, (1.5)] showing

[TABLE]

1.2. Overview of the principal results

The above analysis has driven us to take a fractional-geometrical-functional look at the most important case $p=1$ of the Stein-Weiss inequality (1.1).

Dense subspaces of $H^{s,1}\ \&\ H^{s,1}_{\pm}$

Denote by ${\mathcal{S}}$ the Schwartz class on ${{{\mathbb{R}}}^{n}}$ consisting of functions $f\in C^{\infty}$ such that

[TABLE]

Also, write ${\mathcal{S}}^{\prime}$ for the Schwartz tempered distribution space - the dual of ${\mathcal{S}}$ endowed with the weak-$\ast$ topology.

As detailed in §2, given $s\in(0,1)$, if we let

[TABLE]

then for any

[TABLE]

we can define $\nabla^{s}_{\pm}u$ as a distribution in ${\mathcal{S}}^{\prime}$. This definition and the case $p=1$ of (1.2) motivate us to consider the fractional Hardy-Sobolev space

[TABLE]

Note that

[TABLE]

So, $[\cdot]_{H^{s,1}}$ is properly a norm on quotient space of $H^{s,1}$ modulo the space of all real constants, and consequently this quotient space is a Banach space.

Upon introducing

[TABLE]

we immediately find

[TABLE]

Also, since ${\mathcal{S}}$ is dense in $H^{s,1}$ but it is hard to see the density of ${\mathcal{S}}$ in $H^{s,1}_{\pm}$, we are induced to introduce

[TABLE]

and yet still have

[TABLE]

whose $\mathring{H}^{s,1}_{\pm}$ is a Banach space modulo the space of all real constants.

Correspondingly, for $s\in(0,1)$ let $W^{s,1}$ be the collection of all locally integrable functions $u$ on ${{{\mathbb{R}}}^{n}}$ obeying

[TABLE]

Then the quotient space of $W^{s,1}$ modulo the space of all real constants is equal to the homogeneous Besov space $\dot{\Lambda}^{s}_{1,1}$ (cf. [38]) and is also called Sobolev-Slobodeckij space (cf. [36, p. 36]) or fractional Sobolev space (cf. [23]), and hence

[TABLE]

is dense in $W^{s,1}$. In accordance with [8, Appendix], any function

[TABLE]

can also be approximated by functions in $C_{c}^{\infty}$. Since (cf. [26, 27])

[TABLE]

it follows that

[TABLE]

Thus, both $H^{s,1}$ and $\mathring{H}^{s,1}_{\pm}$ contain $W^{s,1}$. More information on $\{\nabla^{s}_{\pm},H^{s,1},H^{s,1}_{\pm}\}$ is demonstrated in Propositions 2.11-2.12-2.13-2.14.

Tracing laws for $H^{s,1}\ \&\ H^{s,1}_{\pm}$

The previous discussions derive that

[TABLE]

and

[TABLE]

are valid, but

[TABLE]

is not true. In order to understand an essential reason for the truth of (1.5) or (1.6) and the fault of (1.7), we investigate under what condition of a given nonnegative Radon measure $\mu$ (restricting/tracing a function to a lower dimensional manifold) in $\mathbb{R}^{n}$ one has

[TABLE]

Accordingly, we discover such a tracing law that (1.8) is valid if and only if the isocapacitary inequality

[TABLE]

holds, where the right quantity of (1.9) is called \big{\{}H^{s,1},H^{s,1}_{\pm}\big{\}}\ni X-capacity of $K$ and defined by

[TABLE]

In §3, we utilize the fractional Sobolev capacity $\text{Cap}_{W^{s,1}}$ and the Hausdorff capacity $\Lambda^{n-s}_{(\infty)}$ to handle $\text{Cap}_{X\in\{H^{s,1},H^{s,1}_{\pm}\}}$ and its strong or weak capacitary inequality through Theorems 3.3 & 3.5-3.6. Then, we verify (1.8)$\Leftrightarrow$(1.9) in Theorem 3.7.

Duality laws for $H^{s,1}\ \&\ \mathring{H}^{s,1}_{\pm}$

As a by-product of (1.3)-(1.4) and the capacity analysis developed within §3, Theorem 4.3 shows that the dual space $[H^{s,1}]^{\ast}$ can be characterized by the bounded solutions

[TABLE]

of the fractional differential equation

[TABLE]

where

[TABLE]

Also, a similar characterization for

[TABLE]

is presented in Theorem 4.3 in terms of the bounded solutions to the fractional differential equation

[TABLE]

Furthermore, suppose that ${\mathop{\mathrm{missing}}{\,BMO\,}}$ is the well-known John-Nirenberg class of all locally integrable functions $f$ on ${{{\mathbb{R}}}^{n}}$ with bounded mean oscillation (cf. [14])

[TABLE]

where

[TABLE]

and the supremum is taken over all Euclidean balls $B\subset{{{\mathbb{R}}}^{n}}$ with volume $|B|$. Surprisingly and yet naturally, the argument for Theorem 4.3, plus the intrinsic structure of

[TABLE]

reveals (cf. Theorem 4.4) the inclusions

[TABLE]

in the sense of $\textup{in}\ {\mathcal{S}}^{\prime}/\mathcal{P}$ or ${\mathop{\mathrm{missing}}{\,BMO\,}}$, where $L^{\frac{n}{s},\infty}$ denotes the weak Lebesgue $\frac{n}{s}$-space consisting of all Lebesgue measurable functions $f$ on ${{{\mathbb{R}}}^{n}}$ such that

[TABLE]

and ${W}^{1,n}$ is the space of all locally integrable functions $f$ with respect to

[TABLE]

and one has the following decomposition of the canonical ${\mathop{\mathrm{missing}}{\,BMO\,}}$-function (cf. [30, 14])

[TABLE]

with $1_{[-1,1]}$ being the characteristic function of the interval $[-1,1]$. Let us take the space $I_{s}(L^{\frac{n}{s}})$ for example to explain why we require the validity of (1.10) in the sense of ${\mathcal{S}}^{\prime}/\mathcal{P}$. Indeed, if $f\in L^{\frac{n}{s}}$, then $I_{s}f$ may not be pointwisely well defined, but it is a well defined distribution in ${\mathcal{S}}^{\prime}/\mathcal{P}$.

Nevertheless, the importance of (1.10) can be also seen below.

• $\rhd$

Via a different approach, (1.10) reveals the essential structure of the Riesz transform part \vec{R}\cdot\big{(}L^{\infty}\big{)}^{n} of the Fefferman-Stein decomposition (cf. [9, Theorems 2&3], [37] for a constructive proof, and [13, 10] for some related discussions):

[TABLE]

And yet, it is uncertain that I_{s}\big{(}[\mathring{H}^{s,1}_{-}]^{\ast}\big{)} is strictly contained in ${\mathop{\mathrm{missing}}{\,BMO\,}}$ under $n\geq 2$.

• $\rhd$

(1.10) may be treated as a solution to the Bourgain-Brezis question (cf. [6, p.396]) - What are the function spaces $X,W^{1,n}\subset X\subset{\mathop{\mathrm{missing}}{\,BMO\,}}$, such that every $F\in X$ has a decomposition $F=\sum_{j=1}^{n}R_{j}Y_{j}$ where $Y_{j}\in L^{\infty}$?. This question is motivated by the fact that ${W}^{1,n}$ obeys the following decomposition ([6, p.305])

[TABLE]

• $\rhd$

As proved in [22, Theorem 3.5] (solving the open problem in [5, Remark 3.12]), if $\mathrm{BV}$ is the space of all $L^{1}$-functions with bounded variation on $\mathbb{R}^{n}$, then its dual space $[\mathrm{B}V]^{\ast}$ comprises all tempered distributions

[TABLE]

and hence

[TABLE]

This indicates that (1.10) has a limiting case $s\uparrow 1$:

[TABLE]

• $\rhd$

(1.10) improves [29, Corollary 1.5] which proves that

[TABLE]

• $\rhd$

(1.10) derives that (cf. Theorem 4.4) for

[TABLE]

one can get a vector-valued function

[TABLE]

Consequently, this divergence-equation-result is valid for

[TABLE]

Although $W^{1,n=1}$ is a proper subspace of $L^{\infty}$ on $\mathbb{R}$, the foregoing consequence cannot be extended to

[TABLE]

in the following sense that (cf. [6, p.394] or [20])

[TABLE]

Notation

In the foregoing and forthcoming discussions, $U\lesssim V$ (resp.  $U\gtrsim V$) means $U\leq cV$ (resp. $U\geq cV$) for a positive constant $c$ and $U\approx V$ amounts to $U\gtrsim V\gtrsim U$. Moreover, $1_{E}$ stands for the characteristic function of a set $E\subset{{{\mathbb{R}}}^{n}}$, and

[TABLE]

2. Dense subspaces of $H^{s,1}\,\&\,H^{s,1}_{\pm}$

2.1. Initial definitions of $\nabla^{s}_{\pm}$

Note that any $f\in{\mathcal{S}}$ has its Fourier transform

[TABLE]

So the Fourier transform can be naturally extended to ${\mathcal{S}}^{\prime}$ by the dual paring

[TABLE]

Definition 2.1**.**

For $(s,\phi)\in(-n,1]\times{\mathcal{S}}$ let $(-\Delta)^{\frac{s}{2}}\phi$ be determined by the Fourier transform

[TABLE]

Then we have the following comments.

1. (i)

Since $|\xi|^{s}$ has singularity at the origin, it is not true that $(-\Delta)^{\frac{\beta}{2}}\phi\in{\mathcal{S}}$ for general $\phi\in{\mathcal{S}}$. However, if

[TABLE]

then (cf. **[28, Section 2]** or **[7]**)

[TABLE] 2. (ii)

Recall that

[TABLE]

Then

[TABLE] 3. (iii)

As the dual space of ${\mathcal{S}}_{\infty}$ let ${\mathcal{S}}^{\prime}/\mathcal{P}$ be the space $\mathcal{S}^{\prime}$ modulo the space $\mathcal{P}$ of all real-valued polynomials. Then, for any $f\in{\mathcal{S}}^{\prime}/\mathcal{P}$ we can define $(-\Delta)^{\frac{s}{2}}f$ as a distribution in ${\mathcal{S}}^{\prime}/\mathcal{P}$:

[TABLE]

Evidently, $(-\Delta)^{\frac{s}{2}}$ maps ${\mathcal{S}}^{\prime}/\mathcal{P}$ onto ${\mathcal{S}}^{\prime}/\mathcal{P}$ (cf. **[36, pp. 241-242]**).

The $(0,n)\ni\alpha$-th order Riesz potential $I_{\alpha}$ is defined by

[TABLE]

If $f\in{\mathcal{S}}$, then $I_{\alpha}f$ has the integral expression (cf. [33, p. 117])

[TABLE]

Based on Definition 2.1(iii), the definition of $I_{\alpha}f$ is extendable to $f\in{\mathcal{S}}^{\prime}/\mathcal{P}$ and so $I_{\alpha}$ maps ${\mathcal{S}}^{\prime}/\mathcal{P}$ onto ${\mathcal{S}}^{\prime}/\mathcal{P}$.

About $\nabla^{s}_{+}$

Upon following [28, Section 2.1], we can extend the definition of $\nabla^{s}_{+}$ to more general distributions.

Definition 2.2**.**

For $(s,f)\in(0,1)\times{\mathcal{S}}$ set

[TABLE]

Then we have the following comments.

• (i)

If $f\in{\mathcal{S}}_{s}^{\prime}$, then $\nabla^{s}_{+}f$ is defined as a distribution in ${\mathcal{S}}^{\prime}$:

[TABLE]

• (ii)

According to **[28, Proposition 2.4]**, if $f$ belongs to the weighted-$L^{1}$* space*

[TABLE]

and the Hölder space $\mathcal{C}^{s+\epsilon_{x}}$ in a neighborhood of $x\in{{{\mathbb{R}}}^{n}}$ for some $\epsilon_{x}\in(0,1-s]$, then $\nabla^{s}_{+}f$ is continuous at $x$ and it has the integral expression (cf. **[7, Definition 1.1 & Lemma 1.4]**)

[TABLE]

Evidently, this integral expression holds for any $f\in{{\mathcal{S}}}$.

The next lemma shows that $I_{s}$ is the inverse of $\nabla^{s}_{+}$ on ${\mathcal{S}}$, and vice versa.

Lemma 2.3**.**

If $(s,\phi,x)\in(0,1)\times{\mathcal{S}}\times{{{\mathbb{R}}}^{n}}$, then

[TABLE]

Proof.

On the one hand, [33, p. 117,  Lemma 1(a)] and Definition 2.2 derive

[TABLE]

On the other hand, for any $\alpha\in\mathbb{Z}^{n}_{+}$ we use

[TABLE]

to get

[TABLE]

which implies

[TABLE]

Accordingly, $I_{s}\phi\in\mathbb{L}_{s}$ and $I_{s}\phi$ locally satisfies the Lipschitz condition. Now, an application of Definition 2.2(ii) gives that $(-\Delta)^{\frac{s}{2}}I_{s}\phi$ is continuous on ${{{\mathbb{R}}}^{n}}$. Furthermore, since

[TABLE]

we have

[TABLE]

where the second identity is from the Fubini theorem and the last identity is due to the already-proved identification

[TABLE]

Accordingly,

[TABLE]

But nevertheless,

[TABLE]

are continuous on ${{{\mathbb{R}}}^{n}}$, so we arrive at

[TABLE]

∎

About $\nabla^{s}_{-}u$

We begin with the following

Definition 2.4**.**

For $(s,j,\phi)\in(0,1)\times\{1,2,\dots,n\}\times{\mathcal{S}}$ let

[TABLE]

where each $\nabla^{s}_{j}\phi$ is defined via the Fourier transform:

[TABLE]

Lemma 2.5**.**

If $(s,\phi,x)\in(0,1)\times{\mathcal{S}}\times{{{\mathbb{R}}}^{n}}$, then

[TABLE]

Proof.

With

[TABLE]

it follows from Definition 2.4 that any $\phi\in{\mathcal{S}}$ satisfies

[TABLE]

Note that the second equality in the above formula also implies

[TABLE]

Moreover, the integration by parts formula gives

[TABLE]

where $\vec{\nu}$ is the outward unit vector on the surface of the ring

[TABLE]

and ${\mathcal{H}}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure. An application of

[TABLE]

derives

[TABLE]

and

[TABLE]

Consequently, the Lebesgue dominated convergence theorem, along with

[TABLE]

yields the desired integral expression in (2.1):

[TABLE]

∎

Lemma 2.6**.**

If $(s,j)\in(0,1)\times\{1,2,\dots,n\}$, then $\nabla^{s}_{j}$ maps ${\mathcal{S}}$ into ${\mathcal{S}}_{s}$.

Proof.

Suppose

[TABLE]

Since

[TABLE]

the Fourier transform gives

[TABLE]

This, combined with the integral representation of $\nabla^{s}_{j}{D^{\alpha}}\phi$ given in Lemma 2.5, yields

[TABLE]

Clearly,

[TABLE]

Also, the mean-value theorem derives

[TABLE]

Combining the above two estimates gives

[TABLE]

and so

[TABLE]

∎

Lemma 2.6 can be used to extend the definition of $\nabla^{s}_{-}$ to all distributions in ${\mathcal{S}}_{s}^{\prime}$.

Definition 2.7**.**

For $(s,f)\in(0,1)\times{\mathcal{S}}_{s}^{\prime}$ let

[TABLE]

where $\nabla^{s}_{j}\phi$ is defined by

[TABLE]

Like Definition 2.2 made for $\nabla^{s}_{+}$, we have also the integral representing of $\nabla^{s}_{-}f$ whenever $f\in\mathbb{L}_{s}$ has local Hölder regularity.

Lemma 2.8**.**

Let $(s,f,x)\in(0,1)\times\mathbb{L}_{s}\times{{{\mathbb{R}}}^{n}}$. If $f$ has the Hölder continuity of order ${s+\epsilon}$ in a neighborhood $\Omega$ of $x$ for some $\epsilon\in(0,1-s]$, then $\nabla^{s}_{-}f$ is continuous at $x$ and

[TABLE]

Proof.

Without loss of generality, we may assume that $\Omega$ is bounded and naturally $\Omega^{\complement}={{{\mathbb{R}}}^{n}}\setminus\Omega$ is unbounded. An application of both

[TABLE]

and

[TABLE]

derives

[TABLE]

and hence the integral in the right-hand-side of (2.2) converges absolutely.

To show (2.2), we take an arbitrary open set $\Omega_{0}\ni x$ compactly contained in $\Omega$. According to the proof of [28, Proposition 2.4], there exists a sequence $\{f_{k}\}_{k\in{\mathbb{N}}}\subset{\mathcal{S}}$ uniformly bounded in $C^{s+\epsilon}(\Omega)$, converging uniformly to $f$ in $\Omega_{0}$ and also converging to $f$ in the norm of $\mathbb{L}_{s}$. For any $k\in{\mathbb{N}}$, since $f_{k}\in{\mathcal{S}}$, we utilize Lemma 2.5 to write

[TABLE]

From the uniform bound on the $C^{s+\epsilon}$-norm of $f_{k}$ in $\Omega_{0}$ it follows that

[TABLE]

uniformly in $\Omega_{0}$ as $k\to\infty$. Since $\{f_{k}\}_{k\in{\mathbb{N}}}$ converges to $f$ in the norm of $\mathbb{L}_{s}$, it follows easily that

[TABLE]

Accordingly, $\nabla^{s}_{-}f(x)$ must coincide with

[TABLE]

in $\Omega_{0}$ by the uniqueness of the limits. So, (2.2) holds. ∎

Below is more information on $\nabla^{s}_{-}$.

Lemma 2.9**.**

Let $s\in(0,1)$.

1. (i)

If $\phi\in{\mathcal{S}}_{\infty}$, then it holds pointwisely on ${{{\mathbb{R}}}^{n}}$ that

[TABLE] 2. (ii)

If $\phi\in{\mathcal{S}}_{s}^{\prime}$, then $\nabla^{s}_{-}\phi=I_{1-s}\nabla\phi$ in ${\mathcal{S}}^{\prime}/\mathcal{P}$. 3. (iii)

If $\phi\in{\mathcal{S}}$, then all identities in (i) hold almost everywhere on ${{{\mathbb{R}}}^{n}}$.

Proof.

(i) Via the Fourier transform, we see that

[TABLE]

map ${\mathcal{S}}_{\infty}$ into ${\mathcal{S}}_{\infty}$. Then, taking the inverse Fourier transform verifies the assertion in (i).

(ii) Let

[TABLE]

Then by the just-checked (i) and Definition 2.7 we have

[TABLE]

Further, since

[TABLE]

this implication, along with the fact that $I_{1-s}$ maps ${\mathcal{S}}^{\prime}/\mathcal{P}$ onto ${\mathcal{S}}^{\prime}/\mathcal{P}$, derives

[TABLE]

namely,

[TABLE]

(iii) Observe that ${\mathcal{S}}_{\infty}$ is dense in $L^{p}$ whenever $p\in(1,\infty)$. Indeed, this follows easily from the fact that the Calderón reproducing formula of an $L^{p}$-function

[TABLE]

holds in $L^{p}$ (cf. [11, p.8, Theorem (1.2)] for $p=2$ and [24] for general $p$), with $\psi,\varphi\in{\mathcal{S}}_{\infty}$ satisfying

[TABLE]

Thus, if $\phi\in{\mathcal{S}}$, then a discussion similar to (ii) yields that the identity in (i) holds in ${\mathcal{S}}^{\prime}/\mathcal{P}$. Moreover, by the density of ${\mathcal{S}}_{\infty}$ in $L^{2}$ and the duality equality

[TABLE]

we obtain that the identity in (i) holds in $L^{2}$ and hence almost everywhere on ${{{\mathbb{R}}}^{n}}$. ∎

2.2. Dense subspaces of $H^{s,1}\ \&\ H^{s,1}_{+}$

Note that ${\mathcal{S}}_{\infty}$ is dense in $H^{1}$. However, instead of ${\mathcal{S}}_{\infty}$ we may consider the following larger space

[TABLE]

A dense subspace of $H^{1}$

As showed in the coming-up-next Lemma 2.10 whose argument relies on the radial maximal function characterization of the Hardy space $H^{1}$ (cf. [34]), the class defined by (2.3) is a dense subspace of $H^{1}$. To see this, recall that if

[TABLE]

then

[TABLE]

We are led to discover the following density for $H^{1}$.

Lemma 2.10**.**

Let $s\in(0,\infty)$. Then any locally integrable function $f$ on ${{{\mathbb{R}}}^{n}}$ with

[TABLE]

belongs to the Hardy space $H^{1}$. Consequently, ${\mathcal{S}}_{s,0}$ is dense in $H^{1}$. Moreover,

[TABLE]

Proof.

Let $\phi$ and $\{\phi_{t}\}_{t\in(0,\infty)}$ be as in (2.4). By the radial maximal function characterization of $H^{1}$, we only need to show that

[TABLE]

holds for some $\epsilon\in(0,s)$. Indeed,

[TABLE]

However, (2.5) is verified by handling two situations: $|x|\leq 1$ and $|x|>1$.

If $|x|\leq 1$, then

[TABLE]

If $|x|\geq 1$, then by the conditions of $f$ we write

[TABLE]

On the one hand, the mean value theorem gives

[TABLE]

On the other hand,

[TABLE]

Via combining the last three formulae we obtain

[TABLE]

thereby reaching (2.5).

The remaining part of Lemma 2.10 is obvious. ∎

The first and second dense subspaces of $H^{s,1}$

Lemma 2.10 produces the following property.

Proposition 2.11**.**

Let $s\in(0,1)$. Then

1. (i)

$H^{s,1}\cap{\mathcal{S}}_{\infty}=I_{s}({\mathcal{S}}_{\infty}\cap H^{1}).$** 2. (ii)

$I_{s}(H^{1})\subset H^{s,1}$. 3. (iii)

For any $f\in H^{s,1}$ there exists $g\in H^{1}$ such that $f=I_{s}g$ in ${\mathcal{S}}^{\prime}/\mathcal{P}$.

Proof.

(i) For any $\phi\in{\mathcal{S}}_{\infty}$, by the invariant property of ${\mathcal{S}}_{\infty}$ under the action of $I_{s}$ or $(-\Delta)^{\frac{s}{2}}$, we get

[TABLE]

as desired.

(ii) If $f\in I_{s}(H^{1})$, then

[TABLE]

and hence

[TABLE]

Of course, any function in $L^{\frac{n}{n-s}}$ belongs to ${\mathcal{S}}_{s}^{\prime}$. Accordingly, for any $\phi\in{\mathcal{S}}$, by Lemma 2.3 we have

[TABLE]

where in the penultimate equality the Fubini theorem has been applied due to the implication that if

[TABLE]

then

[TABLE]

Therefore, we obtain

[TABLE]

Since $g$ belongs to $H^{1}$, so does $(-\Delta)^{\frac{s}{2}}f$. This proves

[TABLE]

(iii) Recall that both $I_{s}$ and $(-\Delta)^{\frac{s}{2}}$ are one-to-one maps from ${\mathcal{S}}^{\prime}/\mathcal{P}$ to ${\mathcal{S}}^{\prime}/\mathcal{P}$. Thus, we have

[TABLE]

thereby getting

[TABLE]

and so

[TABLE]

∎

Next, we have the following density result.

Proposition 2.12**.**

If $s\in(0,1)$, then

[TABLE]

Moreover, both ${\mathcal{S}}_{\infty}$ and ${\mathcal{S}}$ are dense in $H^{s,1}$.

Proof.

For any $u\in{\mathcal{S}}$, we use Lemma 2.10 to derive

[TABLE]

This proves ${\mathcal{S}}\subset H^{s,1}$; the other inclusions are obvious.

It remains to show the density of ${\mathcal{S}}_{\infty}$ in $H^{s,1}$. If $f\in H^{s,1}$, then

[TABLE]

Due to the density of ${\mathcal{S}}_{\infty}$ in $H^{1}$,

[TABLE]

For any $j\in{\mathbb{N}}$, let

[TABLE]

which actually belongs to ${\mathcal{S}}_{\infty}$ in terms of the Fourier transform. Noticing that

[TABLE]

we have

[TABLE]

Thus, $f\in H^{s,1}$ can be approximated by the ${\mathcal{S}}_{\infty}$-functions $\{f_{j}\}_{j\in{\mathbb{N}}}$. ∎

A dense subspace of $H^{s,1}_{+}$

It is difficult to determine the density of ${\mathcal{S}}$ in $H^{s,1}_{\pm}$. However, we have

Proposition 2.13**.**

If $s\in(0,1)$, then $I_{s}({\mathcal{S}})$ is a dense subspace of $H^{s,1}_{+}$ but

[TABLE]

Proof.

On the one hand, if $f\in I_{s}({\mathcal{S}})$ then

[TABLE]

but Lemma 2.3 implies

[TABLE]

that is, $f\in H^{s,1}_{+}.$ To show the density of $I_{s}({\mathcal{S}})$ in $H^{s,1}_{+}$, given any $f\in H^{s,1}_{+}$ we utilize

[TABLE]

and the density of ${\mathcal{S}}$ in $L^{1}$ to find a sequence

[TABLE]

such that

[TABLE]

Upon defining

[TABLE]

and using Lemma 2.3, we gain the representation

[TABLE]

and the desired convergence

[TABLE]

In other words, $I_{s}({\mathcal{S}})$ is a dense subspace of $H^{s,1}_{+}$.

On the other hand, $I_{s}({\mathcal{S}})$ is not a subspace of $H^{s,1}_{-}$ - otherwise - if $I_{s}({\mathcal{S}})\subset H^{s,1}_{-}$, then this, along with $I_{s}({\mathcal{S}})\subset H^{s,1}_{+}$, would imply $I_{s}({\mathcal{S}})\subset H^{s,1}$ and hence ${\mathcal{S}}\subset H^{1}$ which is impossible. ∎

The third dense subspace of $H^{s,1}$

In addition to Proposition 2.12, we obtain

Proposition 2.14**.**

If $s\in(0,1)$, then

[TABLE]

is a dense subspace of $H^{s,1}$.

Proof.

Proposition 2.12 implies

[TABLE]

So, it suffices to show the density of ${\mathcal{D}}_{0}$ in $H^{s,1}$. Let $f\in H^{s,1}$. Based on Proposition 2.11(iii),

[TABLE]

Note that $H^{1}$ is nothing but the homogeneous Triebel-Lizorkin space $\dot{F}^{0}_{1,2}$. So, the lifting property of $I_{s}$ on the Triebel-Lizorkin spaces (cf. [36, p. 242]) shows

[TABLE]

Therefore,

[TABLE]

Recall that [16, Theorem 1] yields that any element in $\dot{F}^{s}_{1,2}$ can be written as the linear combinations of $\dot{F}^{s}_{1,2}$-atoms, just as the atomic decomposition of the Hardy space $H^{1}$. To be precise, since $\widetilde{f}\in\dot{F}^{s}_{1,2}$, it follows that

[TABLE]

where

[TABLE]

and, based on the remark after [16, Definition (1.6)], every $a_{j}$ is a locally integrable function on ${{{\mathbb{R}}}^{n}}$ with the following three properties:

1. (i)

$a_{j}$ is supported on a ball $B_{j}$; 2. (ii)

$\|a_{j}\|_{\dot{F}^{s}_{2,2}}\leq|B_{j}|^{-\frac{1}{2}}$; 3. (iii)

$\int_{{{\mathbb{R}}}^{n}}a_{j}\,dx=0$.

Again using the lifting property of $(-\Delta)^{\frac{s}{2}}$ (cf. [36, p. 242]) gives

[TABLE]

By [36, p. 242, (2)], any element in $\dot{F}^{0}_{2,2}$ coincides with a function in $L^{2}$ in the sense of ${\mathcal{S}}^{\prime}/\mathcal{P}$. Thus, we know from $(-\Delta)^{\frac{s}{2}}a_{j}\in\dot{F}^{0}_{2,2}$ that $(-\Delta)^{\frac{s}{2}}a_{j}$ coincides with some $L^{2}$-function, denoted by $\widetilde{(-\Delta)^{\frac{s}{2}}a_{j}}$, in the sense of ${\mathcal{S}}^{\prime}/\mathcal{P}$. So, by the density of ${\mathcal{S}}_{\infty}$ in $L^{2}$ (cf. the proof of Lemma 2.9(iii)) and the duality we get

[TABLE]

Let $\psi\in C_{c}^{\infty}$ satisfy

[TABLE]

For any $\epsilon\in(0,\infty)$, define

[TABLE]

Fix an arbitrary small number $\eta\in(0,\infty)$. For any $j\in{\mathbb{N}}$, an application of $(-\Delta)^{\frac{s}{2}}a_{j}\in L^{2}$ produces a sufficiently small $\epsilon_{j}\in(0,\infty)$ such that

[TABLE]

By (2.2), the last inequality is equivalent to that

[TABLE]

Choose $N$ large enough such that

[TABLE]

and define

[TABLE]

Evidently,

[TABLE]

By the argument in [16, p. 239], we know that any $\dot{F}^{s}_{1,2}$-atom $a_{j}$ satisfies $\|a_{j}\|_{\dot{F}^{s}_{1,2}}\lesssim 1.$ The choice of $\epsilon_{j}$ implies that

[TABLE]

is also an $\dot{F}^{s}_{1,2}$-atom, thereby yielding

[TABLE]

Upon recalling

[TABLE]

we obtain

[TABLE]

Finally, using the lifting property of $I_{s}$ (cf. [36, p. 242]) yields

[TABLE]

Due to the arbitrariness of $\eta$, we obtain that $f\in H^{s,1}$ can be approximated by functions in ${\mathcal{D}}_{0}$. ∎

3. Tracing laws for $H^{s,1}\,\&\,H^{s,1}_{\pm}$

3.1. Strong/weak estimates for $\text{Cap}_{X\in\{H^{s,1},H^{s,1}_{\pm}\}}$

This section is devoted to a measure-theoretic study of the capacity living on $X\in\{W^{s,1},H^{s,1},H^{s,1}_{\pm}\}$.

Capacitary concepts

For $\alpha\in(0,n)$, denote by $\Lambda^{\alpha}_{(\infty)}$ the $\alpha$-dimensional Hausdorff capacity:

[TABLE]

for any set $E\subset{{{\mathbb{R}}}^{n}}$ which is covered by a sequence of balls

[TABLE]

Classically, $\Lambda^{\alpha}_{(\infty)}(\cdot)$ is a monotone, countably subadditive set function on the class of all subsets of ${{{\mathbb{R}}}^{n}}$, and enjoys $\Lambda^{\alpha}_{(\infty)}(\emptyset)=0$.

Definition 3.1**.**

For $s\in(0,1)$ and any compact set $K\subset{{{\mathbb{R}}}^{n}}$ define (cf. [38, 4])

[TABLE]

Furthermore, $\text{Cap}_{X}(\cdot)$ is extendable from compact sets to general sets as seen below.

• (i)

If $O\subset{{{\mathbb{R}}}^{n}}$ is open, then

[TABLE]

• (ii)

For an arbitrary set $E\subset{{{\mathbb{R}}}^{n}}$ set

[TABLE]

Thus, the definition of $\text{Cap}_{X}$ on any compact/open set is consistent (cf. [19, Lemma 3.2.4]).

Lemma 3.2**.**

Let

[TABLE]

Then

1. (i)

$\text{Cap}_{X}(\emptyset)=0\ \ \&\ \ \text{Cap}_{X}(B(x,r))=r^{n-s}\text{Cap}_{X}(B(0,1))$. 2. (ii)

$\text{Cap}_{X}(E_{1})\leq\text{Cap}_{X}(E_{2})$* whenever $E_{1}\subset E_{2}\subset{{{\mathbb{R}}}^{n}}$.* 3. (iii)

[TABLE] 4. (iv)

$\text{Cap}_{W^{s,1}}(\cdot)$* is countably subadditive, but $\text{Cap}_{H^{s,1}}(\cdot)$ and $\text{Cap}_{H^{s,1}_{\pm}}(\cdot)$ may not be countably subadditive.*

Proof.

Both (i) and (ii) follow from (3.1).

(iii) First, according to Definition 3.1, we only need to consider these capacities on compact sets. For any $u\in{\mathcal{S}}$, by (1.4), we get

[TABLE]

Noting that

[TABLE]

is given in [23, Theorem 2.1] and [38, (2.1)], we are left to verify

[TABLE]

According to [4, Proposition 3], for any compact set $K$ in ${{{\mathbb{R}}}^{n}}$, the capacity

[TABLE]

satisfies

[TABLE]

By Lemma 2.10, we have

[TABLE]

and the density of ${\mathcal{S}}_{s,0}$ in $H^{1}$. Meanwhile, for any $(f,x)\in{\mathcal{S}}_{s,0}\times{{{\mathbb{R}}}^{n}}$, it is obvious that $I_{s}f(x)$ is well defined and $I_{s}f$ is continuous on ${{{\mathbb{R}}}^{n}}$. Thus, instead of using ${\mathcal{S}}_{\infty}$, we have

[TABLE]

For any $u\in{\mathcal{S}}$ satisfying $u\geq 1$ on $K$ let

[TABLE]

which belongs to ${\mathcal{S}}_{s,0}$ in terms of Lemma 2.10. Then, by Lemma 2.3 we have

[TABLE]

thereby achieving

[TABLE]

Taking the infimum over all such $u\in{\mathcal{S}}$ satisfying $u\geq 1$ on $K$ yields

[TABLE]

Thus,

[TABLE]

This proves (3.2).

(iv) The countable subadditivity of $\text{Cap}_{W^{s,1}}(\cdot)$ follows from [39, Theorem 1(iii)]. Since the test functions used in

[TABLE]

are not assumed to be nonnegative, the capacities under consideration may not be countably subadditive as mentioned in [4]. ∎

Strong estimates for $\text{Cap}_{X\in\{H^{s,1},H^{s,1}_{-}(n>1)\}}$

First of all, an application of Proposition 3.2(iii) and [38, Theorem 1.1] or [23, Theorem 1.3] gives the following strong inequality for $\text{Cap}_{W^{s,1}}$ (cf. [38, Theorem 2.2]):

[TABLE]

Next, we are led by (3.3) to get the strong inequality for $\text{Cap}_{H^{s,1}}$ as seen below.

Theorem 3.3**.**

If $s\in(0,1)$, then

[TABLE]

Proof.

Note that Proposition 3.2(iii) implies

[TABLE]

and [4, Proposition 5] gives that

[TABLE]

In particular, given $u\in{\mathcal{S}}$, we can take

[TABLE]

which belongs to ${\mathcal{S}}_{s,0}$ via Lemma 2.10. Noting that Lemmas 2.3 and 2.9(iii) imply

[TABLE]

we obtain

[TABLE]

as desired. ∎

To establish the strong inequality for $\text{Cap}_{H_{-}^{s,1}(n>1)}$, we require the following lemma which generalizes [4, Proposition 5].

Lemma 3.4**.**

If $n\geq 2$ and $0\leq\beta<\alpha<n$, then

[TABLE]

Proof.

Let $f\in{\mathcal{S}}_{\alpha,0}$. Note that Lemma 2.10 implies

[TABLE]

So, by this and the boundedness of each Riesz transform $R_{j}$ from $H^{1}$ to $L^{1}$, we derive $\|\vec{R}f\|_{L^{1}}<\infty$. Upon applying [4, p.  118, Corollary] we have

[TABLE]

where

[TABLE]

Thus, the desired result follows from showing that

[TABLE]

holds when $\mu$ is a nonnegative Radon measure on ${{{\mathbb{R}}}^{n}}$ with $\||\mu\||_{n-\beta}\leq 1$ - surprisingly - this assertion cannot be extended to the case $\alpha=\beta$ (cf. [31, Theorem 1.3] which solves [30, Open Problem 7.1]).

Upon taking

[TABLE]

we have

[TABLE]

For such $\bar{\alpha},\bar{p}$ and $\bar{\beta}$, we apply

[TABLE]

and [17, Theorem 1.1] (extending the main result in [1]) to derive

[TABLE]

Moreover, it is proved in [25, Theorem A] that under the assumption $n\geq 2$ one has

[TABLE]

From the last two estimates and the fact

[TABLE]

it follows that

[TABLE]

as desired; see also [12, (1.7)] for a similar estimate for an elliptic differential operator $\mathbb{A}[D]$. ∎

Finally, we arrive at the following strong type inequality for $\text{Cap}_{H^{s,1}_{-}}$.

Theorem 3.5**.**

If $0<\hat{s}<s<1<n$, then

[TABLE]

Proof.

Given $u\in{\mathcal{S}}$. Lemmas 3.2 & 2.9 produce

[TABLE]

So, based on the argument for Theorem 3.3 and

[TABLE]

it is enough to verify

[TABLE]

However, this last estimation is established in Lemma 3.4. ∎

Weak estimates for $\text{Cap}_{H^{s,1}_{\pm}}$

Although we do not know whether $\hat{s}$ and $s$ in Theorem 3.5 can coincide, we have the following assertion.

Theorem 3.6**.**

Let $0<s<1\leq n$. Then

[TABLE]

But, if X\in\big{\{}H^{s,1}_{+},H^{s,1}_{-}(n=1)\big{\}} then there is no constant $C>0$ such that

[TABLE]

Proof.

For $(t,u)\in(0,\infty)\times\mathcal{S}$, since

[TABLE]

is open, by the definition of $\text{Cap}_{X}$, for any $\epsilon\in(0,\infty)$ there exists a compact set

[TABLE]

such that

[TABLE]

Let $v=t^{-1}u$. Then

[TABLE]

Accordingly, by definition we have

[TABLE]

which implies

[TABLE]

Letting $\epsilon\to 0$ gives the desired estimate

[TABLE]

Since

[TABLE]

we get

[TABLE]

In order to verify the nonexistence of the capacitary strong estimate for $X$ under consideration, we note that

[TABLE]

which follows from

[TABLE]

In fact,

[TABLE]

can be seen from [2]. This last inequality, along with the fact (from the Fourier transform) that

[TABLE]

and (cf. [21, (1.5)])

[TABLE]

in turn derives

[TABLE]

Both (3.4) and the definition of $\text{Cap}_{H^{s,1}_{\pm}}$ give the iso-capacitary inequality

[TABLE]

We use (3.5) to verify the failure of the capacitary strong estimate for X\in\big{\{}H^{s,1}_{+},H^{s,1}_{-}(n=1)\big{\}}. Suppose otherwise that

[TABLE]

Then, a standard layer-cake method (cf. [19, p.101]) and (3.5) derive a constant $C_{1}$ depending on $C_{0}$ such that any $u\in{\mathcal{S}}$ satisfies

[TABLE]

This contradicts the failure of (1.7) mentioned in §1.2, thereby completing the verification.

∎

3.2. Restrictions/traces of $H^{s,1}\ \&\ H^{s,1}_{\pm}$

Being motivated by [38, Theorem 1.4] for $W^{s,1}$, we establish the coming-up-next restricting/tracing principle.

Theorem 3.7**.**

Let $0<s<1\leq n$, $\mu$ be a nonnegative Radon measure on $\mathbb{R}^{n}$ and

[TABLE]

Then the following two assertions are equivalent:

1. (i)

there exists a positive constant $c$ such that

[TABLE] 2. (ii)

there exists a positive constant $C$ such that

[TABLE]

Moreover, the constants $c$ and $C$ are comparable to each other.

Proof.

The consequence part of Theorem 3.7 follows from (i)$\Leftrightarrow$(ii) and

[TABLE]

So, we are required to validate (i)$\Leftrightarrow$(ii). Two cases are considered for

[TABLE]

Case 1: (i)$\Leftrightarrow$(ii) for $X=H^{s,1}_{\pm}$.

On the one hand, if (i) holds, then the subadditivity of $\mu$, the decomposition

[TABLE]

and Theorem 3.6 derive

[TABLE]

thereby verifying (ii).

On the other hand, suppose that (ii) is valid. For any compact $K\subset{{{\mathbb{R}}}^{n}}$ let

[TABLE]

Then

[TABLE]

Accordingly, by definition we reach (i).

Case 2: (i)$\Leftrightarrow$(ii) for $X=H^{s,1}$.

On the one hand, for any $k\in{\mathbb{Z}}$ the open set $E_{2^{k}}$ has a compact subset $K_{k}$ such that

[TABLE]

Thus, if (i) is valid, then

[TABLE]

Note that for any nonnegative sequence $\{a_{j}\}_{j\in{\mathbb{Z}}}$,

[TABLE]

This in turn gives

[TABLE]

Moreover, by Lemma 3.2(ii) it follows that

[TABLE]

Altogether, we use Theorem 3.3 to obtain

[TABLE]

which implies (ii).

On the other hand, suppose that (ii) is true. Upon letting $K$ be a compact subset of ${{{\mathbb{R}}}^{n}}$ we gain that for any $u\in\mathcal{S}$ with $u\geq 1$ on $K$,

[TABLE]

Via taking the supremum over all such $u\in\mathcal{S}$ with $u\geq 1$ on $K$ we get (i). ∎

4. Duality laws for $H^{s,1}\,\&\,\mathring{H}^{s,1}_{\pm}$

4.1. Adjoint operators of $\nabla^{s}_{\pm}$ via $\{\mathcal{S},{\mathop{\mathrm{missing}}{\,BMO\,}}\}$

This subsection describes the adjoint operators of $\nabla^{s}_{\pm}$ (existing as two basic notions in fractional vector calculus).

Integration-by-parts

Below is a two-fold computation.

• $\rhd$

On the one hand, the dual operator \big{[}(-\Delta)^{\frac{s}{2}}\big{]}^{\ast} of $(-\Delta)^{\frac{s}{2}}$ is itself, i.e.,

[TABLE]

in the sense of

[TABLE]

This is reasonable because of (cf. [32])

[TABLE]

and

[TABLE]

• $\rhd$

On the other hand, if we define

[TABLE]

then it enjoys (cf. [26, Theorem 1.3])

[TABLE]

and (cf. [8, Lemma 2.5])

[TABLE]

Thus $-\text{div}^{s}$ exists as the dual operator \big{[}\nabla^{s}_{-}\big{]}^{\ast} of $\nabla^{s}_{-}$, i.e.,

[TABLE]

Dual pairing for $\{\mathcal{S},{\mathop{\mathrm{missing}}{\,BMO\,}}\}$

We are required to verify that ${\mathop{\mathrm{missing}}{\,BMO\,}}$ can be embedded in a family of relatively bigger spaces.

Lemma 4.1**.**

If $s\in(0,1)$, then ${\mathop{\mathrm{missing}}{\,BMO\,}}\subset{\mathcal{S}}_{s}^{\prime}$.

Proof.

In order to verify $f\in{\mathcal{S}}_{s}^{\prime}$, it suffices to show that $f\in{\mathop{\mathrm{missing}}{\,BMO\,}}$ induces a continuous linear functional on ${\mathcal{S}}_{s}$. To this end, we consider

[TABLE]

Upon writing

[TABLE]

and noting both

[TABLE]

and

[TABLE]

we obtain

[TABLE]

as desired. ∎

Proposition 4.2**.**

For $s\in(0,1)$ one has the following two implications.

1. (i)

If $(f,\phi)\in{\mathop{\mathrm{missing}}{\,BMO\,}}\times{\mathcal{S}}$, then

[TABLE] 2. (ii)

If $\vec{U}=(U_{1},\dots,U_{n})\in(L^{\infty})^{n}$ and $\phi\in{\mathcal{S}}$, then

[TABLE]

Proof.

Note that (i) follows directly from Lemma 4.1 and Definition 2.2(i).

Now we show (ii). For any $j\in\{1,2,\dots,n\}$, it is known that $R_{j}$ maps $L^{\infty}$ functions continuously into ${\mathop{\mathrm{missing}}{\,BMO\,}}$ and that

[TABLE]

follows from Lemma 4.1. So, Definition 2.2(i) derives that every

[TABLE]

By this and the definition of $[\nabla^{s}_{-}]^{\ast}$, we have

[TABLE]

Thus, for $\phi\in{\mathcal{S}}$ we have

[TABLE]

Since $\phi\in{\mathcal{S}}$, Lemma 2.10 yields

[TABLE]

By

[TABLE]

we further obtain

[TABLE]

thereby finding

[TABLE]

∎

4.2. Dualities of $H^{s,1}\ \&\ \mathring{H}^{s,1}_{\pm}$

This subsection is divided into two parts.

Fundamental duality

Below is the expected duality law.

Theorem 4.3**.**

Let $0<s<1\leq n$, $T\in{\mathcal{S}}^{\prime}$ and $\nu$ be a nonnegative Radon measure on $\mathbb{R}^{n}$. Then:

1. (i)

$T\in[{H}^{s,1}]^{\ast}$* if and only if*

[TABLE]

if and only if

[TABLE] 2. (ii)

$\||\nu\||_{n-s}<\infty$* if and only if*

[TABLE]

if and only if

[TABLE] 3. (iii)

$T\in[\mathring{H}^{s,1}_{+}]^{\ast}$* if and only if*

[TABLE] 4. (iv)

$T\in[\mathring{H}^{s,1}_{-}]^{\ast}$* if and only if*

[TABLE]

Proof.

(i) First of all, by using the density of ${\mathcal{S}}_{\infty}$ in both $H^{1}$ and $H^{s,1}$ (cf. Proposition 2.12) and the invariant of ${\mathcal{S}}_{\infty}$ under $I_{s}$ and $(-\Delta)^{\frac{s}{2}}$, we have

[TABLE]

Consequently, an application of the Fefferman-Stein duality and decomposition (cf. [9, Theorem 2 & Theorem 3])

[TABLE]

produces some

[TABLE]

such that

[TABLE]

Next, we utilize (4.1) to show the equivalence in (i). Let $T\in[H^{s,1}]^{\ast}$. For any $\phi\in{\mathcal{S}}$, if we let

[TABLE]

then Lemmas 2.3 & 2.10 imply

[TABLE]

Upon applying (4.1),

[TABLE]

Proposition 4.2(i) and Definition 2.2(i), we arrive at

[TABLE]

This in turn gives

[TABLE]

and so

[TABLE]

Conversely, we assume that

[TABLE]

Then, for any $\psi\in{\mathcal{S}}_{\infty}$, we have

[TABLE]

which, combined with the facts

[TABLE]

and Definition 2.2(i), yields

[TABLE]

Due to the density of ${\mathcal{S}}_{\infty}$ in $H^{1}$ and $[H^{1}]^{\ast}={\mathop{\mathrm{missing}}{\,BMO\,}}$, the last series of identities implies

[TABLE]

Combining this and (4.1) yields

[TABLE]

(ii) Noting that ${\mathcal{S}}_{\infty}$ is dense in both $H^{s,1}$ and $W^{s,1}$ as shown in Proposition 2.12, we apply (1.3) to deduce

[TABLE]

Accordingly, the desired equivalence follows from [18, Proposition 3.2] and

[TABLE]

(iii) Let $T\in{\mathcal{S}}^{\prime}$. If

[TABLE]

then

[TABLE]

where the second equality holds thanks to $L^{\infty}\subset{\mathcal{S}}_{s}^{\prime}$ and Definition 2.2(i). Thus,

[TABLE]

which implies that $T$ induces a bounded linear functional on $\mathring{H}^{s,1}_{+}$ in terms of the density of ${\mathcal{S}}$ in $\mathring{H}^{s,1}_{+}$.

To obtain the converse part, assuming

[TABLE]

we are about to find

[TABLE]

Motivated by the argument in [6, p. 399], we consider the bounded linear operator

[TABLE]

which is actually a closed operator thanks to the definition of $\mathring{H}^{s,1}_{+}$ based on the completeness of $\mathcal{S}$ in $H^{s,1}_{+}$. If

[TABLE]

then

[TABLE]

which implies

[TABLE]

that is, $u=0$ in ${\mathcal{S}}^{\prime}/\mathcal{P}$, or equivalently, $u$ is a polynomial on ${{{\mathbb{R}}}^{n}}$. Further, any $u\in{\mathcal{S}}_{s}^{\prime}$ being a polynomial forces $u$ to be a constant function on ${{{\mathbb{R}}}^{n}}$. In other words, it holds $u=0$ in $\mathring{H}^{s,1}_{+}$. Thus, the operator $A_{+}$ is injective. In the meantime, $A_{+}$ enjoys

[TABLE]

Consequently, $A_{+}$ has a continuous inverse from $L^{1}$ to $\mathring{H}^{s,1}_{+}$. Since the definition of $\mathring{H}^{s,1}_{+}$ (determined by the closure of $\mathcal{S}$ in $H^{s,1}_{+}$) ensures that

[TABLE]

is a closed linear operator, the closed range theorem (see [40, p. 208, Corollary 1]) derives that the transpose of $A_{+}$

[TABLE]

defined by

[TABLE]

is surjective. In particular, since

[TABLE]

we can find

[TABLE]

Consequently, for any $u\in{\mathcal{S}}$, we have

[TABLE]

whence gives

[TABLE]

(iv) Let $T\in{\mathcal{S}}^{\prime}$. If

[TABLE]

then Proposition 4.2(ii) implies

[TABLE]

and hence

[TABLE]

Since ${\mathcal{S}}$ is dense in $\mathring{H}^{s,1}_{-}$, $T$ induces a bounded linear functional on $\mathring{H}^{s,1}_{-}$.

To obtain the converse part, assuming

[TABLE]

we are about to show

[TABLE]

To this end, we consider the bounded linear operator

[TABLE]

• $\rhd$

Suppose

[TABLE]

Since $u\in\mathring{H}^{s,1}_{-}$, it follows that $u\in{\mathcal{S}}_{s}^{\prime}$. For any $\psi\in{\mathcal{S}}_{\infty}$, the Fourier transform implies that

[TABLE]

thereby giving

[TABLE]

This shows that $u=0$ in ${\mathcal{S}}^{\prime}/\mathcal{P}$. In other words, $u$ is a polynomial on ${{{\mathbb{R}}}^{n}}$. However, if a polynomial $u$ is a bounded linear functional on ${\mathcal{S}}_{s}$, then $u$ must be a constant function, which implies that $u=0$ in $\mathring{H}^{s,1}_{-}$. In other words,

[TABLE]

is injective.

• $\rhd$

This last injectiveness and the next identification

[TABLE]

derive that

[TABLE]

has a continuous inverse sending $\mathcal{R}(A_{-})$ to $\mathring{H}^{s,1}_{-}$.

• $\rhd$

Clearly, the closure of $\mathcal{S}$ in $H^{s,1}_{-}$ ensures that $\mathcal{R}(A_{-})$ is closed in $(L^{1})^{n}$. So, from the closed range theorem it follows that the $A_{-}$’s transpose

[TABLE]

defined by

[TABLE]

is surjective. Consequently, for the hypothesis $T\in[\mathring{H}^{s,1}_{-}]^{\ast}$ there exists

[TABLE]

Although it is uncertain that $\vec{U}_{o}\in(L^{\infty})^{n}$, we can utilize the inclusion

[TABLE]

and the classical Hahn-Banach extension theorem to extend $\vec{U}_{o}$ to an element

[TABLE]

such that

[TABLE]

Accordingly, if $\phi\in\mathcal{S}$, then

[TABLE]

and hence

[TABLE]

∎

Fefferman-Stein decomposition & Bourgain-Brezis question for John-Nirenberg space

As a consequence of Theorem 4.3(iv), we surprisingly discover the coming-up-next assertion is indeed a resolution of the Bourgain-Brezis problem (cf. [6, p.396]) asking for any function space $X$ between $W^{1,n}$ and ${\mathop{\mathrm{missing}}{\,BMO\,}}$ such that every $F\in X$ has a representation

[TABLE]

As a subspace of the distribution space ${\mathcal{S}}^{\prime}/\mathcal{P}$, the above-searched space $X$ is nothing but $I_{s}[\mathring{H}^{s,1}_{-}]^{\ast}$ for every number $s\in(0,1)$. Moreover, the well known Fefferman-Stein decomposition (cf. [9, Theorems 2&3]) gives

[TABLE]

Theorem 4.4**.**

Let $s\in(0,1)$ and $n\in{\mathbb{N}}$. Then

[TABLE]

in the sense of

[TABLE]

if and only if

[TABLE]

Moreover, under $n\geq 2$, it holds that

[TABLE]

As a consequence, when $n\geq 2$, given any $Y_{0}\in I_{s}[\mathring{H}^{s,1}_{-}]^{\ast}$, there exists (Y_{1},...,Y_{n})\in\big{(}L^{\infty}\big{)}^{n} such that

[TABLE]

Proof.

Let us first validate

[TABLE]

in (4.3). If $f=I_{s}g$ for some $g\in[\mathring{H}^{s,1}_{-}]^{\ast}$, then

[TABLE]

and hence $I_{s}g$ is a well defined distribution in ${\mathcal{S}}^{\prime}/\mathcal{P}$. Applying Theorem 4.3(iv), we write

[TABLE]

for some

[TABLE]

Consequently, for any $\phi\in{\mathcal{S}}_{\infty}$, upon letting $\psi=I_{s}\phi$, we have

[TABLE]

which gives that

[TABLE]

Due to the density of ${\mathcal{S}}_{\infty}$ in $H^{1}$ (cf. [4]), the above identities also implies

[TABLE]

Now, we are about to show the part

[TABLE]

in (4.3). Given any

[TABLE]

we utilize Theorem 4.3(iv) to derive that

[TABLE]

which in turn gives that

[TABLE]

that is,

[TABLE]

as desired. So, we complete the proof of (4.3).

Next, we show (4.4). Due to the density of ${\mathcal{S}}_{\infty}$ in $H^{1}$, it suffices to show its validity in ${\mathcal{S}}^{\prime}/\mathcal{P}$. To do so, let us begin with verifying

[TABLE]

For any $f\in W^{1,n}$, since $W^{1,n}\subset{\mathop{\mathrm{missing}}{\,BMO\,}}$ (cf. [6]) and ${\mathop{\mathrm{missing}}{\,BMO\,}}\subset{\mathcal{S}}_{s}^{\prime}$ (cf. Lemma 4.1), we know from Definition 2.2(i) that $(-\Delta)^{\frac{s}{2}}f$ is a well-defined distribution in ${\mathcal{S}}^{\prime}$ and

[TABLE]

In particular, if $\psi\in{\mathcal{S}}_{\infty}$, then the Fourier transform gives that

[TABLE]

holds pointwisely, so that we can use the integral by parts formula and the continuity of the mapping $I_{1-s}:\,L^{\frac{n}{n-s}}\to L^{\frac{n}{n-1}}$ (cf. [2]) to derive

[TABLE]

Further, using the density of ${\mathcal{S}}_{\infty}$ in $L^{\frac{n}{n-s}}$ (cf. the proof of Lemma 2.9(iii)), we know that

[TABLE]

Thus, understood in the sense of distributions, it holds $f=I_{s}((-\Delta)^{\frac{s}{2}}f)$ in ${\mathcal{S}}^{\prime}/\mathcal{P}$ and $W^{1,n}\subset I_{s}(L^{\frac{n}{s}})$.

Also, since

[TABLE]

is obvious, we secondly validate

[TABLE]

Upon observing that $L^{\frac{n}{s},\infty}$ exists as the dual of the Lorentz space

[TABLE]

we use the Hölder inequality for weak Lebesgue space (cf. [15, Exercise 1.1.15]) and [29, (1.3)] to derive that for any $(f,g)\in L^{\frac{n}{s},\infty}\times{\mathcal{S}}$ one has

[TABLE]

Combining this and density of ${\mathcal{S}}$ in $\mathring{H}^{s,1}_{-}$ shows that any $f\in L^{\frac{n}{s},\infty}$ can induce a bounded linear functional on $H^{s,1}_{-}$. This proves

[TABLE]

Finally, due to

[TABLE]

we can find a vector-valued function

[TABLE]

such that

[TABLE]

∎

Acknowledgement: We would like to thank D. Spector for his constructive comments on the original version of this paper.