Optimal local embeddings of Besov spaces involving only slowly varying smoothness

TL;DR

This paper establishes optimal local embeddings for Besov spaces with slowly varying smoothness, extending previous results to new target spaces related to small Lebesgue spaces using advanced interpolation and inequality techniques.

Contribution

It introduces new optimal embedding results for Besov spaces with slowly varying smoothness, surpassing prior Lorentz-Karamata space targets through innovative methods.

Findings

Derived new local embedding theorems for Besov spaces

Extended the class of target spaces beyond Lorentz-Karamata spaces

Applied limiting real interpolation and Hardy inequalities effectively

Abstract

The aim of the paper is to establish (local) optimal embeddings of Besov spaces $B^{0,b}_{p,r}$ involving only a slowly varying smoothness $b$. In general, our target spaces are outside of the scale of Lorentz-Karamata spaces and are related to small Lebesgue spaces. In particular, we improve results from [CGO11b], where the targets are (local) Lorentz-Karamata spaces. To derive such results, we apply limiting real interpolation techniques and weighted Hardy-type inequalities.

Equations

$$\Vert t^{-1/r}b(t){\Vert }_{r;(0,1)}=+\infty \text{and}\Vert t^{-1/r}b(t){\Vert }_{r;(1,+\infty )}<+\infty ,$$

$$b_{r,n}(t):=\Vert s^{-1/r}b(s^{1/n}){\Vert }_{r;(t,+\infty )}\text{for all }t\in (0,+\infty ).$$

$${\mathcal{W}}_q(t):=\Vert w(s){\Vert }_{q;(0,t)},t\in (0,1].$$

$$\Vert w(t)f^{\ast }(t){\Vert }_{q;(0,1)}\precsim \Vert f{\Vert }_{B_{p,r}^{0,b}}\text{for all}f\in B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)$$

$${\mathcal{W}}_q(1)+\Vert s^{-\frac{1}{p}-\frac{1}{\rho }}{\mathcal{W}}_q(s){\Vert }_{\rho ;(t,1)}\precsim b_{r,n}(t)\text{for all}t\in (0,1).$$

$$\tilde{b}(t):=\left\{\begin{array}[]{ll}(b_{r,n}(t))^{1-r/q+r/\max\{p,q\}}(b(t^{1/n}))^{r/q-r/\max\{p,q\}}&\mbox{ if }\,r\not=+\infty\\ b_{\infty,n}(t)&\mbox{ if }\,r=+\infty\end{array}\right..$$

$$\Vert t^{1/p-1/q}\tilde{b}(t)f^{\ast }(t){\Vert }_{q;(0,1)}\precsim \Vert f{\Vert }_{B_{p,r}^{0,b}}\text{for all}f\in B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)$$

$$\Vert t^{1/p-1/q}\tilde{b}(t)\kappa (t)f^{\ast }(t){\Vert }_{q;(0,1)}\precsim \Vert f{\Vert }_{B_{p,r}^{0,b}}\text{for all}f\in B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)$$

$$B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)\hookrightarrow L_{p,q;\tilde{b}}^{loc}({\mathrm{I}\mathrm{R}}^n)$$

$$\|f\|_{Z}=\|f\|_{Z_{p,q,n,\bar{b}}}:=\big{\|}t^{-1/q}\bar{b}(t^{1/n})\,\|f^{*}\|_{p;(0,t)}\big{\|}_{q;(0,+\infty)}<+\infty,$$

$$\|f\|_{Z^{loc}}=\|f\|_{Z^{loc}_{p,q,n,\bar{b}}}:=\big{\|}t^{-1/q}\bar{b}(t^{1/n})\,\|f^{*}\|_{p;(0,t)}\big{\|}_{q;(0,1)}<+\infty,$$

$$B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)\hookrightarrow Z_{p,q,n,{\bar{b}}_{(r,q)}}({\mathrm{I}\mathrm{R}}^n)$$

$$b(t):=(1+|\ln t|{)}^{\alpha },t\in (0,+\infty ),\text{with a convenient }\alpha \in \mathrm{I}\mathrm{R}.$$

$$B_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)\ne {\mathcal{B}}_{p,r}^{0,b}({\mathrm{I}\mathrm{R}}^n)$$

$$B_q\in SV(0,+\infty ).$$

$$b_{r,n}(t):=\Vert {\tau }^{-1/r}b({\tau }^{1/n}){\Vert }_{r;(t,+\infty )}\text{for all}t\in (0,+\infty ).$$

$$b_{r,n}\in SV(0,+\infty ).$$

$$\underset{x\to 0_{+}}{\mathrm{limsup}}\frac{\Vert {\tau }^{-1/q}b(\tau ){\Vert }_{q;(x,+\infty )}}{b(x)}=+\infty \text{for any}b\in SV(0,+\infty ).$$

$$\underset{x\to +\infty }{\mathrm{limsup}}\frac{\Vert {\tau }^{-1/q}b(\tau ){\Vert }_{q;(0,x)}}{b(x)}=+\infty \text{for any}b\in SV(0,+\infty ).$$

$$\Vert f{\Vert }_{p,q;b}:=\Vert t^{1/p-1/q}b(t)f^{\ast }(t){\Vert }_{q;(0,+\infty )}<+\infty .$$

$$\Vert f{\Vert }_{p,q;b}^{loc}:=\Vert t^{1/p-1/q}b(t)f^{\ast }(t){\Vert }_{q;(0,1)}<+\infty .$$

$$b(t)=\mbox{\boldmath$\ell$\unboldmath}^{\mbox{\boldmath$\alpha$\unboldmath}}(t):=\prod_{i=1}^{m}\ell_{i}^{\alpha_{i}}(t)\text{ \ for all \ }t\in(0,+\infty)$$

$$\|f\|_{Z_{p,q,n,b}}:=\big{\|}t^{-1/q}b(t^{1/n})\,\|f^{*}\|_{p;(0,t)}\big{\|}_{q;(0,+\infty)}<+\infty.$$

$$\|f\|_{Z_{p,q,n,b}^{loc}}:=\big{\|}t^{-1/q}b(t^{1/n})\,\|f^{*}\|_{p;(0,t)}\big{\|}_{q;(0,1)}<+\infty.$$

$$Z_{p,q,n,b}({\mathrm{I}\mathrm{R}}^n)\hookrightarrow Z_{p,q,n,b}^{loc}({\mathrm{I}\mathrm{R}}^n).$$

$${\omega }_1(f,t{)}_p:=\underset{|h|\le t}{\sup }\Vert {\Delta }_{h}f{\Vert }_p\text{mbox}forallt\ge 0.$$

$$\Vert t^{-1/r}b(t){\Vert }_{r;(0,1)}=+\infty \text{and}\Vert t^{-1/r}b(t){\Vert }_{r;(1,+\infty )}<+\infty .$$

$$\Vert f{\Vert }_{B_{p,r}^{0,b}}:=\Vert f{\Vert }_p+\Vert t^{-1/r}b(t){\omega }_1(f,t{)}_p{\Vert }_{r;(0,+\infty )}$$

$$\Vert f{\Vert }_{B_{p,r}^{0,b}}:=\Vert f{\Vert }_p+\Vert t^{-1/r}b(t){\omega }_1(f,t{)}_p{\Vert }_{r;(0,1)}.$$

$${\int }_t^{+\infty }{\tau }^{-1}{b}^r({\tau }^{1/n})d\tau \approx {\int }_t^2{\tau }^{-1}{b}^r({\tau }^{1/n})d\tau \text{for all}t\in (0,1).$$

Full text

The research has been supported by the grant P201-18-00580S of the Grant Agency of the Czech Republic and by Centre of Mathematics of the University of Coimbra – UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

1. Introduction

Classical Besov spaces $B^{s,b}_{p,r}$ play an important role in numerous parts of mathematics. However, it has gradually become clear that, to solve some problems, Besov spaces with smoothness which can be more finely turned, i.e., Besov spaces with generalized smoothness, are essential. These spaces have been studied especially by the Soviet mathematical school (cf. [KL87, Section 8]) and a lot of attention has been paid to optimal embeddings and to growth envelopes of such spaces (see, e.g., [Gol87], [GK03], [Net87], [FL06], [CF06], [CM04b],[CM04a], [CH05], [GO05], [Mou01], [GO07], [Tri06, Chapter 1], [Gol07], [CGO08], [CGO11b], etc.)

Besov spaces $B^{0,b}_{p,r}$ involving the zero classical smoothness and logarithmic smoothness $b$ are particular cases of Besov spaces with generalized smoothness. They appear (probably for the first time) already in [DRS79] in a connection with the weak-type interpolation. During the recent years these spaces have attracted an increasing attention (see [CGO08], [CD14], [GOTT14], [CDSFM15], [CD15b], [CD16], [CDT16], [Dom16], [Dom17b], [BC18], [CDK18], etc.). The problem of optimal embeddings in this setting is much harder to study.

In [CGO11b] the authors have characterized (with easily verifiable conditions) local embeddings of Besov spaces $B^{0,b}_{p,r}$ involving the zero classical smoothness and involving only a slowly varying smoothness $b$ into classical Lorentz spaces. These results have been then applied to establish sharp local embeddings of Besov spaces in question into Lorentz-Karamata spaces and to determinate the growth envepoles of spaces $B^{0,b}_{p,r}$. In particular, the following three theorems (adapted to our notation, cf. Remarks 2.2 and 2.3 below) have been proved there.111*)* Besov spaces are defined by means of the modulus of continuity, we refer to Section 2 for precise definition.)

Theorem 1.1** ([CGO11b, Theorem 3.2 (i)]).**

*Let $n\in{\rm I\kern-1.69998ptN}$, $1\leq p<+\infty$, $1\leq r\leq q\leq+\infty$. Let $b$ be a slowly varying function on the interval $(0,+\infty)$ *(notation $b\in SV(0,+\infty)$) satisfying

[TABLE]

and let $b_{r,n}$ be defined by

[TABLE]

Put $\rho=+\infty$ if $p\leq q$ and define $\rho$ by $\frac{1}{\rho}=\frac{1}{q}-\frac{1}{p}$ if $q<p$. Assume that $w$ is a non-negative measurable function on $(0,1)$ and

[TABLE]

Then

[TABLE]

if and only if

[TABLE]

Theorem 1.2** ([CGO11b, Theorem 3.3]).**

Let $n\in{\rm I\kern-1.69998ptN}$, $1\leq p<+\infty$, $\,1\leq r\leq+\infty$, $0<q\leq+\infty$ and let $b\in SV(0,+\infty)$ satisfy (1.1). Let $b_{r,n}$ be giving by (1.2) and define, for all $t\in(0,+\infty)$,

[TABLE]

Then

[TABLE]

if and only if $q\geq r$.

Theorem 1.3** ([CGO11b, Theorem 3.4]).**

Let $n\in{\rm I\kern-1.69998ptN}$, $1\leq p<+\infty$, $1\leq r\leq q\leq+\infty$ and let $\,b\in SV(0,+\infty)\,$ satisfy (1.1). Define $b_{r,n}$ and $\tilde{b}$ by (1.2) and (1.5), respectively. Let $\kappa$ be a non-negative and non-increasing function on $(0,1)$. Then

[TABLE]

if and only if $\kappa$ is bounded.

Theorems 1.2 and 1.3 describe the sharp continuous embeddings of the Besov space $B^{0,b}_{p,r}({\rm I\kern-1.69998ptR}^{n})$ into the local Lorentz-Karamata space $L_{p,q;\tilde{b}}^{loc}({\rm I\kern-1.69998ptR}^{n})$ (we refer to Section 2 for the precise definition of spaces in question). Namely, these theorems imply that

[TABLE]

and that this embedding is sharp within the scale of local Lorentz-Karamata spaces.

For characterization of compact embeddings of spaces $B^{0,b}_{p,r}({\rm I\kern-1.69998ptR}^{n})$ into Lorentz-Karamata spaces we refer to [CGO11a].

The aim of this paper is to improve embedding (1.8). Namely, we are going to prove (cf. Theorems 3.1 and 3.7 of Section 3) that the target space in (1.8) can be replaced by the better space $Z({\rm I\kern-1.69998ptR}^{n})=Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ if $p\neq q$, where $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ is the set of all measurable $f$ functions on ${\rm I\kern-1.69998ptR}^{n}$ satisfying

[TABLE]

with $\bar{b}=\bar{b}_{(r,q)}$ given by (3.1) below .

Moreover, the target space in (1.8) can still be replaced by $Z^{loc}({\rm I\kern-1.69998ptR}^{n})=Z^{loc}_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$, if $p\neq q$, which is a local version of $Z({\rm I\kern-1.69998ptR}^{n})$, defined by replacing the interval $(0,+\infty)$ by $(0,1)$ in the (quasi-)norm (1.9), that is, $Z^{loc}_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ is the set of all measurable $f$ functions on ${\rm I\kern-1.69998ptR}^{n}$ satisfying

[TABLE]

with $\bar{b}=\bar{b}_{(r,q)}$ given by (3.1) below .

Furthermore, we show (cf. Theorems 3.4 and 3.5 below) that the embedding

[TABLE]

is locally sharp with respect to the slowly varying function $\bar{b}_{(r,q)}$ and that embedding (1.10) with $q=r$ is optimal among all embeddings (1.10) with $1\leq r\leq q<+\infty$.

To prove our results, we make use of limiting interpolation, embedding theorems for spaces from two different scales of interpolation spaces (cf. Theorems 4.8, 4.10 below, which are of independent interest), weighted inequalities, and some results from [CGO11b].

In the particular case when $n=1$ and $b$ is of logarithmic type, the embedding of the subspace ${\mathcal{P}}_{2\pi}({\rm I\kern-1.69998ptR})\subset B_{p,r}^{0,b}({\rm I\kern-1.69998ptR})$ of $2\pi$-periodical functions into the space $Z_{p,q,1,\bar{b}}({\rm I\kern-1.69998ptR})$ follows from [CD15a, Theorem 4.4], where the authors used approximation spaces (including their limiting forms), reiteration of approximation constructions, the Nikolskiĭ inequality for trigonometric polynomials and limiting interpolation.

As in [CGO11b], also in our paper Besov spaces are defined by means of the modulus of continuity. Note that some authors use the Fourier-analytical approach to define Besov spaces with the zero classical smoothness and involving the logarithmic smoothnees $b$, where

[TABLE]

However, if we denote the resulting space by ${\mathcal{B}}^{0,b}_{p,r}({\rm I\kern-1.69998ptR}^{n})$, with $b$ given by (1.11), then, in general,

[TABLE]

(see [ST95], [CL13], [CD15b]).

When $b$ is given by (1.11), and $n\in{\rm I\kern-1.69998ptN}$, then an analogue of embedding (1.10) with $B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ replaced by the subspace ${\mathcal{P}}({\rm I\kern-1.69998ptR}^{n})\subset B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ of periodical functions has been proved in [Dom16] (where also the corresponding result for the the subspace ${\mathcal{P}}({\rm I\kern-1.69998ptR}^{n})\subset{\mathcal{B}}_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ of periodical functions can be found, see also [Dom17a]).

The paper is organized as follows: Section 2 contains notation and preliminaries. In Section 3 we present our main results on embeddings of Besov spaces involving only slowly varying smoothness, their sharpness and local optimality. In Section 4 we collect and prove some auxiliary results. The proofs of our main results from Section 3 are given in Sections 5, 6 and 7.

2. Notation, definitions and basic properties

As usual, ${\rm I\kern-1.69998ptR}^{n}$ denotes the Euclidean $n$-dimensional space. Throughout the paper $\mu_{n}$ is the $n$-dimensional Lebesgue measure in ${\rm I\kern-1.69998ptR}^{n}$ and $\Omega$ is a domain in ${\rm I\kern-1.69998ptR}^{n}$. We denote by $\chi_{\Omega}$ the characteristic function of $\Omega$ and put $|\Omega|_{n}=\mu_{n}(\Omega)$. The family of all extended scalar-valued (real or complex) $\mu_{n}$-measurable functions on $\Omega$ is denoted by ${\mathcal{M}}(\Omega)$ while ${\mathcal{M}}_{0}(\Omega)$ stands for the class of functions in ${\mathcal{M}}(\Omega)$ that are finite $\mu_{n}$-a.e. on $\Omega$ and ${\mathcal{M}}^{+}(\Omega)$ denotes the subset of ${\mathcal{M}}(\Omega)$ consisting of all functions which are non-negative $\mu_{n}$-a.e. on $\Omega$. When $\Omega$ is an interval $(a,b)\subseteq\mathbb{R}$, we denote these sets by $\mathcal{M}(a,b)$, $\mathcal{M}_{0}(a,b)$ and $\mathcal{M}^{+}(a,b),$ respectively. By $\mathcal{M}_{0}^{+}(a,b;\downarrow)$ we mean the subset of $\mathcal{M}_{0}^{+}(a,b)$ containing all non-increasing functions on the interval $(a,b)$ and by $\mathcal{M}_{0}^{+}(a,b;\uparrow)$ we mean the subset of $\mathcal{M}_{0}^{+}(a,b)$ containing all non-decreasing functions on the interval $(a,b)$. The symbol ${\mathcal{W}}(a,b)$ stands for the class of weight functions on $(a,b)\subseteq{\rm I\kern-1.69998ptR}$ consisting of all $\mu_{1}$-measurable functions which are positive and finite $\mu_{1}$-a.e. on $(a,b)$. The non-increasing rearrangement of $f\in{\mathcal{M}}({\rm I\kern-1.69998ptR}^{n})$ is the function $f^{*}$ defined by $f^{*}(t):=\inf\left\{\lambda\geq 0:|\{x\in{\rm I\kern-1.69998ptR}^{n}:|f(x)|\!>\!\lambda\}|_{n}\leq t\right\}$ for all $t\geq 0$. By $f^{**}$ we denote the maximal function of $f^{*}$ given by $f^{**}(t):=t^{-1}\int_{0}^{t}f^{*}(\tau)\,d\tau$, $t\in(0,+\infty)$. The maximal operator $f\mapsto f^{**}$ is subadditive (cf. [BS88, p. 54]).

By $c$, $C$, $c_{1}$, $C_{1}$, $c_{2}$, $C_{2}$, etc. we denote positive constants independent of appropriate quantities. For two non-negative expressions (i.e. functions or functionals) ${\mathcal{A}}$, ${\mathcal{B}}$, the symbol ${\mathcal{A}}\precsim{\mathcal{B}}$ (or ${\mathcal{A}}\succsim{\mathcal{B}}$) means that ${\mathcal{A}}\leq c\,{\mathcal{B}}$ (or $c\,{\mathcal{A}}\geq{\mathcal{B}}$). If ${\mathcal{A}}\precsim{\mathcal{B}}$ and ${\mathcal{A}}\succsim{\mathcal{B}}$, we write ${\mathcal{A}}\approx{\mathcal{B}}$ and say that ${\mathcal{A}}$ and ${\mathcal{B}}$ are equivalent. Throughout the paper we use the abbreviation $\text{LHS}(*)$ ($\text{RHS}(*)$) for the left- (right-) hand side of relation $(*)$. We adopt the convention that $a/(+\infty)=0$, $a/0=+\infty$ and $(+\infty)^{a}=+\infty$ for all $a\in(0,+\infty)$. If $p\in[1,+\infty]$, the conjugate number $p^{\prime}$ is given by $1/p+1/p^{\prime}=1$. In the whole paper $\|.\|_{p;(c,d)},\,p\in(0,+\infty]$, denotes the usual $L_{p}$-(quasi-)norm on the interval $(c,d)\subseteq{\rm I\kern-1.69998ptR}$.

We say that a positive, finite and Lebesgue-measurable function $b$ is slowly varying on $(0,+\infty)$, and write $b\in SV(0,+\infty)$, if, for each $\varepsilon>0$, $t^{\varepsilon}b(t)$ is equivalent to a non-decreasing function on $(0,+\infty)$ and $t^{-\varepsilon}b(t)$ is equivalent to a non-increasing function on $(0,+\infty)$. Here we follow the definition of $SV(0,+\infty)$ given in [GOT05]; for other definitions see, for example, [BGT87], [EE04], [EKP00], and [Nev02]. The family $SV(0,+\infty)$ includes not only powers of iterated logarithms and the broken logarithmic functions of [EO00] but also such functions as $t\mapsto\exp\left(\left|\log t\right|^{a}\right),$ $a\in(0,1).$ (The last mentioned function has the interesting property that it tends to infinity more quickly than any positive power of the logarithmic function).

The replacement of the interval $(0,+\infty)$ in the definition of the class $SV(0,+\infty)$ by the interval $(0,1)$ yields the definition of the class $SV(0,1)$.

Let $q\in(0,+\infty]$, $b\in SV(0,+\infty)$, $B_{q}(t):=\|\tau^{-1/q}b(\tau)\|_{q;(0,t)}$, $t\in(0,+\infty)$, and let $B_{q}(1)<+\infty$. Then, by [GO07, Lemma 2.1 (v)],

[TABLE]

If $n\in{\rm I\kern-1.69998ptN}$, $r\in(0,+\infty]$ and $b\in SV(0,+\infty)$, then we put

[TABLE]

Suppose $b_{r,n}(1)<+\infty$. By [GO07, Lemma 2.1 (ii), (v)],

[TABLE]

By [CGO11b, Lemma 2.2 (8)],

[TABLE]

By [GO07, Lemma 2.1 (i))] and the previous result, it also follows that,

[TABLE]

More properties and examples of slowly varying functions can be found in [Zyg57, Chapter V, p. 186], [BGT87], [EKP00], [Mar00], [Nev02], [GOT05] and [GNO10].

Let $n\in{\rm I\kern-1.69998ptN}$, $p,q\in(0,+\infty]$, $b\in SV(0,+\infty)$.The Lorentz-Karamata (LK) space $L_{p,q;b}({\rm I\kern-1.69998ptR}^{n})$ is defined to be the set of all functions $f\in{\mathcal{M}}({\rm I\kern-1.69998ptR}^{n})$ such that

[TABLE]

The local Lorentz-Karamata space $L_{p,q;b}^{loc}({\rm I\kern-1.69998ptR}^{n})$ consists of all functions $f\in{\mathcal{M}}({\rm I\kern-1.69998ptR}^{n})$ such that

[TABLE]

Particular choices of $b$ give well-known spaces. If $m\in{\rm I\kern-1.69998ptN}$, \mbox{\boldmath\alpha\unboldmath}=(\alpha_{1},\dots,\alpha_{m})\in{\rm I\kern-1.69998ptR}^{m} and

[TABLE]

(where $\ell(t)=\ell_{1}(t):=1+\left|\log t\right|,\text{ }\ell_{i}(t):=\ell_{1}(\ell_{i-1}(t))\text{ if }i>1$), then the LK-space $L_{p,q;b}({\rm I\kern-1.69998ptR}^{n})$ is the generalized Lorentz-Zygmund space L_{p,q,\mbox{\boldmath\alpha\unboldmath}} introduced in [EGO97] and endowed with the (quasi-)norm \|f\|_{p,q;\mbox{\boldmath\alpha\unboldmath};{\rm I\kern-1.18999ptR}^{n}}, which in turn becomes the Lorentz-Zygmund space $L_{p,q}(\log L)^{\alpha_{1}}({\rm I\kern-1.69998ptR}^{n})$ of Bennett and Rudnick [BR80] when $m=1$. If \mbox{\boldmath\alpha\unboldmath}=(0,\dots,0), we obtain the Lorentz space $L_{p,q}({\rm I\kern-1.69998ptR}^{n})$ endowed with the (quasi-)norm $\|.\|_{p,q;{\rm I\kern-1.18999ptR}^{n}}$, which is just the Lebesgue space $L_{p}({\rm I\kern-1.69998ptR}^{n})$ equipped with the (quasi-)norm $\|.\|_{p}$ when $p=q$; if $p=q$ and $m=1$, we obtain the Zygmund space $L_{p}(\log L)^{\alpha_{1}}({\rm I\kern-1.69998ptR}^{n})$ endowed with the (quasi-)norm $\|.\|_{p;\alpha_{1}}$.

Let $n\in{\rm I\kern-1.69998ptN}$, $1<p<+\infty$, $1\leq q\leq+\infty$, $b\in SV(0,+\infty)$. The space $Z_{p,q,n,b}({\rm I\kern-1.69998ptR}^{n})$ is defined to be the set of all functions $f\in{\mathcal{M}}({\rm I\kern-1.69998ptR}^{n})$ such that

[TABLE]

We can define as well the local space $Z_{p,q,n,b}^{loc}({\rm I\kern-1.69998ptR}^{n})$ as the set of all functions $f\in{\mathcal{M}}({\rm I\kern-1.69998ptR}^{n})$ for which

[TABLE]

From (2.5) and (2.6) it is clear that

[TABLE]

We refer to [FK04], [CF05] and [FFG18] for the connection of these spaces with small Lebesgue spaces.

Let $h\in{\rm I\kern-1.69998ptR}^{n}$. The first difference operator $\Delta_{h}$ is defined on scalar functions $f$ on ${\rm I\kern-1.69998ptR}^{n}$ by $\Delta_{h}f(x)=f(x+h)-f(x)$ for all $x\in{\rm I\kern-1.69998ptR}^{n}$.

Let $p\in[1,+\infty)$. The modulus of smoothness of a function $f$ in $L_{p}({\rm I\kern-1.69998ptR}^{n})$ is given by

[TABLE]

Definition 2.1**.**

Let $n\in{\rm I\kern-1.69998ptN}$, $1\leq p<+\infty$, $1\leq r\leq+\infty$ and let $b\in SV(0,+\infty)$ be such that

[TABLE]

The Besov space $B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ consists of those $f\in L_{p}({\rm I\kern-1.69998ptR}^{n})$ for which the norm

[TABLE]

is finite.

Remark 2.2**.**

• (i)

An equivalent norm on $B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ is given by the functional

[TABLE]

We refer to [CGO11b, Remark 2.5 (iii)] for more details.

• (ii)

The assumption $\|t^{-1/r}b(t)\|_{r;(1,+\infty)}<+\infty$ is natural. Otherwise the space $B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$ is trivial (that is, it consists only of the zero element). We again refer to [CGO11b, Remark 2.5 (iii)] for more details.

• (iii)

Note also that only the case when $\|t^{-1/r}b(t)\|_{r;(0,1)}=+\infty$ is of interest. Otherwise $B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})\equiv L_{p}({\rm I\kern-1.69998ptR}^{n})$, cf. [CGO11b, Remark 2.5 (i)].

• (iv)

An equivalent norm results on $B^{0,b}_{p,r}({\rm I\kern-1.69998ptR}^{n})$ if the modulus of smoothness $\omega_{1}(f,\cdot)_{p}\,$ in (2.8) is replaced by the $k$-th order modulus of smoothness $\,\omega_{k}(f,\cdot)_{p}\,$, where $\,k\in\{2,3,4,\ldots\}$, cf. [CGO11b, Remark 2.5 (ii)].

Remark 2.3**.**

Assumption (2.7) implies that

[TABLE]

Indeed, since

[TABLE]

and since the reverse estimate is trivial, we see that (2.9) holds. **

3. Main Results

Theorem 3.1**.**

Let $n\in{\rm I\kern-1.69998ptN}$, $1<p<+\infty$, $1\leq r,q<+\infty$, and let $b\in SV(0,+\infty)$ be such that (2.7) holds. If  $\bar{b}=\bar{b}_{(r,q)}\in SV(0,+\infty)$ is given by

[TABLE]

where

[TABLE]

then

[TABLE]

if and only if $q\geq r$.

Corollary 3.2**.**

Let all the assumptions of  Theorem 3.1 be satisfied and $r\leq q$. Let $\widetilde{b}=\widetilde{b}_{(r,q,n,p)}\in SV(0,+\infty)$ be given by

[TABLE]

with $b_{r}$ from (3.2). Then

[TABLE]

Remark 3.3**.**

• (i)

Theorem 3.1 gives in general a better result than Corollary 3.2 since, if $1<p<+\infty$, $r\leq q$ and $q\neq p$, then the space $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ is strictly smaller than the Lorentz-Karamata space $L_{p,q;\widetilde{b}}({\rm I\kern-1.69998ptR}^{n})$ (cf. Remark 3.8 below). By Theorem 3.7 below, the target spaces in (3.3) and in (3.5) coincide if $1<p<+\infty$, $r\leq q$ and $q=p$.

• (ii)

Embedding (3.5) implies that

[TABLE]

The embedding (3.6) was proved in [CGO11b, Theorem 3.3] by a completely different method. In [CGO11b, Theorem 3.4] it was also shown that the target space $L_{p,r;\widetilde{b}}^{loc}({\rm I\kern-1.69998ptR}^{n})$, with $\widetilde{b}=\widetilde{b}_{(r,r,n,p)}$, given by (3.4) with $q$ replaced by $r$, is optimal among the Lorentz-Karamata spaces $L_{P,Q;B}^{loc}({\rm I\kern-1.69998ptR}^{n})$, with $1<P<+\infty$, $1\leq r\leq Q<+\infty$, $B\in SV(0,+\infty)$, for which the embedding (3.6) holds.

Theorem 3.4**.**

Let all the assumptions of  Theorem 3.1 be satisfied and $r\leq q$. Then the embedding

[TABLE]

is optimal among all the embeddings

[TABLE]

Concerning the sharpness of embedding (3.3) with respect to the function $\bar{b}$, we have the following result.

Theorem 3.5**.**

Let all the assumptions of  Theorem 3.1 be satisfied and $r\leq q$. Let $\kappa\in{{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)}\cap SV(0,+\infty)$ be such that

[TABLE]

with $\bar{b}$ given by (3.1). Then the function $\kappa$ is bounded on the interval $(0,+\infty)$.

Remark 3.6**.**

If all the assumptions of  Theorem 3.1 are satisfied and $r\leq q$, then embedding (3.8) can be replaced by

[TABLE]

Indeed, (3.8) implies (3.9). On the other hand, (3.9) means that

[TABLE]

Moreover, since $\kappa$ is non-increasing, (3.3) also implies that

[TABLE]

Consequently, (3.10) and (3.11) imply (3.8). **

The next result show that if $q\neq p$, then the target space $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ in (3.3) is strictly smaller than the target space $L_{p,q;\widetilde{b}}({\rm I\kern-1.69998ptR}^{n})$ in (3.5).

Theorem 3.7**.**

Let all the assumptions of  of Corollary 3.2 be satisfied and $r\leq q$. Assume that $\bar{b},\widetilde{b}\in SV(0,+\infty)$ are given by (3.1) and (3.4), respectively. If the space $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ is defined as in Theorem 3.1, then

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

Remark 3.8**.**

Let all the assumptions of Corollary 3.2 be satisfied. Since then $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$ and $L_{p,q;\widetilde{b}}({\rm I\kern-1.69998ptR}^{n})$ are r.i. Banach function spaces, [BS88, Chap. I, p.7, Theorem 1.8] and (3.14) imply that

[TABLE]

This, together with (3.12), gives

[TABLE]

4. Auxiliary assertions

Lemma 4.1**.**

If  $1\leq p<+\infty$, $1\leq r\leq q<+\infty$, $b\in SV(0,+\infty)$ and $\bar{b}\in SV(0,+\infty)$ is given by

[TABLE]

then

[TABLE]

• Proof.

Emebdding (4.1) means that, for all $f\in B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$,

[TABLE]

Thus, (4.1) holds if, for all $f\in B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$,

[TABLE]

Since $\omega_{1}(f,\cdot)_{p}\in\mathcal{M}_{0}^{+}(0,+\infty;\uparrow)$ for any $f\in B_{p,r}^{0,b}({\rm I\kern-1.69998ptR}^{n})$, estimate (4.2) will be satisfied provided that

[TABLE]

where

[TABLE]

Putting

[TABLE]

we obtain by [HS93, Proposition 2.1] that inequality (4.3) holds if

[TABLE]

Since condition (4.5) is satisfied when $w$ and $v$ are given by (4.4), the result follows. $\square$

We shall also need the result mentioned in [BS88, Chap. V, Corollary 4.20, p. 346], which states that

[TABLE]

Definition 4.2**.**

Let $(X_{0},X_{1})$ be a compatible couple of quasi-Banach spaces.

• *     *(i)

For each $t\in(0,+\infty)$, $K(\cdot,t;X_{0},X_{1})$ is the Peetre’s $K$-functional defined by

[TABLE]

Sometimes, we denote $K(f,t;X_{0},X_{1})$ simply by $K(f,t)$.

• *     *(ii)

For $0\leq\theta\leq 1$, $0<q\leq+\infty$, and $b\in SV(0,+\infty)$, we put

[TABLE]

where

[TABLE]

If $b\equiv 1$ on $(0,+\infty)$, we write $(X_{0},X_{1})_{\theta,q}$ instead of $(X_{0},X_{1})_{\theta,q,b}$.

• *     *(iii)

For $0<\sigma<1$, $0<r,q\leq+\infty$, and $a,b\in SV(0,+\infty)$, we define the space

[TABLE]

where

[TABLE]

(The superscript and the subscript ${\mathcal{L}}$ in (4.9) and (4.10), respectively, is an indication of the fact that the local (quasi-)norm in (4.10) is taken from the left end of the interval $(0,+\infty)$.)

Next we collect some weighted inequalities, which are needed in the rest of the paper. To prove Theorem 4.8 below (involving the embedding $\overline{X}_{\theta,r,b,q,a}^{\mathcal{L}}\hookrightarrow\overline{X}_{\theta,r,d}$), we shall make use of the following assertions.

Theorem 4.3** ([Lai93, Theorem 2.1]).**

Let $1\leq P\leq Q<+\infty$, $\varphi:{\rm I\kern-1.69998ptR}_{+}\times{\rm I\kern-1.69998ptR}_{+}\longrightarrow{\rm I\kern-1.69998ptR}_{+}$, and $v,w\in{\mathcal{W}}(0,+\infty)$. Then the inequality

[TABLE]

holds for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$ if and only if, for all $z\in(0,+\infty)$,

[TABLE]

(The constant C in both inequalities is the same.)**

Lemma 4.4** ([EO00, Lemma 3.4]).**

Let $s\in(0,1)$, $\varphi,w\in{\mathcal{W}}(0,+\infty)$ and define $v\in{\mathcal{W}}(0,+\infty)$ by

[TABLE]

Then, for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty)$,

[TABLE]

To prove Theorem 4.10 mentioned below (which is a counterpart of Theorem 4.8), we shall make use of the following assertions.

Theorem 4.5** ([Lai93, Theorem 2.2]).**

Let $0<Q\leq P\leq 1$, $\varphi:{\rm I\kern-1.69998ptR}_{+}\times{\rm I\kern-1.69998ptR}_{+}\longrightarrow{\rm I\kern-1.69998ptR}_{+}$, and $v,w\in{\mathcal{W}}(0,+\infty)$. Then the inequality

[TABLE]

holds for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$ if and only if, for all $z\in(0,+\infty)$,

[TABLE]

(The constant C in both inequalities is the same.)**

Theorem 4.6** ([OK90, Theorem 5.9, p. 63]).**

Let $1\leq P\leq Q\leq+\infty$ and $v,w\in{\mathcal{W}}(0,+\infty)$. Then, the inequality

[TABLE]

holds for all $g\in{\mathcal{M}}_{0}^{+}(0,+\infty)$ if and only if

[TABLE]

To study the relations among the target spaces in Theorem 3.1 and in Corollary 3.2, we shall need the following result.

Theorem 4.7** ([Saw90, Theorem 2]).**

Suppose $1<P\leq Q<+\infty$ and $v,w\in{\mathcal{W}}(0,+\infty)$. Then the inequality

[TABLE]

holds for all $g\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$ if and only if there are constants $A$ and $B$ such that

[TABLE]

and

[TABLE]

for all $z\in(0,+\infty)$.

Moreover, if $C$ is the best constant in (4.19), then $C\approx A+B$.

The next result extends [EO00, Theorem 4.7].

Theorem 4.8**.**

Let $(X_{0},X_{1})$ be a compatible couple of quasi-Banach spaces. If  $\theta\in[0,1)$, $0<r,q<+\infty$ and $a,b\in SV(0,+\infty)$, then

[TABLE]

where

[TABLE]

• Proof.

Embedding (4.22) means that, for all $f\in\overline{X}_{\theta,r,b,q,a}^{\mathcal{L}}$,

[TABLE]

Put $Q:=\frac{r}{q}$ and $h(t):=[K(f,t)/t]^{q}$, $t\in(0,+\infty)$. Then $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$ and inequality (4.24) can be rewritten as

[TABLE]

that is

[TABLE]

where $v(x):=x^{(1-\theta)r-1}\,d^{r}(x)$, $w(x):=x^{-1}\,\left(\frac{b(x)}{a(x)}\right)^{r}$, $x\in(0,+\infty)$, and $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$.

• I.

Assume first that $Q=\frac{r}{q}\in[1,+\infty)$. Put $P:=Q$ and

[TABLE]

Then inequality (4.25) coincides with inequality (4.11). Since one can verify that (4.12) is satisfied in our case, Theorem 4.3 implies that inequality (4.25) holds for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$. Thus, embedding (4.22) is satisfied if $0<q\leq r<+\infty$.

• II.

Assume now that $Q=\frac{r}{q}\in(0,1)$. Put $s:=Q$ and $\varphi(x):=x^{(1-\theta)q-1}\,a^{q}(x)$, $x\in(0,+\infty)$. Then inequality (4.25) is of the same form as inequality (4.14). Since in our case one can verify that

[TABLE]

Lemma 4.4 implies that inequality (4.25) is satisfied for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$. Consequently, embedding (4.22) holds if $0<r<q<+\infty$.

$\square$

Remark 4.9**.**

If we assume that the compatible couple $(X_{0},X_{1})$ of (quasi-) Banach spaces is ordered in the sense that $X_{1}\hookrightarrow X_{0}$, then one can define the spaces $(X_{0},X_{1})_{\theta,q,b}$ and $(X_{0},X_{1})_{\theta,r,b,q,a}^{\mathcal{L}}$ just as in Definition 4.2, except that the role of the interval $(0,+\infty)$ is played by the interval $(0,1)$.

Moreover, one can show that when an ordered couple $(X_{0},X_{1})$ is considered, Theorem 4.8 remain true provided that the functions $a,b\in SV(0,1)$ are extended to the interval $(0,2)$ by setting $a(x){\approx}b(x){\approx}1$, $x\in[1,2)$, and the function $d$ is given by

[TABLE]

Now we are going to prove a counterpart of Theorem 4.8, which extends [EO00, Lemma 4.3].

Theorem 4.10**.**

Let $(X_{0},X_{1})$ be a compatible couple of quasi-Banach spaces. If  $\theta\in[0,1)$, $0<r,q<+\infty$ and $a,b\in SV(0,+\infty)$, then

[TABLE]

where

[TABLE]

• Proof.

Embedding (4.27) means that, for all $f\in\overline{X}_{\theta,r,d}$,

[TABLE]

Put $Q:=\frac{r}{q}$ and $h(t):=[K(f,t)/t]^{q}$, $t\in(0,+\infty)$. Then $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$ and inequality (4.29) can be rewritten as

[TABLE]

• I.

Assume first that $Q:=\frac{r}{q}\in(0,1]$. Remark that (4.30) can be formulated as

[TABLE]

where $v(x):=x^{(1-\theta)r-1}\,d^{r}(x)$, $w(x):=x^{-1}\,\left(\frac{b(x)}{a(x)}\right)^{r}$, $x\in(0,+\infty)$, and $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$. Put $P:=Q$ and

[TABLE]

Then inequality (4.31) coincides with inequality (4.15). Using properties of slowly varying functions, one can verify that (4.16) is satisfied in our case. Thus, Theorem 4.5 implies that inequality (4.31) holds for all $h\in{\mathcal{M}}_{0}^{+}(0,+\infty;\downarrow)$. Consequently, embedding (4.27) is satisfied if $0<r\leq q<+\infty$.

• II.

Suppose now that $Q=\frac{r}{q}\in(1,+\infty)$. Putting $g(\tau):=\tau^{(1-\theta)q-1}\,a^{q}(\tau)h(\tau)$, $\tau\in(0,+\infty)$, we have $g\in{\mathcal{M}}_{0}^{+}(0,+\infty)$ and inequality (4.30) can be rewritten as

[TABLE]

i.e., as (4.17), with $P:=Q$, $v(t):=t^{\frac{r}{q}-1}\left(\frac{d(t)}{a(t)}\right)^{r}$, $w(t):=t^{-1}\left(\frac{b(t)}{a(t)}\right)^{r}$, $t\in(0,+\infty)$. One can verify that, for all $x\in(0,+\infty)$,

[TABLE]

and

[TABLE]

Consequently, condition (4.18) is satisfied, which means that inequality (4.30) holds for all $g\in{\mathcal{M}}_{0}^{+}(0,+\infty)$. Hence, embedding (4.27) holds as well if $0<q<r<+\infty$.

$\square$

Remark 4.11**.**

If  $r=q$, then Theorems 4.8 and 4.10 imply that

[TABLE]

with $d$ given by (4.23) (or by (4.28)), for all $\theta\in[0,1)$, $0<r<+\infty$ and $a,b\in SV(0,+\infty)$.

Theorem 4.12** ([BS88, Chap. V, Corollary 4.13, p. 341]).**

Suppose $1\leq p\leq+\infty$ and $r\in{\rm I\kern-1.69998ptN}$. If  $0<\theta<1$ and $1\leq q\leq+\infty$, then

[TABLE]

with equivalent norms.

We refer to [BS88, p. 310], in the Banach setting, and to [EOP02, Definition 3.12], in the general setting, for the definition of intermediate spaces of $(X_{0},X_{1})$ of class 0 or class 1.

Theorem 4.13** ([GOT05, Theorem 3.5]).**

Let $0<\theta_{0}<\theta_{1}<1$, $0<q,q_{0},q_{1}\leq+\infty$ and let $b,b_{0},b_{1}\in SV(0,+\infty)$. Put $\overline{X}_{\theta_{i}}:=(X_{0},X_{1})_{\theta_{i},q_{i},b_{i}}$ for $i\in\{0,1\}$. Suppose that $\overline{X}_{i}$, $i\in\{0,1\}$, are intermediate spaces between $X_{0}$ and $X_{1}$ of class $i$. Then

[TABLE]

Lemma 4.14** ([GOT05, Lemma 5.2]).**

Let $0<\theta<1$, $0<q\leq+\infty$, $0<s<+\infty$, and let $b\in SV(0,+\infty)$. Then, for all $f\in L_{s}+L_{\infty}$ and all $t\in(0,+\infty]$,

[TABLE]

and, for all $f\in L_{s,\infty}+L_{\infty}$ and all $t\in(0,+\infty]$,

[TABLE]

5. Proofs of Theorem 3.1 and Corollary 3.2

To prove the sufficient part of our first main result (i.e. Theorem 3.1), we shall use the following assertion.

Theorem 5.1**.**

Let $n\in{\rm I\kern-1.69998ptN}$, $1<p<+\infty$, $1\leq q<+\infty$, and let $\bar{b}\in SV(0,+\infty)$ be such that

[TABLE]

Then

[TABLE]

where

[TABLE]

and

[TABLE]

• Proof.

Putting

[TABLE]

we see that $d\in SV(0,+\infty)$ and

[TABLE]

Using the trivial embedding $L_{p}({\rm I\kern-1.69998ptR}^{n})\hookrightarrow L_{p}({\rm I\kern-1.69998ptR}^{n})$ and (4.6), we arrive at

[TABLE]

Choose $r\in{\rm I\kern-1.69998ptN}$ such that $\theta:=\frac{n}{p}\frac{1}{r}<1$. Then $B_{p,1}^{n/p}({\rm I\kern-1.69998ptR}^{n})=B_{p,1}^{\theta r}({\rm I\kern-1.69998ptR}^{n})$. Together with (4.33), this yields

[TABLE]

and, on applying (4.34) of Theorem 4.13, with $X_{0}=L_{p}({\rm I\kern-1.69998ptR}^{n})$ and $X_{1}=W_{p}^{r}({\rm I\kern-1.69998ptR}^{n})$, we obtain

[TABLE]

By [BS88, Chap. V, Theorem 4.12, p. 339],

[TABLE]

for all $f\in L_{p}({\rm I\kern-1.69998ptR}^{n})$ and all $t\in(0,+\infty)$. Thus, for all $f\in L_{p}({\rm I\kern-1.69998ptR}^{n})$,

[TABLE]

Moreover, $N_{1}\approx N_{11}+N_{12}$, where

[TABLE]

and, by (5.6),

[TABLE]

Therefore,

[TABLE]

Furthermore, by (5.4),

[TABLE]

Together with (5.7) and (5.8), this shows that the norm in the space $X({\rm I\kern-1.69998ptR}^{n})$ is equivalent to

[TABLE]

Hence, $X({\rm I\kern-1.69998ptR}^{n})=B_{p,q}^{0,\bar{b}}({\rm I\kern-1.69998ptR}^{n})$.

Now we are going to determine the space $Z({\rm I\kern-1.69998ptR}^{n})$. Since, by [BS88, Chap. V, Theorem 1.9, p. 300], $L_{p}({\rm I\kern-1.69998ptR}^{n})=(L_{1}({\rm I\kern-1.69998ptR}^{n}),L_{\infty}({\rm I\kern-1.69998ptR}^{n}))_{\frac{1}{p^{\prime}},p}$, we see that

[TABLE]

and on using (4.35) of Theorem 4.13, with $X_{0}=L_{1}({\rm I\kern-1.69998ptR}^{n})$ and $X_{1}=L_{\infty}({\rm I\kern-1.69998ptR}^{n})$, we obtain

[TABLE]

Hence,

[TABLE]

Making use of the fact that $d(t^{1/p})=\bar{b}(t^{1/n})$ for all $t\in(0,+\infty)$, we arrive at

[TABLE]

Finally, applying (4.36) of Lemma 4.14, we obtain

[TABLE]

and the proof is complete. $\square$

• Proof of the Sufficient Part of Theorem

3.1. By Lemma 4.1,

[TABLE]

where

[TABLE]

Using a change of variables and (2.7), we get

[TABLE]

and

[TABLE]

Therefore, by Theorem 5.1,

[TABLE]

where

[TABLE]

Combining now (5.10) and (5.13), we obtain the result. $\square$

Remark 5.2**.**

Note that if $n\in{\rm I\kern-1.69998ptN}$, $1\leq r<+\infty$ and $a\in SV(0,+\infty)$, then, for all $t\in(0,+\infty)$,

[TABLE]

i.e.,

[TABLE]

• Proof of Corollary

3.2. By the sufficient part of Theorem 3.1, embedding (3.3) holds, with $\bar{b}$ given by (3.1).

If $d$ is defined by (5.4), then, by (5.9),

[TABLE]

Thus, by Theorem 4.8,

[TABLE]

where

[TABLE]

Since $d(t^{1/p})=\bar{b}(t^{1/n})$ for any $t\in(0,+\infty)$, (5.17) can be rewritten as

[TABLE]

First we calculate $\bar{b}_{q,n}$. Using (5.14), (3.1) and a change of variables, we obtain, for all $t\in(0,+\infty)$,

[TABLE]

i.e.,

[TABLE]

Thus,

[TABLE]

and, on using (3.1) and (3.4), we arrive at

[TABLE]

Therefore, by (3.3) and (5.16),

[TABLE]

Since one can easily show that

[TABLE]

together with (5.20) and (5.21), this gives (3.5). $\square$

• Proof of the Necessity Part of Theorem

3.1. Assume now that (3.3) holds. Then, as in the proof of Corollary 3.2, one can show that

[TABLE]

with $\widetilde{b}$ defined by (3.4). Consequently,

[TABLE]

Then, together with (5.14), Theorem 1.2 gives $q\geq r$. $\square$

6. Proofs of Sharpness and Local Optimality

Lemma 6.1**.**

Let $n\in{\rm I\kern-1.69998ptN}$, $1<p<+\infty$, $1\leq r\leq q<+\infty$, and let $b\in SV(0,+\infty)$ satisfy (2.7). If $\,\bar{b}_{(r,r)},\bar{b}_{(r,q)}\in SV(0,+\infty)$ are given by (3.1), then

[TABLE]

• Proof.

Embedding (6.1) means that, for all $f\in Z_{p,r,n,\bar{b}_{(r,r)}}({\rm I\kern-1.69998ptR}^{n})$, 333) Note that $\bar{b}_{(r,r)}=b$ and so $Z_{p,r,n,\bar{b}_{(r,r)}}({\rm I\kern-1.69998ptR}^{n})=Z_{p,r,n,b}({\rm I\kern-1.69998ptR}^{n})$.)

[TABLE]

which can be rewritten as

[TABLE]

where, for all $t\in(0,+\infty)$,

[TABLE]

Since $g\in\mathcal{M}^{+}(0,+\infty;\uparrow)$, on putting

[TABLE]

we obtain by [HS93, Proposition 2.1] that inequality (6.2) holds for all $g\in\mathcal{M}^{+}(0,+\infty;\uparrow)$ if and only if

[TABLE]

As $V^{1/r}(t)\approx\bar{b}_{r}(t^{1/n})$ and $W^{1/q}(t)\approx\bar{b}_{r}(t^{1/n})$ for all $t\in(0,+\infty)$, condition (6.3) is satisfied. Consequently, (6.1) holds. $\square$

• Proof of Theorem

3.4. The result is an immediate consequence of Theorem 3.1 and Lemma 6.1. $\square$

• Proof of Theorem

3.5. Putting $\bar{B}=\kappa\bar{b}$ (with $\bar{b}$ given by (3.1)), then one can show (as in the proof of Corollary 3.2) that

[TABLE]

where (cf. (5.18) and (5.14)), for all $t\in(0,+\infty)$,

[TABLE]

Hence, by (3.8) and (6.4),

[TABLE]

In particular, this gives

[TABLE]

Now we intend to use Theorem 1.1 to verify the validity of (6.6). To this end, we are going to estimate the quantity

[TABLE]

where

[TABLE]

and $\rho=+\infty$ if $p\leq q$, and $\rho$ is defined by $\frac{1}{\rho}=\frac{1}{q}-\frac{1}{p}$ if $q<p$. Since $\widetilde{B}\in SV(0,+\infty)$,

[TABLE]

which implies that

[TABLE]

If $p\leq q$, then, by (6.5),

[TABLE]

If $q<p$, then, by (6.5) and a simple calculation,

[TABLE]

Put $A=0$ if $\rho=+\infty$ and $A=(\bar{B}_{q}(1))$ if $\rho<+\infty$. First, using L’Hopital’s Rule, one can see that

[TABLE]

Together with (5.19) and (5.14), this implies that (1.4) cannot hold unless $\kappa(0_{+})<+\infty$, that is, unless $\kappa$ is bounded. $\square$

7. Relations among the target spaces in the main results

• Proof of Theorem

3.7. First, by (5.16), (5.20) and (5.22),

[TABLE]

Moreover, by (5.15),

[TABLE]

Hence, it follows from (5.20), (5.22) and (4.32) that $Z_{p,q,n,\bar{b}}({\rm I\kern-1.69998ptR}^{n})=L_{p,q;\widetilde{b}}({\rm I\kern-1.69998ptR}^{n})$ if $q=p$.

Now we intend to prove that the opposite embedding to (3.12) does not hold if $q\neq p$. That is, we are going to prove (3.14), which is

[TABLE]

On the contrary, suppose that

[TABLE]

if $p\neq q$, $1<p<+\infty$, $1\leq r\leq q<+\infty$ and $b\in SV(0,+\infty)$ safisfies (2.7).444*)* Recall that $r$ is involved in the definitions of $\bar{b}$ and $\widetilde{b}$.) Then, according to the proof of Theorem 3.1 (cf. (5.11) and (5.12)), $\bar{b}$ satisfies

[TABLE]

Embeding (7.1) means that

[TABLE]

which is equivalent to

[TABLE]

Hence, putting $g:=(f^{*})^{p}$ and using [BS88, Chap. III, Corollary 7.8, p. 86], we obtain

[TABLE]

Set $Q:=\frac{q}{p}$. Then (7.3) is equivalent to

[TABLE]

• I.

Assume first that $Q=\frac{q}{p}\in(0,1)$, i.e., $q<p$. Inequality (7.4) can be rewritten as

[TABLE]

where, for all $x\in(0,+\infty)$,

[TABLE]

If we put $P:=Q$ and

[TABLE]

then (7.5) coincides with inequality (4.15). This fact and Theorem 4.5 then imply that

[TABLE]

However, we are going to show that this is not the case. First,

[TABLE]

Second,

[TABLE]

Since (3.1) implies that (cf. (5.18), (5.19) and (5.20))

[TABLE]

we see that

[TABLE]

Therefore, by (7.7), (7.8) and (7.10), inequality (7.6) is equivalent to

[TABLE]

which can be rewritten as

[TABLE]

However, this is a contradiction since, by (2.1),

[TABLE]

• II.

Assume now that $Q=\frac{q}{p}\in(1,+\infty)$, i.e., $p<q$. Inequality (7.4) can be rewritten as (4.19) with $P=Q$ and

[TABLE]

This fact and Theorem 4.7 then imply that both conditions (4.20) and (4.21) are satisfied. However, we are going to show that this is not the case.

Let us check condition (4.21). First,

[TABLE]

Second,

[TABLE]

Since now $q>p$, (7.9) implies that

[TABLE]

Thus, defining the function $a\in SV(0,+\infty)$ by

[TABLE]

we obtain from (7.11) and (7.12) that

[TABLE]

Since, by (2.2),

[TABLE]

condition (4.21) does not hold, which is a contradiction.

Consequently, in both cases I and II, (3.14) holds. $\square$