Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $k$-modulus of smoothness

Abstract

We use an estimate of the $k$-modulus of smoothness of a function $f$ such that the norm of its distributional gradient $|\nabla ^kf|$ belongs locally to the Lorentz space $L^{n/k, 1}({\mathbb {R}}^n),\,k \in {\mathbb {N}},\,k\le n$, and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces $X({\mathbb {R}}^n)$. Target spaces of our embeddings are generalized Hölder spaces defined by means of the $k$-modulus of smoothness $(k\in {\mathbb {N}})$. General results are illustrated with examples.

1 Introduction

Sobolev-type spaces and their embeddings play a significant role in mathematical analysis and in applications. Recently, a lot of attention has been devoted to embeddings of Sobolev-type spaces $W^kX({\mathbb {R}}^n)$, where $X({\mathbb {R}}^n)$ is a rearrangement-invariant Banach function space (r.i. BFS) and $k \in {\mathbb {N}}$; especially, in the study of regularity of PDEs, embeddings of Sobolev-type spaces $W^kX({\mathbb {R}}^n)$ with Hölder-type spaces as targets are of great importance. When Hölder-type spaces are defined by means of the first-order modulus of smoothness, to derive such embeddings, one can use a result due to R. A. DeVore and R. C. Sharpley (cf. [11, Lemma 2]). This result states that a function $f$, such that the norm of its distributional gradient $|\nabla f|$ belongs locally to the Lorentz space $L^{n,1}({\mathbb {R}}^n)$, can be redefined on a set of measure zero so that $f$ is continuous on ${\mathbb {R}}^n$ and the modulus of smoothness $\omega (f,\cdot )$ of $f$ satisfies the inequality

$$\begin{aligned} \omega (f,t)\precsim \int \limits _0^{t^n}s^{\frac{1}{n}-1}|\nabla f|^*(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1) \end{aligned}$$

(1.1)

(here $|\nabla f|^*$ denotes the non-increasing rearrangement of $|\nabla f |$).

Such an approach cannot be used when we consider fractional Sobolev-type spaces with order of smoothness less than 1 (since then, the symbol $\nabla f$ is not meaningful for all $f$ from the Sobolev space in question). This was the case of [17], where we have characterized all continuous embeddings of the Bessel potential space $H^{\sigma }X({\mathbb {R}}^n)$ with order of smoothness less than one, modelled upon an r.i. BFS $X({\mathbb {R}}^n)$, into generalized Hölder spaces (defined by means of the first-order modulus of smoothness) and we have found the optimal embedding.

For this purpose, we have proved (cf. [17, Theorem 1, p. 655]) that if $\sigma \in (0,1),\,X=X({\mathbb {R}}^n)$ is an r. i. BFS and the Bessel potential kernel $g_{\sigma }$ belongs to the associate space of $X$, then

$$\begin{aligned} \omega (f*g_{\sigma },t)\precsim \int \limits _0^{t^n}s^{\frac{\sigma }{n}-1}f^*(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1)\quad \text {and every}\quad f\in X, \end{aligned}$$

(1.2)

where $f^*$ denotes the non-increasing rearrangement of $f$. Moreover, we have proved a reverse form of estimate (1.2).

In [19], estimate (1.2) and its reverse form have been extended to the case when $\sigma \in (0, n)$. Namely, we have proved (cf. [19, Theorem 4.1]) that if $\sigma \in (0,n),\,X=X({\mathbb {R}}^n$) is an r. i. BFS and the Bessel potential kernel $g_{\sigma }$ belongs to the associate space of $X$, then $f*g_{\sigma }$ is bounded and uniformly continuous on ${\mathbb {R}}^n$ and

$$\begin{aligned} \omega _k(f*g_{\sigma },t)\precsim \int \limits _0^{t^n}s^{\frac{\sigma }{n}-1}f^*(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1)\quad \text {and every}\quad f\in X, \end{aligned}$$

(1.3)

where $k\ge [\sigma ]+1$.

Moreover, estimate (1.3) is sharp in the sense that given $k\in {\mathbb {N}}$, there are (small enough) $\delta \in (0,1)$ and (big enough) $\alpha >0$ such that

$$\begin{aligned} \omega _k(\overline{f}*g_{\sigma },t)\succsim \int \limits _0^{t^n}s^{\frac{\sigma }{n}-1}f^*(s)\,\mathrm{d}s \quad \text {for all} \quad t\in (0,1)\quad \text {and every}\quad f\in X, \end{aligned}$$

(1.4)

where

$$\begin{aligned} \overline{f}(x):=f^*(\beta _n|x|^n) \chi _{C_\alpha (0,\delta )}(x),\qquad x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n, \end{aligned}$$

(1.5)

$C_\alpha (0,\delta ):=C_\alpha \cap B_n(0,\delta )$ with $C_\alpha :=\{y\in {\mathbb {R}}^n:\,y_1>0,\; y_1^2>\alpha \sum _{i=2}^ny_i^2\},\,B_n(0,\delta ):=\{y\in {\mathbb {R}}^n:\,|y|<\delta \}$, and $\beta _n:=|B_n(0,1)|_n$.

Estimates (1.3) and (1.4) have been applied in [18] to establish necessary and sufficient conditions for embeddings of Bessel potential spaces $H^{\sigma }X({\mathbb {R}}^n)$ with order of smoothness $\sigma \in (0, n)$ into Hölder-type spaces defined by means of the $k$-modulus of smoothness $(k\in {\mathbb {N}})$ and to find the optimal embeddings of the spaces in question. In general, such embeddings give better results than those with Hölder-type spaces defined by means of the first-order modulus of smoothness (see [18]).

Since Sobolev spaces $W^kX({\mathbb {R}}^n),\,k \in {\mathbb {N}}$, coincide with Bessel potential spaces $H^{k}X({\mathbb {R}}^n)$ when the Boyd’s indices of the r.i. BFS $X:=X({\mathbb {R}}^n)$ belong to $(0,1)$ and when the space $X$ has absolutely continuous norm, we can transfer such embedding results to Sobolev spaces $W^kX({\mathbb {R}}^n)$. However, if the r.i. BFS $X$ does not possess such properties, the mentioned embeddings are not available for Sobolev spaces $W^kX({\mathbb {R}}^n)$ in the existing literature.

The aim of this paper is to fill in this hole and to establish necessary and sufficient conditions for continuous embeddings of both non-homogeneous Sobolev spaces $W^kX({\mathbb {R}}^n)$ and homogeneous Sobolev spaces $\dot{W}^kX({\mathbb {R}}^n)$, modelled upon rearrangement-invariant Banach function spaces $X({\mathbb {R}}^n)$, into generalized Hölder spaces defined by means of the $k$-modulus of smoothness $(k\in {\mathbb {N}})$.

Theorem 4.1 mentioned below states that (we refer to Sect. 2 for notation and precise definitions) if $k,n\in {\mathbb {N}},\,k\le n,\,X=X({\mathbb {R}}^n)$ is an r.i. BFS and $G:=G(0,1)$ is a quasi-Banach lattice of functions over $(0,1)$ satisfying conditions (2.8), (2.9) below, and the Fatou property, then the following statements are equivalent:

$$\begin{aligned}&\dot{W}^kX\hookrightarrow \dot{\Lambda }_{\infty ,G}^k,\end{aligned}$$

(1.6)

$$\begin{aligned}&W^kX\hookrightarrow \dot{\Lambda }_{\infty ,G}^k,\end{aligned}$$

(1.7)

$$\begin{aligned}&\Bigg \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Bigg \Vert _G\precsim \Vert f\Vert _X\quad \text {for all }f\in X. \end{aligned}$$

(1.8)

This means that Theorem 4.1 reduces our original problems (1.6) and (1.7) to inequality (1.8) involving Hardy-type operator considered on the class of non-increasing functions.

The case $k>n$ will be investigated in a forthcoming paper.

To find necessary and sufficient conditions for embeddings in question, one has to characterize inequality (1.8). Such characterizations are given in Sect. 5 for several cases important in applications (cf. Theorems 5.1,  5.4,  5.6, 5.7 below). Using these characterizations, it is possible to determine sharp and optimal targets of given embeddings (cf. remarks in Sect. 5; note also that the notions of sharpness and optimality are recalled in Remark  5.9 below). For example, the choice $k=1$ and $G:=L^{\infty }_{\psi }(0,1)$ ($\psi$ is a convenient weight) in Theorem  4.1 (cf. Theorem  5.1, Remark  5.2 below) corresponds to [7, Theorem 1.3]. In the recent paper [8, Theorem 3.4], the authors studied embeddings of the type $W^mX(\Omega )\hookrightarrow \Lambda _{\infty ,G}^1(\Omega )$, with $2\le m\le n$. However, our approach admits much more general quasi-Banach lattices $G(0,1)$, than that of $L^{\infty }_{\psi }(0,1)$, and target spaces $\dot{\Lambda }_{\infty ,G}^k$, with $k>1$. Consequently, we obtain much finer results.

A special attention is devoted to the case when $X$ is the Lorentz–Karamata space $L_{p,q;b}$ (cf. Corollaries  5.8,  5.12,  5.15, 5.18). Then, a characterization of the given embedding is not so complicated as in the situation when $X$ is the classical Lorentz space $\Lambda ^q(\varphi )$ (cf. Theorem 5.7). Nevertheless, even when $X=L_{p,q;b}$ one has to distinguish four different cases: $p\in (\frac{n}{k}, +\infty ),\,p=+\infty ,\,p=\frac{n}{k}$ with $k<n$, and $p=\frac{n}{k}$ with $k=n$ (i.e., $p=1$) since conditions characterizing the embeddings in question are mutually different. For example, the Brézis–Wainger result (cf. [4, Corollary 5]) on “almost” Lipschitz continuity of functions from $W^{k+1,n/k}({\mathbb {R}}^n),\,n,k\in {\mathbb {N}},\,k+1\le n$, is a consequence of a better embedding whose target is a Zygmund-type space $\Lambda _{\infty ,n/k}^{2,Id(\cdot )}(\overline{{\mathbb {R}}^n})$ (cf. Example 5.11 below); we refer to [18, Section 2] for the definition of spaces $\Lambda _{\infty ,r}^{k,\lambda (\cdot )}(\overline{{\mathbb {R}}^n})$. Note also that we have proved an analogous result for homogeneous Sobolev spaces $\dot{W}^{k+1,n/k}({\mathbb {R}}^n)$ defined as the set of functions on ${\mathbb {R}}^n$ such that the norm of its distributional gradient $|\nabla ^{k+1} f|$ belongs to $L^{n/k}({\mathbb {R}}^n)$ (see, again, Example 5.11 below).

To prove Theorem 4.1, we first use a variant of estimate (1.1), proved in [23], which states that if $n,k\in {\mathbb {N}},\,k\le n$, and $f$ is a function such that the norm of its distributional gradient $|\nabla ^k f|$ belongs locally to the Lorentz space $L^{n/k,1}({\mathbb {R}}^n)$, then $f$ can be redefined on a set of measure zero so that $f$ is continuous on ${\mathbb {R}}^n$ and the $k$-modulus of smoothness $\omega _k(f,\cdot )$ satisfies

$$\begin{aligned} \omega _k(f,t)\precsim \int \limits _0^{t^n}s^{\frac{k}{n}-1}|\nabla ^k f|^*(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1) \end{aligned}$$

(1.9)

(see Theorem 2.1 below). Second, we prove a reverse form of estimate (1.9), which states that if $k,n\in {\mathbb {N}}$, and $g \in \mathcal {M}^{+}(0,1;\downarrow )\cap C([0,1])$, then there is a nonnegative continuous function $F_g$ satisfying

$$\begin{aligned}&F_g\in \bigcap _{p\in [1, +\infty ]}^{} W^{k,p}({\mathbb {R}}^n),\\&\omega _k(F_g,t)\succsim \int \limits _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1), \end{aligned}$$

and

$$\begin{aligned} |\nabla ^m F_g|^{**}(t)\precsim \,g^{**}(t) \quad \text {for all} \quad m\in \{0,1,\ldots ,k\} \quad \text {and}\quad t>0 \end{aligned}$$

(see Theorem 3.1 below).

The paper is organized as follows. Section 2 contains notation and preliminaries. In Sect. 3, we establish a reverse form of estimate (1.9) while in Sect. 4, we present the reduction Theorem 4.1. Characterization of inequality (1.8) for particular cases of spaces is given in Sect. 5. This section also contains some examples of embeddings and remarks on their sharpness and optimality.

2 Notation, definitions, and preliminaries

As usual, ${\mathbb {R}}^n$ denotes the Euclidean $n$-dimensional space. Throughout the paper, $\mu _n$ is the $n$-dimensional Lebesgue measure in ${\mathbb {R}}^n$ and $\Omega$ is a domain in ${\mathbb {R}}^n$. 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 that are nonnegative $\mu _n$-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 $\mathcal {M}^{+}(a,b;\downarrow )$, we mean the subset of $\mathcal {M}^{+}(a,b)$ containing all non-increasing functions on the interval $(a,b)$. The symbol ${\mathcal W}(a,b)$ stands for the class of weight functions on $(a,b)\subseteq {\mathbb {R}}$ consisting of all $\mu _1$-measurable functions that are positive and finite $\mu _1$-a.e. on $(a,b)$. The non-increasing rearrangement of $f\in {\mathcal M}(\Omega )$ is the function $f^*$ defined by $f^*(t):=\inf \left\{ \lambda \ge 0:|\{x\in \Omega :|f(x)|\!>\!\lambda \}|_n\le t \right\}$ for all $t\ge 0$. Note that if $|\Omega |_n<+\infty$, then $f^*(t)=0$ for all $t\ge |\Omega |_n$. By $f^{**}$, we denote the maximal function of $f^*$ given by $f^{**}(t):=t^{-1}\int _0^tf^*(\tau )\,d\tau ,\,t>0$. The maximal operator ${f\mapsto f^{**}}$ is subadditive (cf. [2, p. 54]).

Given a non-empty domain $\Omega$ in ${\mathbb {R}}^n$, a rearrangement-invariant Banach function space (r. i. BFS) $X:=X(\Omega )$ is a Banach space of Lebesgue-measurable functions on $\Omega$, endowed with a norm $\Vert \cdot \Vert _X$, that satisfies the lattice property and the Fatou property, contains all characteristic functions of finite measure subsets of $\Omega$, all functions from $X$ are locally integrable, that is, $X$ is continuously embedded into $L^1(E)$, for all sets $E\subset \Omega$ with $|E|_n<+\infty$, and is such that if $f\in X$ and $g^*=f^*$, then $g\in X$ and $\Vert f\Vert _X=\Vert g\Vert _X$. By $X'$, we denote the associate space of $X$. It is well known that there exists an r.i. space $\overline{X}=\overline{X}(0,+\infty )$ (called Luxemburg’s representation space of $X$) such that $\Vert f\Vert _X =\Vert f^*\Vert _{\overline{X}}$ for all $f\in X$. We refer to [2] for details.

By $c,\,C,\,c_1,\,C_1,\,c_2,\,C_2$, we denote positive constants independent of appropriate quantities. For two nonnegative 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 ${\mathcal A}\le c\, {\mathcal B}$ (or $c\,{\mathcal A}\ge {\mathcal B}$). 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 $a/(+\infty )=0$ and $a/0=+\infty$ for all $a>0$. If $p\in [1,+\infty ]$, the conjugate number $p'$ is given by $1/p+1/p'=1$. In the whole paper, $\Vert .\Vert _{p;(c,d)},\, p\in (0,+\infty ]$, denotes the usual $L^p$-(quasi-)norm on the interval $(c,d)\subseteq {\mathbb {R}}$.

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 [21]; for other definitions see, for example, [3, 12, 14, 25]. The family $SV(0,+\infty )$ includes not only powers of iterated logarithms and the broken logarithmic functions of [15] 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).

Let $q\in (0,+\infty ],\,b\in SV(0,+\infty ),\,B_q(t):=\Vert \tau ^{-1/q}b(\tau )\Vert _{q;(0,t)},\,t>0$, and let $B_q(1)<+\infty$. Then, by Gurka and Opic [22, Lemma 2.1 (v)],

$$\begin{aligned} B_q\in SV(0,+\infty ). \end{aligned}$$

(2.1)

Moreover, by Gurka and Opic [22, Lemma 2.1(iv)],

$$\begin{aligned} \Vert b(\tau )\Vert _{1;(0,t)}\approx t b(t)\quad \text {for all }t>0. \end{aligned}$$

(2.2)

Thus, without loss of generality, we shall assume that

$$\begin{aligned} \text {all slowly varying functions in question are continuous on}~(0,+\infty ). \end{aligned}$$

(2.3)

More properties and examples of slowly varying functions can be found in [3, 14, 18, 21, 24, 25], [30, Chapter V, p. 186].

Let $p,q\in (0,+\infty ],\,b\in SV(0,+\infty )$ and let $\Omega$ be a non-empty Lebesgue-measurable subset of ${\mathbb {R}}^n$. The Lorentz–Karamata (LK) space $L_{p,q;b}(\Omega )$ is defined to be the set of all functions $f\in {\mathcal M}(\Omega )$ such that

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

(2.4)

If $\Omega ={\mathbb {R}}^n$, we simply write $\Vert \cdot \Vert _{p,q;b}$ instead of $\Vert \cdot \Vert _{p,q;b;{\mathbb {R}}^n}$.

Particular choices of $b$ give well-known spaces. If $m\in {\mathbb {N}},\varvec{\alpha }=(\alpha _1,\dots ,\alpha _m)\in {\mathbb {R}}^m$ and

$$\begin{aligned} b(t)=\varvec{\ell }^{\varvec{\alpha }}(t):=\prod _{i=1}^{m}\ell _{i}^{\alpha _{i}}(t)\quad \text {for all }t>0 \end{aligned}$$

(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 $L_{p,q;b}(\Omega )$ is the generalized Lorentz–Zygmund space $L_{p,q,\varvec{\alpha }}$ introduced in [13] and endowed with the (quasi-)norm $\Vert f\Vert _{p,q;\varvec{\alpha };\Omega }$, which in turn becomes the Lorentz–Zygmund space $L^{p,q}(\log L)^{\alpha _{1}}(\Omega )$ of Bennett and Rudnick [1] when $m=1$. If $\varvec{\alpha }=(0,\dots , 0)$, we obtain the Lorentz space $L^{p,q}(\Omega )$ endowed with the (quasi-)norm $\Vert .\Vert _{p,q;\Omega }$, which is just the Lebesgue space $L^p(\Omega )$ equipped with the (quasi-)norm $\Vert .\Vert _{p;\Omega }$ when $p=q$; if $p=q$ and $m=1$, we obtain the Zygmund space $L^{p}(\log L)^{\alpha _1}(\Omega )$ endowed with the (quasi-)norm $\Vert .\Vert _{p;\alpha _1;\Omega }$.

Given a domain $\Omega$ in ${\mathbb {R}}^n,\,k\in {\mathbb {N}}$ and an r.i. BFS $X:=X(\Omega )$, we denote by $W^kX(\Omega )$ the corresponding Sobolev space, that is, the space of all $f$ in $X(\Omega )$ whose distributional derivatives $D^{\alpha }f,\,|\alpha |\le k$, belong to $X(\Omega )$. This space is equipped with the norm

$$\begin{aligned} \Vert f\Vert _{W^kX(\Omega )}:=\sum _{|\alpha |\le k}\Vert D^{\alpha }f\Vert _{X(\Omega )} \approx \sum _{m=0}^k\Vert \,|\nabla ^{m}f|\,\Vert _{X(\Omega )}, \end{aligned}$$

where $\displaystyle {|\nabla ^{m}f|:=\sum _{|\alpha |= m} |D^{\alpha }f|}$. If $X(\Omega )=L^p(\Omega )$, then we set $W^{k,p}(\Omega ):=W^kL^p(\Omega )$.

The homogeneous Sobolev space $\dot{W}^kX(\Omega )$ is the space of all $f$ in ${\mathcal M}(\Omega )$ whose distributional derivatives $D^{\alpha }f,\,|\alpha |= k$, belong to $X(\Omega )$. This space is equipped with the semi-norm

$$\begin{aligned} \Vert f\Vert _{\dot{W}^kX(\Omega )}=\sum _{|\alpha |= k}\Vert D^{\alpha }f\Vert _{X(\Omega )} \approx \Vert \,|\nabla ^{k}f|\,\Vert _{X(\Omega ).} \end{aligned}$$

If $X(\Omega )=L^p(\Omega )$, then we put $\dot{W}^{k,p}(\Omega ):=\dot{W}^kL^p(\Omega )$.

Let $\Omega$ be a domain in ${\mathbb {R}}^n$. The space of all continuous functions on $\Omega$ is denoted by $C(\Omega )$. The subspace of $C(\Omega )$ consisting of all bounded and continuous functions on $\Omega$ is denoted by $C_B(\Omega )$ and it is equipped with the $L^{\infty }(\Omega )$-norm. By $C_B^m(\Omega )$, we denote the space of all functions that, together with all partial derivatives of orders $|\alpha |\le m$, are bounded and continuous on $\Omega$.

Let $h\in {\mathbb {R}}^n$. The first difference operator $\Delta _h$ is defined on scalar functions $f$ on ${\mathbb {R}}^n$ by $\Delta _hf(x)=f(x+h)-f(x),\,x\in {\mathbb {R}}^n$. Higher-order differences are defined inductively by

$$\begin{aligned} \Delta _h^{k+1}f(x)=\Delta _h(\Delta _h^kf)(x),\quad x\in {\mathbb {R}}^n,\; k\in {\mathbb {N}}. \end{aligned}$$

The k-modulus of smoothness ($k\in {\mathbb {N}}$) of a function $f$ in $C({\mathbb {R}}^n)$ is given by

$$\begin{aligned} \displaystyle {\omega _k(f,t):=\sup _{|h|\le t}\Vert \Delta _h^kf\Vert _{L^{\infty }({\mathbb {R}}^n)}\quad \text{ for } \text{ all } t\ge 0}. \end{aligned}$$

If $k=1$, we write $\omega (f,t)$ instead of $\omega _1(f,t)$. By DeVore and Lorentz [9, ineq. (7.8), p. 45],

$$\begin{aligned} \omega _k(f,\lambda t)\le (\lambda +1)^k \omega _k(f, t),\;\;t>0,\;\lambda >0. \end{aligned}$$

(2.5)

If $\omega _k(f,t)=+\infty$ for some $t$, then, by (2.5), $\omega _k(f,t)=+\infty$ for all $t>0$.

Estimate (2.5) also implies that

$$\begin{aligned} \frac{\omega _k(f,t_2)}{t_2^k}\le 2^k \,\frac{\omega _k(f,t_1)}{t_1^k},\;\;0<t_1<t_2. \end{aligned}$$

(2.6)

Thus, putting

$$\begin{aligned} \widetilde{\omega }_k(f,t):=\omega _k(f,t)/t^k, \quad \overline{\omega }_k(f,t):=\sup _{s\in [t,+\infty )}\widetilde{\omega }_k(f,s) \text{ for } \text{ all } t>0, \end{aligned}$$

and using (2.6), one can prove that

$$\begin{aligned} \widetilde{\omega }_k(f,\cdot )\text { is equivalent to the non-increasing function }\overline{\omega }_k(f,\cdot )\text { on }(0,+\infty ). \end{aligned}$$

(2.7)

Let $k\in {\mathbb {N}}$ and let $G=G(0,1)$ be a quasi-Banach lattice of functions over $(0,1)$ such that

$$\begin{aligned} \left\| \chi _{(0,1)}(t)\right\| _{G(0,1)}=+\infty \end{aligned}$$

(2.8)

and

$$\begin{aligned} \left\| t^k\chi _{(0,1)}(t)\right\| _{G(0,1)}<+\infty . \end{aligned}$$

(2.9)

The generalized Hölder space $\dot{\Lambda }_{\infty ,G}^{k}:=\dot{\Lambda }_{\infty ,G}^{k}({\mathbb {R}}^n)$ consists of all functions $f\in C({\mathbb {R}}^n)$ for which the quasi-semi-norm

$$\begin{aligned} \Vert f\Vert _{\dot{\Lambda }_{\infty ,G}^{k}}:= \left\| \omega _k(f,t) \right\| _{G(0,1)} \end{aligned}$$

is finite.

If (2.9) does not hold, then $\left\| \omega _k(f,t)\right\| _{G(0,1)}$ is finite if and only if $f$ is a polynomial of degree less or equal to $k-1$.

Note that $\omega _k(f,t)< +\infty$ for all $t\in (0,1)$ and all $f\in \dot{\Lambda }_{\infty ,G}^{k}$. Indeed, assume that $f\in \dot{\Lambda }_{\infty ,G}^{k}$. Then, by (2.7) and (2.9),

$$\begin{aligned} +\infty&> \left\| \omega _k(f,t)\right\| _{G(0,1)}\ge \left\| t^k\frac{\omega _k(f,t)}{t^k}\right\| _{G(0,\delta )}\succsim \frac{\omega _k(f,\delta )}{\delta ^k}\left\| t^k\chi _{(0,\delta )}\right\| _{G(0,1)}\qquad \end{aligned}$$

(2.10)

for any $\delta \in (0,1)$. Hence, $\omega _k(f,\delta )<+ \infty$ for any $\delta \in (0,1)$.

We shall need the following assertion proved in [23, (1.6)].

Theorem 2.1

Let $n,k\in {\mathbb {N}},\,k\le n$. Let $f$ be a function such that the norm of its distributional gradient $|\nabla ^k f|$ belongs locally to the Lorentz space $L^{n/k,1}({\mathbb {R}}^n)$. Then, $f$ can be redefined on a set of measure zero so that $f$ is continuous on ${\mathbb {R}}^n$ and the modulus of smoothness $\omega _k(f,\cdot )$ satisfies

$$\begin{aligned} \omega _k(f,t)\le c \int \limits _0^{t^n}s^{\frac{k}{n}-1}|\nabla ^k f|^*(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1), \end{aligned}$$

(2.11)

where $c$ is a positive constant independent of $f$.

3 Lower estimate of the modulus of smoothness

The main result of this section is Theorem 3.1 below, which concerns the reverse form of inequality (2.11).

Theorem 3.1

Let $k,n\in {\mathbb {N}}$, and let $g\in \mathcal {M}^{+}(0,1;\downarrow )\cap C([0,1])$. Then, there is a nonnegative continuous function $F_g\in \displaystyle {\bigcap _{p\in [1,+\infty ]}}W^{k,p}({\mathbb {R}}^n)$ satisfying

$$\begin{aligned} \omega _k(F_g,t)\succsim \int \limits _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \quad \text {for all}\quad t\in (0,1), \end{aligned}$$

(3.1)

and

$$\begin{aligned} |\nabla ^m F_g|^{**}(t)\precsim \,g^{**}(t) \quad \text {for all} \quad m\in \{0,1,\ldots ,k\} \quad \text {and}\quad t>0. \end{aligned}$$

(3.2)

Proof

Step 1. In this step of the proof, our intent is to establish inequality (3.6) mentioned below. This inequality coincides with [8, (4.6), p. 1950]. For a convenience of the reader, we give its proof here.

The domain $\Omega$ given by

$$\begin{aligned} \Omega := \Biggl \{x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n: 0<x_1< 1, \, \sum _{i=2}^nx_i< x_1, \,x_i > 0,\, i=2,\ldots ,n\Biggr \} \end{aligned}$$

has minimally smooth boundary. Then, by Stein [29, Chap. VI, Sect. 3, pp. 180–192], there exists an extension operator $E$ mapping functions on $\Omega$ to functions on ${\mathbb {R}}^n$ such that $Ef(x)=f(x),\,x\in \Omega$, and $E:W^{k,p}(\Omega ) \rightarrow W^{k,p}({\mathbb {R}}^n)$ is continuous for all $p\in [1,+\infty ]$ and $k\in {\mathbb {N}}_0$. In particular,

$$\begin{aligned} E: W^{k,1}(\Omega )\longrightarrow W^{k,1}({\mathbb {R}}^n) \quad \text {and} \quad E: W^{k,\infty }(\Omega )\longrightarrow W^{k,\infty }({\mathbb {R}}^n). \end{aligned}$$

Consequently,

$$\begin{aligned} K(t,Ef;W^{k,1}({\mathbb {R}}^n),W^{k,\infty }({\mathbb {R}}^n))\precsim K(t,f;W^{k,1}(\Omega ),W^{k,\infty }(\Omega )) \end{aligned}$$

(3.3)

for all $t>0$ and $f\in W^{k,1}(\Omega )+W^{k,\infty }(\Omega )$, where the symbol $K$ stands for the Peetre $K$-functional.

The $K$-functional for the pair $(W^{k,1}({\mathbb {R}}^n),W^{k,\infty }({\mathbb {R}}^n))$ has been computed by DeVore and Scherer [10, Theorem 1],

$$\begin{aligned} K(f,t;W^{k,1}({\mathbb {R}}^n),W^{k,\infty }({\mathbb {R}}^n)) \approx \sum _{|\alpha |\le k}\int \limits _0^t(D^{\alpha }f)^*(s)\,\mathrm{d}s \approx \sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}f|)^*(s)\,\mathrm{d}s\qquad \end{aligned}$$

(3.4)

for all $t>0$ and all $f\in W^{k,1}({\mathbb {R}}^n)+W^{k,\infty }({\mathbb {R}}^n)$. They also proved that when ${\mathbb {R}}^n$ is replaced by a domain $\Omega$ in ${\mathbb {R}}^n$ with a minimally smooth boundary, then

$$\begin{aligned} K(f,t;W^{k,1}(\Omega ),W^{k,\infty }(\Omega ))\approx \sum _{|\alpha |\le k}\int \limits _0^t(D^{\alpha }f)^*(s)\,\mathrm{d}s \approx \sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}f|)^*(s)\,\mathrm{d}s\qquad \end{aligned}$$

(3.5)

for all $t>0$ and all $f\in W^{k,1}(\Omega )+W^{k,\infty }(\Omega )$.

By (3.4) and (3.5), (3.3) is equivalent to

$$\begin{aligned} \sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}Ef|)^*(s)\,\mathrm{d}s\precsim \sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}f|)^*(s)\,\mathrm{d}s \end{aligned}$$

(3.6)

for all $t>0$ and $f\in W^{k,1}(\Omega )+W^{k,\infty }(\Omega )$.

Step 2. Given $k,n\in {\mathbb {N}}$, and $g\in \mathcal {M}^{+}(0,1;\downarrow )\cap C([0,1])$, let $g_k$ be defined by

$$\begin{aligned} g_k(\tau ):= \int \limits _{\tau }^1\int \limits _{\tau _1}^1\cdots \int \limits _{\tau _{k-1}}^1 g(\tau _k^n)\, d\tau _k\cdots d\tau _2 \, d\tau _1 \quad \text {for}\quad \tau \in [0,1], \end{aligned}$$

(3.7)

and let

$$\begin{aligned} G_k(x):=g_k(x_1)\quad \text {for all}\,\, \quad x=(x_1,\ldots ,x_n)\in \overline{\Omega }. \end{aligned}$$

(3.8)

Note that $G_k\in \mathcal {M}^{+}(\Omega )\cap C_B^k(\Omega )$. Consequently, $G_k\in \displaystyle {\bigcap _{p\in [1,+\infty ]}}W^{k,p}(\Omega )$. Moreover, if $t\in (0,\frac{1}{k})$ and $h=(t,0,\ldots ,0)$, then

$$\begin{aligned} |(\Delta ^k_hG_k)(0, \ldots , 0)| \!&= \! \left| \sum _{i=0}^k {k \atopwithdelims ()i} (-1)^{k-i} G_k(ih) \right| \nonumber \\&= \!\left| \sum _{i=0}^k {k \atopwithdelims ()i} (-1)^{k-i} g_k(it) \right| = |\Delta ^k_tg_k(0)|. \end{aligned}$$

(3.9)

Using induction, one can easily prove that if $k\in {\mathbb {N}}$ and $\varphi \in C^k([0,1])$, then

$$\begin{aligned} \Delta ^k_t\varphi (y)=\int \limits _0^t\int \limits _0^t\cdots \int \limits _{0}^{t}\varphi ^{(k)}(y+y_1+y_2+\cdots +y_k)\,\mathrm{d}y_k\cdots \mathrm{d}y_2\, \mathrm{d}y_1, \end{aligned}$$

for all $y\in [0,1)$ and $t\in (0,\frac{1-y}{k})$. Together with (3.7) and a change of variables, this gives

$$\begin{aligned} |\Delta ^k_tg_k(0)|&= \left| \int \limits _0^t\int \limits _0^t\cdots \int \limits _{0}^{t}g_k^{(k)}(y_1+y_2+\cdots +y_k)\,\mathrm{d}y_k\cdots \mathrm{d}y_2\, \mathrm{d}y_1\right| \\&= \int \limits _0^t\int \limits _0^t\cdots \int \limits _{0}^{t}g((y_1+y_2+\cdots +y_k)^n)\,\mathrm{d}y_k\cdots \mathrm{d}y_2\, \mathrm{d}y_1\\&= \int \limits _0^t\int \limits _{y_1}^{y_1+t}\cdots \int \limits _{y_{k-1}}^{y_{k-1}+t} g(y_k^n)\, \mathrm{d}y_k\cdots \mathrm{d}y_2 \, \mathrm{d}y_1 \quad \text {for all}\quad t\in \left( 0,\frac{1}{k}\right) . \end{aligned}$$

Hence,

$$\begin{aligned}&|\Delta ^k_tg_k(0)| \ge \int \limits _0^t\int \limits _{y_1}^{t}\cdots \int \limits _{y_{k-1}}^{t} g(y_k^n)\, \mathrm{d}y_k\cdots \mathrm{d}y_2 \mathrm{d}y_1 \\&= \frac{1}{(k-1)!}\int \limits _0^t y_k^{k-1}g(y_k^n)\, \mathrm{d}y_k \\&\approx \int \limits _0^{t^n} s^{\frac{k}{n}-1} g(s)\, \mathrm{d}s \quad \text {for all } t\in \left( 0,\frac{1}{k}\right) . \end{aligned}$$

On using this estimate in (3.9), we obtain that

$$\begin{aligned} |(\Delta ^k_hG_k)(0, \ldots , 0)| \succsim \int \limits _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \quad \text {for} \quad h=(t,0,\ldots ,0)\text { and }t\in \left( 0,\frac{1}{k}\right) .\qquad \end{aligned}$$

(3.10)

Let the symbol $\delta _{ij}$ stands for the Kronecker delta, i.e., $\delta _{ij}=1$ when $i=j$ and $\delta _{ij}=0$ when $i\ne j$. If $x\in \Omega$, then

$$\begin{aligned} \displaystyle \frac{\partial G_k}{\partial x_i}(x)&= \delta _{1i}\,g_k^{\prime }(x_1),\\ \displaystyle \frac{\partial ^2 G_k}{\partial x_i\partial x_j}(x)&= \delta _{1i}\delta _{1j}\,g_k^{\prime \prime }(x_1),\\ \vdots&\\ \displaystyle \frac{\partial ^k G_k}{\partial x_{i_1}\ldots \partial x_{i_k}}(x)&= \delta _{1i_1}\ldots \delta _{1i_k} \,g_k^{(k)}(x_1), \end{aligned}$$

which implies that

$$\begin{aligned} |\nabla ^mG_k(x)|=|g_k^{(m)}(x_1)| \quad \text {for all } x\in \Omega \text { and } m\in \{1,\ldots ,k\}. \end{aligned}$$

(3.11)

Since $g$ is non-increasing on $(0,1)$, we easily obtain from (3.7) that

$$\begin{aligned} |g_k^{(m)}(t)|\le g(t^n)\quad \text {for all } t\in (0,1) \quad \text {and}\quad m\in \{0,\ldots ,k-1\}, \end{aligned}$$

and

$$\begin{aligned} |g_k^{(k)}(t)|= g(t^n)\quad \text {for all } t\in (0,1). \end{aligned}$$

Using these facts in (3.11), we get

$$\begin{aligned} |\nabla ^mG_k(x)|\le g(x_1^n) \quad \text {for all } x\in \Omega \quad \text {and}\quad m\in \{0,\ldots ,k\}. \end{aligned}$$

Since $g$ is non-increasing, $g\in C([0,1])$ and $|\Omega |_n=\frac{1}{n!}$, we obtain

$$\begin{aligned} |\nabla ^mG_k|^{*}(t)\le (g(x_1^n))^*(t)=g^*(n!\,t)\le g^*(t) \quad \text {for all } t\in (0,1/{n!}) \text { and } m\in \{0,\ldots ,k\}. \end{aligned}$$

Moreover, since for any $m\in \{0,\ldots ,k\}$ and $t\ge 1/{n!},\,|\nabla ^mG_k|^{*}(t)=0$, we see that

$$\begin{aligned} |\nabla ^mG_k|^{*}(t)\le g^*(t) \quad \text {for all }t\ge 1/{n!} \text { and } m\in \{0,\ldots ,k\}. \end{aligned}$$

Therefore,

$$\begin{aligned} |\nabla ^mG_k|^{*}(t)\le g^*(t) \quad \text {for all } t>0 \text { and } m\in \{0,\ldots ,k\} \end{aligned}$$

and, hence,

$$\begin{aligned} |\nabla ^mG_k|^{**}(t)\le g^{**}(t) \quad \text {for all } t >0\text { and } m\in \{0,\ldots ,k\}. \end{aligned}$$

(3.12)

Step 3. Putting $F_g:=EG_k$, we see that $F_g\in \displaystyle {\bigcap _{p\in [1,+\infty ]}}W^{k,p}({\mathbb {R}}^n)$. In particular, $F_g\in W^{k,\infty }({\mathbb {R}}^n)$. Thus, $F_g$ can be modified on a set of measure zero so that it is continuous on ${\mathbb {R}}^n$ (cf. [29, Chap V, Sect. 2, Theorem 2 (iii), p. 124]). Now, by (3.6) and (3.12), for any $m\in \{0,1,\ldots ,k\}$,

$$\begin{aligned} \int \limits _0^t(|\nabla ^{m}F_g|)^*(s)\le \sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}F_g|)^*(s)\,\mathrm{d}s&\precsim&\sum _{m=0}^k\int \limits _0^t(|\nabla ^{m}G_k|)^*(s)\,\mathrm{d}s\\&\precsim&tg^{**}(t)\quad \text {for all } t>0, \end{aligned}$$

and (3.2) follows.

Since $F_g \in C_B({\mathbb {R}}^n),\,{F_g|}_{\Omega }=G_k$ and $G_k\in C(\overline{\Omega })$, using $h=(t,0,\ldots ,0)$, with $t\in (0,\frac{1}{k})$, and estimate (3.10), we arrive at

$$\begin{aligned}&\omega _k(F_g,t)\!\ge \! |(\Delta ^k_hF_g)(0, \ldots , 0)|\! \nonumber \\&=\!|(\Delta ^k_hG_k)(0, \ldots , 0)| \! \succsim \! \!\int \limits _0^{t^n}\!\!\!s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s\quad \text {for all}\quad \,\, t\in \left( 0,\frac{1}{k}\right) . \end{aligned}$$

(3.13)

Moreover, if $t\in [\frac{1}{k},1)$, then

$$\begin{aligned} \omega _k(F_g,t)\ge \omega _k(F_g,\frac{1}{2k}) \succsim \int \limits _0^{(2k)^{-n}}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s. \end{aligned}$$

(3.14)

Since $2kt \approx 1$ if $t\in [\frac{1}{k},1)$ and the function $t \mapsto \frac{1}{t^k}\int _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s$ belongs to $\mathcal {M}^{+}(0,+\infty ;\downarrow )$, we get

$$\begin{aligned} \int \limits _0^{(2k)^{-n}}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \approx (2kt)^k \int \limits _0^{(2k)^{-n}}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \ge \int \limits _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s\quad \text {for all}\,\,\quad t\in \left[ \frac{1}{k},1\right) . \end{aligned}$$

Combining the last estimate with (3.14), we obtain that

$$\begin{aligned} \omega _k(F_g,t)\succsim \int \limits _0^{t^n}s^{\frac{k}{n}-1}g(s)\,\mathrm{d}s \quad \text {for all} \quad t\in \left[ \frac{1}{k},1\right) . \end{aligned}$$

(3.15)

Thus, estimate (3.1) follows from (3.13) and (3.15). $\square$

4 Reduction theorem

In this section, we reduce our original problems on embeddings to Hardy-type inequality for non-increasing functions.

Theorem 4.1

Let $k,n\in {\mathbb {N}},\,k\le n$, let $X:=X({\mathbb {R}}^n)$ be an r.i. Banach function space and let $G:=G(0,1)$ be a quasi-Banach lattice of functions over $(0,1)$ satisfying (2.8), (2.9) and the Fatou property. Then, the following statements are equivalent:

$$\begin{aligned}&\dot{W}^kX\hookrightarrow \dot{\Lambda }_{\infty ,G}^k,\end{aligned}$$

(4.1)

$$\begin{aligned}&W^kX\hookrightarrow \dot{\Lambda }_{\infty ,G}^k,\end{aligned}$$

(4.2)

$$\begin{aligned}&\left\| \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\right\| _G\precsim \Vert f\Vert _X\quad \text {for all }f\in X. \end{aligned}$$

(4.3)

Proof

(i) (4.3)$\implies$(4.1): Suppose that (4.3) holds. If $f\in \dot{W}^kX$, then $|\nabla ^k f|\in X$ and by (4.3), with $f$ replaced by $|\nabla ^k f|$,

$$\begin{aligned} \Bigg \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} (|\nabla ^k f|)^*(s)\,\mathrm{d}s\Bigg \Vert _G\precsim \Vert \,|\nabla ^k f|\,\Vert _X. \end{aligned}$$

(4.4)

Moreover, using (4.3), (2.9) and the fact that $t\mapsto \frac{1}{t^k}\int _0^{t^n}s^{\frac{k}{n}-1} (|\nabla ^k f|)^*(s)\,\mathrm{d}s$ is a non-increasing function, we can proceed as in (2.10) to prove that $\dot{W}^kX\hookrightarrow \dot{W}^k L_{loc}^{n/k,1}({\mathbb {R}}^n)$. Hence, $f$ can be redefined on a set of measure zero so that $f$ is continuous on ${\mathbb {R}}^n$ and (2.11) holds. Now, (2.11) and (4.4) yield

$$\begin{aligned} \Vert \omega _k(f,t)\Vert _G\precsim \Vert \,|\nabla ^k f|\,\Vert _X. \end{aligned}$$

Thus, $\dot{W}^kX\hookrightarrow \dot{\Lambda }_{\infty ,G}^k$, which gives (4.1).

(ii) (4.1)$\implies$(4.2): Assume that (4.1) holds. Since $W^kX\hookrightarrow \dot{W}^kX$, embedding (4.2) follows immediately.

(iii) (4.2)$\implies$ (4.3): Suppose now that (4.2) holds. Then,

$$\begin{aligned} \Vert \omega _k(f,t)\Vert _G\precsim \sum _{m=0}^k\Vert \,|\nabla ^{m}f|\,\Vert _{X}\quad \text {for all}\,\,\quad f \in W^k X. \end{aligned}$$

(4.5)

Let $f\in X$. Suppose, in addition, that $f^*$ is continuous on $[0,1]$. Let $F_{f^*}$ be given by Theorem 3.1, with $g$ replaced by $f^*\chi _{[0,1]}$. By (3.2), for any $m\in \{0,1,\ldots ,k\}$,

$$\begin{aligned} |\nabla ^m F_{f^*}|^{**}(t)\precsim \,f^{**}(t) \quad \text {for all}\quad t>0. \end{aligned}$$

Therefore, [2, Chapter II, Sect. 4, Corollary 4.7, p. 61] implies that

$$\begin{aligned} \Vert \,|\nabla ^{m}F_{f^*}|\,\Vert _{X}\precsim \Vert f\Vert _X\quad \text { for any } m\in \{0,1,\ldots ,k\}. \end{aligned}$$

(4.6)

Thus, $F_{f^*}$ also belongs to $W^k X$. Consequently, by (4.5),

$$\begin{aligned} \Vert \omega _k(F_{f^*},t)\Vert _G\precsim \sum _{m=0}^k\Vert \,|\nabla ^{m}F_{f^*}|\,\Vert _{X}. \end{aligned}$$

Together with (3.1) and (4.6), this shows that

$$\begin{aligned} \Bigg \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Bigg \Vert _G\precsim \Vert f\Vert _X= \Vert f^*\Vert _{\overline{X}}, \end{aligned}$$

(4.7)

where $\overline{X}$ is a representation space of $X$, and the proof is complete in the case that $f^*$ is continuous on [0,1]. In a general case, when $f^*\in \mathcal {M}^{+}(0,1;\downarrow )$, we can construct a non-decreasing sequence $\{f_m\}_{m\in {\mathbb {N}}},\,f_m\in \mathcal {M}^{+}(0,1;\downarrow )\cap C([0,1]),\,m\in {\mathbb {N}}$, such that $f_m \nearrow f^*$ as $m\rightarrow +\infty$. Then, by the Fatou property of $\overline{X},\,\Vert f_m\Vert _{\overline{X}}\nearrow \Vert f^*\Vert _{\overline{X}}$ as $m\rightarrow +\infty$. Moreover, by the Monotone Convergence Theorem, $\int _0^{t^n}s^{\frac{k}{n}-1}f_m^*(s)\,\mathrm{d}s\nearrow \int _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s$ as $m\rightarrow +\infty$, for each $t\in (0,1)$. Finally, by the Fatou property of $G,\, \Vert \int _0^{t^n}s^{\frac{k}{n}-1}f_m^*(s)\,\mathrm{d}s\Vert _G\nearrow \Vert \int _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Vert _G$ as $m\rightarrow +\infty$ and the result follows from (4.7). $\square$

Remark 4.2

Our approach can be used to obtain similar results as those of Theorem 4.1 when $X:=X(\Omega )$ is an r.i. BFS on a domain in ${\mathbb {R}}^n$ with minimally smooth boundary.

5 Characterizations of embeddings in particular cases

To find necessary and sufficient conditions for embeddings in question, one has to characterize inequality (1.8). Here, we present some of such characterizations.

We start by recalling a few definitions.

Let $k\in {\mathbb {N}},\,r\in (0,+\infty ]$ and let ${\mathcal L}_r^k$ be the class of all continuous functions $\mu :(0,1)\rightarrow (0,+\infty )$ such that

$$\begin{aligned} \left\| t^{-1/r}\frac{1}{\mu (t)}\right\| _{r;(0,1)}=+\infty \end{aligned}$$

(5.1)

and

$$\begin{aligned} \left\| t^{-1/r}\frac{t^k}{\mu (t)}\right\| _{r;(0,1)}<+\infty . \end{aligned}$$

(5.2)

When $r=+\infty$, we simply write ${\mathcal L}^k$ instead of ${\mathcal L}_r^k={\mathcal L}_\infty ^k$.

If we take $G:=L^r_{\psi }(0,1)$ (the weighted Lebesgue space consisting of those measurable functions $g$ defined on the interval (0,1) such that $\Vert g\psi \Vert _{r;(0,1)}<+\infty$), where $\psi (t):=t^{-1/r}\mu (t)^{-1},\,t\in (0,1)$, then we denote the space $\dot{\Lambda }_{\infty ,G}^{k}({\mathbb {R}}^n)$ by $\dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n)$. Note that (2.8) and (2.9) are satisfied if and only if $\mu \in {\mathcal L}_r^k$.

According to Gogatishvili et al. [18, the end of Section 2], if $r=+\infty$, we shall assume, without loss of generality, in the definition of $\dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}(\overline{{\mathbb {R}}^n})$ that all elements $\mu$ of ${\mathcal L}^k$ are continuous non-decreasing functions on the interval (0,1) such that $\displaystyle \lim _{t\rightarrow 0_+}\mu (t)=0.$

I. First, we apply our results to the case when $G:=L^{\infty }_{1/\mu }(0,1)$, with $\mu \in {\mathcal L}^k$. In such a case $\dot{\Lambda }_{\infty ,G}^k=\dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}({\mathbb {R}}^n)$ and we characterize inequality (4.3) for any r.i. BFS $X$ (cf. (5.4) below).

Theorem 5.1

Let $k,n\in {\mathbb {N}},\,k\le n$, let $X:=X({\mathbb {R}}^n)$ be an r.i. Banach function space and let $\mu \in {\mathcal L}^k$. Then, the following statements are equivalent:

$$\begin{aligned}&\dot{W}^kX\hookrightarrow \dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}({\mathbb {R}}^n),\nonumber \\&W^kX\hookrightarrow \dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}({\mathbb {R}}^n),\nonumber \\&\Bigg \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Bigg \Vert _{L^{\infty }_{1/\mu }(0,1)}\precsim \Vert f\Vert _X\quad \text {for all}\,\,\quad f\in X,\end{aligned}$$

(5.3)

$$\begin{aligned}&\sup _{t\in (0,1)}\frac{1}{\mu (t)}\left\| \chi _{(0,t^n)}(s)s^{\frac{k}{n}-1}\right\| _{\overline{X}'} <+\infty . \end{aligned}$$

(5.4)

Proof

By Theorem 4.1 applied to $G:=L_{1/\mu }^{\infty }(0,1)$, it is sufficient to prove that conditions (5.3) and (5.4) are equivalent.

Since the space $({\mathbb {R}}^n,\mu _n)$ is resonant, see [2, Corollary II.4.4 (the norm of associated space)], an exchange of suprema gives

$$\begin{aligned} \sup _{t\in (0,1)}\frac{1}{\mu (t)} \;\left\| \chi _{(0,t^n)}(s)s^{\frac{k}{n}-1}\right\| _{\overline{X}'}&= \sup _{t\in (0,1)}\frac{1}{\mu (t)} \sup _{\Vert f\Vert _X\le 1}\int \limits _0^{t^n}s^{\frac{k}{n}-1}f^*(s)\,\mathrm{d}s \\&= \sup _{\Vert f\Vert _X\le 1}\sup _{t\in (0,1)}\frac{1}{\mu (t)}\int \limits _0^{t^n}s^{\frac{k}{n}-1}f^*(s)\,\mathrm{d}s \\&= \sup _{\Vert f\Vert _X\le 1}\Big \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Big \Vert _{L^{\infty }_{1/\mu }(0,1)}, \end{aligned}$$

which shows that (5.3) and (5.4) are equivalent. $\square$

Remark 5.2

When $k=1$, we recover the result of [7, Theorem 1.3], where the corresponding spaces are defined on a cube $Q\subset {\mathbb {R}}^n$. We remark that the technique used in their proof cannot be extended to the case when $k>1$.

Example 5.3

Let $k,n\in {\mathbb {N}},\,k\le n$ and let $X:=L_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)$, with $\alpha < 0$. Then, by Opic and Pick [26, Theorem 7.1], $X$ is (equivalent to) a Banach function space and, by Opic and Pick [26, Theorem 6.2 (v)],

$$\begin{aligned} \overline{X}'=L_{1,1;-\alpha }(0,1)\cap L^1(0,+\infty ). \end{aligned}$$

Thus, (5.4) is satisfied if and only if¹

$$\begin{aligned} \mu (t)\succsim t^k\ell ^{-\alpha }(t)\quad \text {for all}\quad t\in (0,1). \end{aligned}$$

Taking $\mu (t):= t^k\ell ^{-\alpha }(t)\text { for all }t\in (0,1)$, we obtain from Theorem 5.1 that

$$\begin{aligned} \dot{W}^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}({\mathbb {R}}^n) \quad \text {and}\quad W^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,\infty }^{k,\mu (\cdot )}({\mathbb {R}}^n). \end{aligned}$$

(5.5)

Together with the facts $\dot{W}^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\subset C_B(B)$ and $W^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\hookrightarrow C_B(B)$ (cf. Theorem 2.1), which hold for any ball $B$ in ${\mathbb {R}}^n$, (5.5) implies that

$$\begin{aligned} \dot{W}^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\subset \Lambda _{\infty ,\infty }^{k,\mu (\cdot )}(\overline{B}) \quad \text {and}\quad W^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,\infty }^{k,\mu (\cdot )}(\overline{B}). \end{aligned}$$

We refer to [20, Section 2] for the definition of the space $\Lambda _{\infty ,\infty }^{k,\mu (\cdot )}(\overline{B})$. Assume in addition that $k>1$. Then, using Marchaud’s inequality for a domain in ${\mathbb {R}}^n$ with minimally smooth boundary (see, for example, [20, Theorem 3.3]), and proceeding as in the proof of [18, Theorem 3.7], we get an analogue of [18, Theorem 3.7] for a domain in ${\mathbb {R}}^n$ with minimally smooth boundary. Consequently,

$$\begin{aligned} \Lambda _{\infty ,\infty }^{k,\mu (\cdot )}(\overline{B})\hookrightarrow \Lambda _{\infty ,\infty }^{1,Id(\cdot )}(\overline{B}). \end{aligned}$$

This means that the functions from the spaces $\dot{W}^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)$ and $W^kL_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)$ are locally Lipschitz in ${\mathbb {R}}^n$ provided that $1<k\le n$ and $\alpha < 0$.

II. Second, we apply our results to the case when $G:=L^{1}_{\psi }(0,1)$, with $\psi (t):=(t\mu (t))^{-1},\,t\in (0,1)$, where $\mu \in {\mathcal L}_1^k$. In such a case $\dot{\Lambda }_{\infty ,G}^k=\dot{\Lambda }_{\infty ,1}^{k,\mu (\cdot )}({\mathbb {R}}^n)$ and, again, we characterize inequality (4.3) for any r.i. BFS $X$ (cf. (5.7) below).

Theorem 5.4

Let $k,n\in {\mathbb {N}},\,k\le n$, let $X:=X({\mathbb {R}}^n)$ be an r.i. Banach function space and let $\mu \in {\mathcal L}_1^k$. Then, the following statements are equivalent:

$$\begin{aligned}&\dot{W}^kX\hookrightarrow \dot{\Lambda }_{\infty ,1}^{k,\mu (\cdot )}({\mathbb {R}}^n),\nonumber \\&W^kX\hookrightarrow \dot{\Lambda }_{\infty ,1}^{k,\mu (\cdot )}({\mathbb {R}}^n),\nonumber \\&\Big \Vert \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\Big \Vert _{L^{1}_{1/(t\mu (t))}(0,1)}\precsim \Vert f\Vert _X\quad \text {for all }f\in X,\end{aligned}$$

(5.6)

$$\begin{aligned}&\left\| \chi _{(0,1)}(s)s^{\frac{k}{n}-1} \int \limits _{s^{1/n}}^1 (t\,\mu (t))^{-1}\, \mathrm{d}t \right\| _{\overline{X}'} <+\infty . \end{aligned}$$

(5.7)

Proof

By Theorem 4.1 applied to $G=L^{1}_{\psi }(0,1)$, with $\psi (t):=(t\mu (t))^{-1},\,t\in (0,1)$, it is sufficient to prove that conditions (5.6) and (5.7) are equivalent.

Suppose first that (5.7) holds. By Fubini’s Theorem,

$$\begin{aligned} \int \limits _0^1 \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s \frac{1}{t\mu (t)}\, \mathrm{d}t&= \int \limits _0^1 s^{\frac{k}{n}-1} f^*(s)\int \limits _{s^{1/n}}^1 \frac{1}{t\mu (t)}\,\mathrm{d}t\;\mathrm{d}s. \end{aligned}$$

(5.8)

Now, Hölder’s inequality and (5.7) imply that

$$\begin{aligned} \mathrm{RHS}(5.8)&\le \Vert f^*\Vert _{\overline{X}} \;\left\| \chi _{(0,1)}(s)s^{\frac{k}{n}-1}\int \limits _{s^{1/n}}^1 \frac{1}{t\mu (t)}\,\mathrm{d}t\right\| _{\overline{X}'} \precsim \Vert f^*\Vert _{\overline{X}} \quad \text{ for } \text{ all } \;\;f\in X, \nonumber \end{aligned}$$

which, together with (5.8), gives (5.6).

Assume now that (5.6) holds. Since $\chi _{(0,1)}(s)s^{\frac{k}{n}-1}\int _{s^{1/n}}^1 (t\mu (t))^{-1}\,\mathrm{d}t \in \mathcal {M}^{+}(0,+\infty ;\downarrow )$, using [2, Corollary 4.4], (5.8), and (5.6), we arrive at

$$\begin{aligned} \mathrm{LHS}(5.7)&=\sup _{\Vert f\Vert \le 1} \int \limits _0^{\infty }\chi _{(0,1)}(s)s^{\frac{k}{n}-1} f^*(s)\int \limits _{s^{1/n}}^1 (t\mu (t))^{-1}\,\mathrm{d}t \,\mathrm{d}s\\&=\sup _{\Vert f\Vert \le 1}\int \limits _0^1 \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s \frac{1}{t\mu (t)}\, \mathrm{d}t\\&=\sup _{\Vert f\Vert \le 1}\left\| \int \limits _0^{t^n}s^{\frac{k}{n}-1} f^*(s)\,\mathrm{d}s\right\| _{L^{1}_{1/(t \mu (t))}(0,1)}\precsim 1, \end{aligned}$$

and (5.7) is verified. $\square$

Example 5.5

Let $k,n\in {\mathbb {N}},\,k< n$ and let $X:=L_{n/k,1;\alpha }({\mathbb {R}}^n)$, with $\alpha > 0$. Then, by Opic and Pick [26, Theorem 6.6 (i)],

$$\begin{aligned} \overline{X}'=L_{(n/k)',\infty ;-\alpha }(0,+\infty )=L_{n/(n-k),\infty ;-\alpha }(0,+\infty ). \end{aligned}$$

Thus, (5.7) is satisfied if and only if

$$\begin{aligned} \sup _{s\in (0,1)}\ell ^{-\alpha } (t)\int \limits _{s^{1/n}}^1 (\mu (t))^{-1}t^{-1}\, \mathrm{d}t <+\infty . \end{aligned}$$

(5.9)

In (5.9), the optimal choice is $\mu (t):= \ell ^{1-\alpha }(t)\text { for all }t\in (0,1)$. Consequently, by Theorem 5.4,

$$\begin{aligned} \dot{W}^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,1}^{k,\mu (\cdot )}({\mathbb {R}}^n) \quad \text {and}\quad W^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,1}^{k,\mu (\cdot )}({\mathbb {R}}^n). \end{aligned}$$

(5.10)

Together with the facts $\dot{W}^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\subset C_B(B)$ and $W^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\hookrightarrow C_B(B)$ (cf. Theorem 2.1), which hold for any ball $B$ in ${\mathbb {R}}^n$, (5.10) implies that

$$\begin{aligned} \dot{W}^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\subset \Lambda _{\infty ,1}^{k,\mu (\cdot )}(\overline{B}) \quad \text {and}\quad W^kL_{n/k,1;\alpha }({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,1}^{k,\mu (\cdot )}(\overline{B}) \end{aligned}$$

(we refer to [20, Section 2] for the definition of the space $\Lambda _{\infty ,1}^{k,\mu (\cdot )}(\overline{B})$). Moreover, using Marchaud’s inequality for a domain in ${\mathbb {R}}^n$ with minimally smooth boundary (see, for example, [20, Theorem 3.3]), and proceeding as in the proof of [18, Theorem 3.7], we get an analogue of [18, Corollary 3.8] for a domain in ${\mathbb {R}}^n$ with minimally smooth boundary. Consequently,

$$\begin{aligned} \Lambda _{\infty ,1}^{k,\mu (\cdot )}(\overline{B})= \Lambda _{\infty ,1}^{1,\mu (\cdot )}(\overline{B}). \end{aligned}$$

This means that we can reduce the order of the modulus of continuity in this case.

We refer to Corollary 5.18 and Remark 5.19 (i) for an analogue of Example 5.5 with $k=n$.

III. Next, we apply Theorem 4.1 to the case when $X$ is the classical Lorentz space $\Lambda ^q(\varphi )$, the set of $\mu _n$-measurable functions $f$ on ${\mathbb {R}}^n$ such that

$$\begin{aligned} \Vert f\Vert _{\Lambda ^q(\varphi )}:=\left( \int \limits _0^{+\infty }{(f^{*}(t))}^q \varphi (t)\,\mathrm{d}t\right) ^{1/q} <+\infty \text {if}\quad \,\, q\in [1,+\infty ),\\ \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{\Lambda ^q(\varphi )}&:=\mathop {\mathrm{ess\,sup}}_{t\in (0,+\infty )}f^{*}(t) \varphi (t)<+\infty \text {if}\quad \,\, q=+\infty , \end{aligned}$$

where $\varphi$ is a weight from $L_{loc}^1[0,+\infty )$.

If $1<q<+\infty$, then, by Sawyer [27, Theorem 4], $\Lambda ^q(\varphi )$ is (equivalent to) an r.i. Banach function space if and only if $\varphi$ satisfies the $B_q$ condition:

$$\begin{aligned} r^q\int \limits _r^{+\infty }{\frac{\varphi (t)}{t^q}}\,\mathrm{d}t\precsim \int \limits _0^r\varphi (t)\,\mathrm{d}t\quad \text {for all }\quad \,\,r>0. \end{aligned}$$

(5.11)

If $q=1$, then, by Carro et al. [5, Theorem 2.3], $\Lambda ^1(\varphi )$ is an r.i. Banach function space if and only if $\varphi$ satisfies the $B_{1,\infty }$ condition:

$$\begin{aligned} \frac{1}{r}\int \limits _0^{r}\varphi (t)\,\mathrm{d}t \precsim \frac{1}{s}\int \limits _0^s\varphi (t)\,\mathrm{d}t\quad \text {for all }\quad \,\,0<s\le r<+\infty . \end{aligned}$$

(5.12)

When $q=+\infty$, then $\Lambda ^\infty (\varphi )=\Lambda ^\infty (\tilde{\varphi })$, with $\displaystyle \tilde{\varphi }(t):=\mathop {\mathrm{ess\,sup}}_{0<s<t}\varphi (s),\,t>0$, and thus we shall assume, without loss of generality, that $\varphi$ is a non-decreasing function on $(0,+\infty )$. The space $\Lambda ^\infty (\varphi )$ is (equivalent to) an r.i. Banach function space if and only if

$$\begin{aligned} \int \limits _t^{+\infty }\frac{\varphi (s)}{s^2}\,\mathrm{d}s\precsim \frac{\varphi (t)}{t}, \quad t>0. \end{aligned}$$

(5.13)

Note that if $\varphi$ is a smooth non-decreasing function on $[0,+\infty )$ with $\varphi (0)=0$, condition (5.13) means that $\varphi '\in B_1$ (cf. [28]).

Let $b\in SV(0,+\infty ),\,1\le p\le +\infty$. If $1\le q<+\infty$ and $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, then $\Lambda ^q(\varphi )$ coincides with the Lorentz–Karamata space $L_{p,q;b}({\mathbb {R}}^n)$. When $q\in (1,+\infty )$, condition (5.11) is satisfied if and only if $p>1$. When $q=1$, condition (5.12) is satisfied if and only if either $p>1$ or $p=1$ and $b$ is equivalent to a non-increasing function on $(0,+\infty )$. If $q=+\infty$ and $\varphi (t):=t^{{1/p}}\,b(t) ,\,t>0$, then $\Lambda ^q(\varphi )$ coincides with the Lorentz–Karamata space $L_{p,q;b}({\mathbb {R}}^n)=L_{p,\infty ;b}({\mathbb {R}}^n)$.

In the following theorem, we suppose that $X:=\Lambda ^\infty (\varphi )$ and we characterize inequality (4.3) for a general quasi-Banach lattice from Theorem 4.1.

Theorem 5.6

Let $k,n\in {\mathbb {N}},\,k\le n$. Let $\varphi$ be a non-decreasing weight, finite a.e. on $(0,+\infty )$, satisfying (5.13) and let $G:=G(0,1)$ be a quasi-Banach lattice of functions over (0,1) satisfying (2.8), (2.9) and the Fatou property. Then, the following statements are equivalent:

$$\begin{aligned}&\dot{W}^k\Lambda ^\infty (\varphi )\hookrightarrow \dot{\Lambda }_{\infty ,G}^{k}({\mathbb {R}}^n),\end{aligned}$$

(5.14)

$$\begin{aligned}&W^k\Lambda ^\infty (\varphi )\hookrightarrow \dot{\Lambda }_{\infty ,G}^{k}({\mathbb {R}}^n),\end{aligned}$$

(5.15)

$$\begin{aligned}&\left\| \int \limits _0^{t^n} s^{\frac{k}{n}-1}f^{*}(s)\,\mathrm{d}s\right\| _{G} \precsim \;\Vert f\Vert _{\Lambda ^\infty (\varphi )}\quad \text {for all } \; f\in \Lambda ^\infty (\varphi ),\end{aligned}$$

(5.16)

$$\begin{aligned}&\left\| \int \limits _0^{t^n} s^{\frac{k}{n}-1}(\varphi (s))^{-1}\,\mathrm{d}s\right\| _{G}< +\infty . \end{aligned}$$

(5.17)

Proof

The proof is elementary and we leave the details to the reader. $\square$

The next theorem provides a complete characterization of (4.3) when $X:=\Lambda ^q(\varphi ),\,q\in [1,+\infty ]$, and $G:=L^r_{\psi }(0,1)$, with $\psi (t):=t^{-1/r}\mu (t)^{-1},\,t\in (0,1),\,\mu \in {\mathcal L}_r^k,\,r\in (0,+\infty ]$, i.e., $\dot{\Lambda }_{\infty ,G}^{k}({\mathbb {R}}^n)=\dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n)$.

Theorem 5.7

Let $k,n\in {\mathbb {N}},\,k\le n,\,q\in [1,+\infty ],\,r\in (0,+\infty ]$ and let $\mu \in {\mathcal L}_r^k$. Let $\varphi$ be a weight from $L_{loc}^1[0,+\infty )$ and suppose that $\varphi$ satisfies the $B_q$ condition if $1<q<+\infty$, the $B_{1,\infty }$ condition if $q=1$, and (5.13) if $q=+\infty$. Then, the following statements are equivalent:

$$\begin{aligned}&\dot{W}^k\Lambda ^q(\varphi )\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

(5.18)

$$\begin{aligned}&W^k\Lambda ^q(\varphi )\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

(5.19)

$$\begin{aligned}&\left\| t^{-\frac{1}{r}}(\mu (t))^{-1}\int \limits _0^{t^n} s^{\frac{k}{n}-1}f^{*}(s)\,\mathrm{d}s\right\| _{r;(0,1)} \precsim \;\Vert f\Vert _{\Lambda ^q(\varphi )}\quad \text {for all } \; f\in \Lambda ^q(\varphi ). \end{aligned}$$

(5.20)

Moreover,

• (i) If  $1\le q\le r<+\infty$, then inequality (5.20) holds if and only if

  $$\begin{aligned}&\sup _{t\in (0,1)}\left\| s^{k-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(0,t)} \left( \int \limits _0^{t^n} \varphi (s)\,\mathrm{d}s \right) ^{-\frac{1}{q}} <+\infty \end{aligned}$$

  and

  $$\begin{aligned}&\sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} \left\| (\varphi (s))^{\frac{1}{q'}}s^{\frac{k}{n}} \left( \int \limits _0^{s} \varphi (\xi )\,d\xi \right) ^{-1} \right\| _{q';(0,{t^n})} <+\infty . \end{aligned}$$

• (ii) If  $1\le q< +\infty$ and $r=+\infty$, then inequality (5.20) holds if and only if

  $$\begin{aligned}&\sup _{t\in (0,1)}(\mu (t))^{-1} \left\| s^{\frac{k}{n}-\frac{1}{q'}} \left( \int \limits _0^{s} \varphi (\xi )\,d\xi \right) ^{-\frac{1}{q}} \right\| _{q';(0,{t^n})} <+\infty . \end{aligned}$$

• (iii)   If  $1\le q <+\infty ,\,0<r<q$ and $u=(rq)/(q-r)$, then inequality (5.20) holds if and only if

  $$\begin{aligned}&\int \limits _0^1\left\| s^{k-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(0,t)}^{u-r} \left( \int \limits _0^{t^n} \varphi (s)\,\mathrm{d}s\right) ^{-\frac{u}{q}} \,t^{rk-1}(\mu (t))^{-r} \,\mathrm{d}t <+\infty \end{aligned}$$

  and

  $$\begin{aligned}&\int \limits _0^1\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)}^{u-r} \left\| (\varphi (s))^{\frac{1}{q'}}s^{\frac{k}{n}} \left( \int \limits _0^{s} \varphi (\xi )\,d\xi \right) ^{-1} \right\| _{q';(0,{t^n})}^u\! t^{-1}(\mu (t))^{-r}\,\mathrm{d}t <+\infty . \end{aligned}$$

• (iv)   If  $q =+\infty$ and $0<r\le +\infty$, then inequality (5.20) holds if and only if

  $$\begin{aligned}&\left\| t^{-\frac{1}{r}}(\mu (t))^{-1} \int \limits _0^{t^n} s^{k/n-1} (\varphi (s))^{-1}\,\mathrm{d}s \right\| _{r;(0,1)} <+\infty . \end{aligned}$$

Proof

By Theorem 4.1 applied to $X:=\Lambda ^q(\varphi )$ and to $G:=L^{r}_{v}(0,1)$, with $v(t):=t^{-1/r}\mu (t)^{-1},\,t\in (0,1)$, statements (5.18)–(5.20) are mutually equivalent.

On the other hand, one can verify (using change of variables and choosing $g$ such that $g^*(t):=f^*(t^{n/k})$) that (5.20) is equivalent to

$$\begin{aligned} \left\| t^{1-\frac{1}{r}}(\mu (t^{\frac{1}{k}}))^{-1}g^{**}(t)\right\| _{r;(0,1)} \precsim \;\Vert g\Vert _{\Lambda ^q(\tilde{\varphi })}\quad \text {for all } \; g\in \Lambda ^q(\tilde{\varphi }), \end{aligned}$$

where $\tilde{\varphi }(t):=\varphi (t^{\frac{n}{k}})t^{\frac{n}{k}-1},\,t>0$. The results in (i)–(iii) are now a consequence of [6, Theorems 4.1, 4.2] or [16, Theorem 3.1]) with weights $v(t):=\tilde{\varphi }(t),\,t>0,\, w(t):=t^{r-1}(\mu (t^{\frac{1}{k}}))^{-r}\chi _{(0,1)}(t),\,t>0$, if $r<+\infty$, and $w(t):=t\,(\mu (t^{\frac{1}{k}}))^{-1}\chi _{(0,1)}(t),\,t>0$, if $r=+\infty$. In the case (ii), using integration by parts when $q>1$ and exchange of suprema when $q=1$, one can show that the conditions given in the mentioned references are equivalent to the one in (ii). The case (iv) follows easily from Theorem 5.6. $\square$

The next corollaries of Theorem 5.7 concern the case when $X:=L_{p,q;b}({\mathbb {R}}^n)$ provided that $p\in [\frac{n}{k},+\infty ]$. In the first corollary, $p\in (\frac{n}{k},+\infty )$; in the second one, $p=+\infty$; in the third corollary, $p=\frac{n}{k}$ with $k<n$; and in the last one, $p=\frac{n}{k}$ with $k=n$ (i.e., $p=1$).

Corollary 5.8

Let $k,n\in {\mathbb {N}},\,k\le n,\,p\in (\frac{n}{k},+\infty ),\,q\in [1,+\infty ],\,b\in SV(0,+\infty ),\,r\in (0,+\infty ]$ and $\mu \in {\mathcal {L}}_r^k$.

• (i) If $1\le q\le r\le +\infty$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.21)
  $$\begin{aligned}&W^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.22)
  $$\begin{aligned}&\sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} t^{k-\frac{n}{p}}(b(t^n))^{-1}<+\infty . \end{aligned}$$

  (5.23)

• (ii) If $\;0<r<q\le +\infty$ and $\frac{1}{u}:=\frac{1}{r}-\frac{1}{q}$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.24)
  $$\begin{aligned}&W^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.25)
  $$\begin{aligned}&\int \limits _0^1\Big (\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)}\, t^{k-\frac{n}{p}}(b(t^n))^{-1}\Big )^{u}\,\frac{\mathrm{d}t}{t}<+\infty . \end{aligned}$$

  (5.26)

Proof

The result is a consequence of Theorem 5.7. To see it, we distinguish several cases:

(i-1) If $1\le q\le r<+\infty$, then the second condition in Theorem 5.7 (i) is equivalent to (5.23). Moreover, in our case when $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, the second condition implies the first one in Theorem 5.7 (i). (For a convenience of the reader, note that to prove it, one uses the estimate $s^{rk}\approx \int _0^s\sigma ^{rk-1}\,d\sigma$, the Fubini theorem and the estimate $\Vert s^{-\frac{1}{r}}(\mu (s))^{-1}\Vert _{r;(t,1)}\precsim t^{\frac{n}{p}-k}b(t^n)$, which follows from (5.23).)

(i-2) If $1\le q<+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (ii) is equivalent to (5.23).

(i-3) If $q=+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (iv) is equivalent to (5.23).

(ii-1) If $1\le q<+\infty$ and $0<r<q$, then, using integration by parts, one can show that the second condition in Theorem 5.7 (iii) is equivalent to (5.26). Moreover, in our case when $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, the second condition implies the first one in Theorem 5.7 (iii).

(ii-2) If $0<r<q=+\infty$, then the condition in Theorem 5.7 (iv) is equivalent to (5.26). $\square$

Remark 5.9

To make comments on sharpness and optimality of embeddings, we recall these notions.

The statement that an embedding

$$\begin{aligned} W^kL_{p,q;b}({\mathbb {R}}^n) \hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n), \end{aligned}$$

(5.27)

with $r\ge q$ and $\mu =\lambda :=\lambda (k,p,q,r,b)\in {\mathcal L}_r^k$, is sharp (with respect to $\mu \in {\mathcal L}_r^k$) means:

• (i) Embedding (5.27) holds with $\mu =\lambda$.

• (ii) If embedding (5.27) holds with $r\ge q$ and some $\mu \in {\mathcal L}_r^k$, then $\dot{\Lambda }_{\infty ,r}^{k,\lambda (\cdot )}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n)$.

Similarly, the statement that embedding (5.27) is optimal with $r=q$ and $\mu =\lambda :=\lambda (k,p,q,b)\in {\mathcal L}_q^k$ means:

• (iii) Embedding (5.27) holds with $r=q$ and $\mu =\lambda$.

• (iv) If embedding (5.27) holds with some $r\in (0,+\infty )$ and $\mu \in {\mathcal L}_r^k$, then $\dot{\Lambda }_{\infty ,q}^{k,\lambda (\cdot )}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n)$. (This means that the space $\dot{\Lambda }_{\infty ,q}^{k,\lambda (\cdot )}({\mathbb {R}}^n)$ is the smallest target for the embedding (5.27) in the scale of spaces $\dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n)$, with $r\in (0,+\infty )$ and $\mu \in {\mathcal L}_r^k$.)

An analogous explanation of sharpness and optimality concerns the embedding

$$\begin{aligned} \dot{W}^kL_{p,q;b}({\mathbb {R}}^n) \hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n). \end{aligned}$$

Remark 5.10

In Corollary 5.8, the sharp embeddings are obtained by taking $r\ge q$ and $\mu (t)=\lambda (t):=t^{k-\frac{n}{p}}(b(t^n))^{-1},\,t\in (0,1)$. Moreover, we obtain the optimal embedding by putting $r=q$. Note that $\lambda \in {\mathcal {L}}_r^k$ for any $r\in (0,+\infty ]$ (recall that $b$ is continuous, cf. (2.3)). Moreover, taking $b\equiv 1$, we get the optimal embedding

$$\begin{aligned} \dot{W}^kL_{p,q}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\lambda (\cdot )}({\mathbb {R}}^n), \end{aligned}$$

with $\lambda (t):=t^{k-\frac{n}{p}},\,t\in (0,1)$, provided that $k,n\in {\mathbb {N}},\,k\le n,\,p\in (\frac{n}{k}, +\infty )$ and $q\in [1,+\infty ]$. This result coincides with the first part of [23, Proposition 6.1].

Example 5.11

If $k,n \in {\mathbb {N}},\,k+1\le n$ and $\alpha \in {\mathbb {R}}$, then, from Corollary 5.8 and Remark 5.10, we obtain the optimal embeddings

$$\begin{aligned} W^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\!\hookrightarrow \! \dot{\Lambda }_{\infty ,n/k}^{k+1,\lambda (\cdot )}({\mathbb {R}}^n)\quad {and}\quad \dot{W}^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\!\hookrightarrow \! \dot{\Lambda }_{\infty ,n/k}^{k+1,\lambda (\cdot )}({\mathbb {R}}^n), \end{aligned}$$

where

$$\begin{aligned} \lambda (t):=t\,\ell ^{-\alpha }(t)\quad \text {for all} \quad t\in (0,1). \end{aligned}$$

Since also

$$\begin{aligned} W^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\hookrightarrow L^{\infty }({\mathbb {R}}^n) \end{aligned}$$

and

$$\begin{aligned} \dot{W}^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\hookrightarrow L^{\infty }(B)\quad \text {for any ball}\, B\subset {\mathbb {R}}^n, \end{aligned}$$

we get

$$\begin{aligned} W^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{{\mathbb {R}}^n}) \end{aligned}$$

(5.28)

and

$$\begin{aligned} \dot{W}^{k+1}L^{n/k}(\log L)^{\alpha }({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{B})\quad \text {for any ball}\, B\subset {\mathbb {R}}^n. \end{aligned}$$

(5.29)

Put $p=\frac{n}{k}$ and let $\alpha \le 1-\frac{k}{n}$. By Gogatishvili et al. [18, Section 3] and its analogues with ${\mathbb {R}}^n$ replaced by $B$,

$$\begin{aligned} \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{{\mathbb {R}}^n})=\Lambda _{\infty ,n/k}^{2,\lambda (\cdot )}(\overline{{\mathbb {R}}^n})\hookrightarrow \Lambda _{\infty ,r}^{1,\lambda _{pr}(\cdot )}(\overline{{\mathbb {R}}^n})\hookrightarrow \Lambda _{\infty ,\infty }^{1,\lambda _{p\infty }(\cdot )}(\overline{{\mathbb {R}}^n}) \end{aligned}$$

(5.30)

and

$$\begin{aligned} \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{B})=\Lambda _{\infty ,n/k}^{2,\lambda (\cdot )}(\overline{B})\hookrightarrow \Lambda _{\infty ,r}^{1,\lambda _{pr}(\cdot )}(\overline{B})\hookrightarrow \Lambda _{\infty ,\infty }^{1,\lambda _{p\infty }(\cdot )}(\overline{B})\quad \text {for any ball}\, B\subset {\mathbb {R}}^n, \end{aligned}$$

with

$$\begin{aligned} \lambda _{pr}(t):=t\;\left( \ell (t)\right) ^{1-k/n+1/r-\alpha },\quad t\in (0,1),\;r\in [n/k,+\infty ]. \end{aligned}$$

Taking $\alpha =0$, we arrive at

$$\begin{aligned} W^{k+1,n/k}({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,n/k}^{k+1,Id(\cdot )}(\overline{{\mathbb {R}}^n}) \quad \text {and}\quad \dot{W}^{k+1,n/k}({\mathbb {R}}^n)\hookrightarrow \Lambda _{\infty ,n/k}^{k+1,Id(\cdot )}(\overline{B}). \end{aligned}$$

Consequently, the Brézis–Wainger result (cf. [4, Corollary 5]) on “almost” Lipschitz continuity of functions from $W^{k+1,n/k}({\mathbb {R}}^n)$ is a consequence of the better embedding (5.28) whose target space is the Zygmund-type space $\Lambda _{\infty ,n/k}^{2,Id(\cdot )}(\overline{{\mathbb {R}}^n})$ (cf. (5.30)), where the symbol $Id$ stands for the identity map on the interval $(0,1)$.

If $p=\frac{n}{k}$ and $\alpha > 1-\frac{k}{n}$, then, by Gogatishvili et al. [18, Theorem 3.7] and its analogues with ${\mathbb {R}}^n$ replaced by $B,$

$$\begin{aligned} \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{{\mathbb {R}}^n})\hookrightarrow \Lambda _{\infty ,\infty }^{1,Id(\cdot )}(\overline{{\mathbb {R}}^n})=Lip(\overline{{\mathbb {R}}^n}) \end{aligned}$$

and

$$\begin{aligned} \Lambda _{\infty ,n/k}^{k+1,\lambda (\cdot )}(\overline{B})\hookrightarrow \Lambda _{\infty ,\infty }^{1,Id(\cdot )}(\overline{B})=Lip(\overline{B})\quad \text {for any ball}\, B\subset {\mathbb {R}}^n. \end{aligned}$$

Corollary 5.12

Let $k,n\in {\mathbb {N}},\,k\le n,\,q\in [1,+\infty ],\,b\in SV(0,+\infty ),\,r\in (0,+\infty ]$ and $\mu \in {\mathcal {L}}_r^k$.

• (i) If  $1\le q\le r\le +\infty$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{\infty ,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.31)
  $$\begin{aligned}&W^kL_{\infty ,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.32)
  $$\begin{aligned}&\sup _{t\in (0,1)} \left\| s^{k-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(0,t)} \left\| s^{-\frac{1}{q}}b(s)\right\| _{q;(0,t^n)}^{-1}<+\infty . \end{aligned}$$

  (5.33)

• (ii) If $\;0<r<q\le +\infty$ and $\frac{1}{u}:=\frac{1}{r}-\frac{1}{q}$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{\infty ,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.34)
  $$\begin{aligned}&W^kL_{\infty ,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.35)
  $$\begin{aligned}&\int \limits _0^1\Big (\left\| s^{k-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(0,t)}^{r/q}\, \left\| s^{-\frac{1}{q}}b(s)\right\| _{q;(0,t^n)}^{-1}\Big )^{u}t^{kr-1}(\mu (t))^{-r}\,\mathrm{d}t<+\infty .\qquad \end{aligned}$$

  (5.36)

Proof

The result is a consequence of Theorem 5.7. To see it, we distinguish several cases:

(i-1) If $1\le q\le r<+\infty$, then the first condition in Theorem 5.7 (i) is equivalent to (5.33). Moreover, in our case when $\varphi (t):=t^{-1}\,b^q(t) ,\,t>0$, the first condition implies the second one in Theorem 5.7 (i).

(i-2) If $1\le q<+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (ii) is equivalent to (5.33).

(i-3) If $q=+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (iv) is equivalent to (5.33).

(ii-1) If $1\le q<+\infty$ and $0<r<q$, then the first condition in Theorem 5.7 (iii) is equivalent to (5.36). Moreover, in our case when $\varphi (t):=t^{-1}\,b^q(t) ,\,t>0$, the first condition implies the second one in Theorem 5.7 (iii).

(ii-2) If $0<r<q=+\infty$, then the condition in Theorem 5.7 (iv) is equivalent to (5.36). $\square$

Remarks 5.13

• (i) When $q<+\infty$, then in Corollary 5.12 the sharp embeddings are obtained by taking $r\ge q$ and $\mu :=\lambda _{qr}$, where

  $$\begin{aligned} \lambda _{qr}(t):=t^k(b(t^n))^{-q/r}\left( \int \limits _0^{t^n}b^q(s)\, \frac{\mathrm{d}s}{s}\right) ^{\frac{1}{r}-\frac{1}{q}}, \quad t\in (0,1), \end{aligned}$$

  (5.37)

  provided that $\left\| s^{-\frac{1}{q}}b(s)\right\| _{q;(0,1)}<+\infty$. Note that $\lambda _{qr}\in {\mathcal {L}}_r^k$ for any $r\in (0,+\infty ]$ (recall that $b$ is continuous, cf. (2.3)). Moreover, we obtain the optimal embedding by putting $r=q$.

• (ii) When $q=+\infty$, we assume without loss of generality that $b$ is equivalent to non-decreasing function. Then, in Corollary 5.12, the optimal embedding is obtained by taking $\mu :=\lambda _{\infty \infty }$, where

  $$\begin{aligned} \lambda _{\infty \infty }(t):= t^{k}(b(t^n))^{-1}, \quad t\in (0,1), \end{aligned}$$

  and $r=+\infty$ (note that $\lambda _{\infty \infty }\in {\mathcal {L}}^k$).

In Example 5.3, we have put $X:=L_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)$, with $\alpha < 0$. In the next example, we take $X:=L_{\infty ,q;\alpha }({\mathbb {R}}^n)$, with $q\in [1,+\infty )$ and $\alpha < -1/q$ (which is a smaller space than $L_{\infty ,\infty ;\alpha }({\mathbb {R}}^n)$).

Example 5.14

Let $k, n\in {\mathbb {N}},\,k\le n,\,1\le q\le r \le +\infty$, $\alpha <-1/q,\,\mu \in {\mathcal L}_r^k$ and let $b(t):=\ell ^{\alpha }(t),\,t\in (0,+\infty )$. Then, by Corollary 5.12 (i), embeddings (5.31) and (5.32) hold provided that

$$\begin{aligned} \sup _{t\in (0,1)}\left\| s^{k-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(0,t)} \ell ^{-\alpha -\frac{1}{q}}(t)<+\infty . \end{aligned}$$

(5.38)

Hence, by Remarks 5.13, taking $\mu (t)=\lambda _{qr}(t):=t^k (\ell (t))^{-\alpha -\frac{1}{q}+\frac{1}{r}},\,t\in (0,1)$, we obtain sharp embeddings

$$\begin{aligned} \dot{W}^kL_{\infty ,q;\alpha }({\mathbb {R}}^n) \hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\lambda _{qr}(\cdot )}({\mathbb {R}}^n) \qquad \hbox {and}\qquad W^kL_{\infty ,q;\alpha }({\mathbb {R}}^n) \hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\lambda _{qr}(\cdot )}({\mathbb {R}}^n), \end{aligned}$$

which are optimal if $r=q$.

Corollary 5.15

Let $k,n\in {\mathbb {N}},\,k<n,\,p=\frac{n}{k},\,q\in [1,+\infty ],\,r\in (0,+\infty ]$, $\mu \in {\mathcal {L}}_r^k$ and let $b\in SV(0,+\infty )$ be such that $\Vert t^{-\frac{1}{q'}}(b(t))^{-1}\Vert _{q';(0,1)}<+\infty$.

• (i) If  $1\le q\le r\le +\infty$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.39)
  $$\begin{aligned}&W^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.40)
  $$\begin{aligned}&\sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} \left\| s^{-\frac{1}{q'}}(b(s))^{-1} \right\| _{q';(0,t^n)} <+\infty . \end{aligned}$$

  (5.41)

• (ii) If  $0<r<q\le +\infty ,\,q> 1$ and $\frac{1}{u}:=\frac{1}{r}-\frac{1}{q}$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.42)
  $$\begin{aligned}&W^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.43)
  $$\begin{aligned}&\displaystyle {\int \limits _0^{1}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)}^{u} \, \left\| s^{-\frac{1}{q'}}(b(s))^{-1} \right\| _{q';(0,t^n)}^{u-q'}\, (b(t^n))^{-q'}\,\frac{\mathrm{d}t}{t}<+\infty . }\qquad \end{aligned}$$

  (5.44)

• (iii) If  $q=1,\,0<r<1$ and $\frac{1}{u}:=\frac{1}{r}-1$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.45)
  $$\begin{aligned}&W^kL_{p,q;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{k,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.46)
  $$\begin{aligned}&\int \limits _0^1\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)}^{ur} \left\| (b(s))^{-1} \right\| _{\infty ;(0,t^n)}^{u}\,(\mu (t))^{-r}\,\frac{\mathrm{d}t}{t} <+\infty . \end{aligned}$$

  (5.47)

Proof

The result is a consequence of Theorem 5.7. To see it, we distinguish several cases:

(i-1) If $1\le q\le r<+\infty$, then the second condition in Theorem 5.7 (i) is equivalent to (5.41). Moreover, in our case when $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, the second condition implies the first one in Theorem 5.7 (i).

(i-2) If $1\le q<+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (ii) is equivalent to (5.41).

(i-3) If $q=+\infty$ and $r=+\infty$, then the condition in Theorem 5.7 (iv) is equivalent to (5.41).

(ii-1) If $1< q<+\infty$ and $0<r<q$, then, using integration by parts, one can show that the second condition in Theorem 5.7 (iii) is equivalent to (5.44). Moreover, in our case when $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, the second condition implies the first one in Theorem 5.7 (iii).

(ii-2) If $0<r<q=+\infty$, then, using integration by parts, one can show that the condition in Theorem 5.7 (iv) is equivalent to (5.44).

(iii) If $q=1$ and $0<r<1$, then the second condition in Theorem 5.7 (iii) is equivalent to (5.47). Moreover, in our case when $\varphi (t):=t^{{q/p}-1}\,b^q(t) ,\,t>0$, the second condition implies the first one in Theorem 5.7 (iii). $\square$

Remark 5.16

When $1=q\le r\le +\infty$ in Corollary 5.15, on exchanging suprema, we can rewrite (5.41) as

$$\begin{aligned} \sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} (b(t^n))^{-1} <+\infty . \end{aligned}$$

By (2.1) and (2.2), we can also rewrite (5.41) as

$$\begin{aligned} \sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} t^{-1}\int \limits _0^t\sup _{s\in (0,y)}(b(s^n))^{-1}\,\mathrm{d}y <+\infty \end{aligned}$$

and use this to calculate the optimal $\mu$ in Corollary 5.15.

Remarks 5.17

• (i) When $q>1$, then in Corollary 5.15 the sharp embeddings are obtained by taking $r\ge q$ and $\mu :=\lambda _{qr}$, where

  $$\begin{aligned} \lambda _{qr}(t):=(b(t^n))^{q'/r}\left( \int \limits _0^{t^n}(b(s))^{-q'}\, \frac{\mathrm{d}s}{s}\right) ^{\frac{1}{q'}+\frac{1}{r}}, \quad t\in (0,1). \end{aligned}$$

  (5.48)

  Note that $\lambda _{qr}\in {\mathcal {L}}_r^k$ for any $r\in (0,+\infty ]$ (recall that $b$ is continuous, cf. (2.3)). Moreover, we obtain the optimal embedding by putting $r=q$.

• (ii) When $q=1$ and $b(t)\rightarrow +\infty$ as $t\rightarrow 0_+$, then in Corollary 5.15 the embeddings are sharp if $r\ge 1$ and $\mu \in {\mathcal {L}}_r^k$ is given by $\mu :=\lambda _{1r}$, where

  $$\begin{aligned} \lambda _{1r}(t)\!&:= \! t^{-1}\left( \int \limits _0^t\sup _{s\in (0,y)}(b(s^n))^{-1}\,\mathrm{d}y \right) ^{1+\frac{1}{r}} \nonumber \\&\quad \times \left( t \sup _{s\in (0,t)} (b(s^n))^{-1}-\int \limits _0^t\sup _{s\in (0,y)}(b(s^n))^{-1}\,\mathrm{d}y \right) ^{-\frac{1}{r}}, \quad t\in (0,1).\qquad \end{aligned}$$

  (5.49)

  Moreover, the embedding is optimal if $r=1$. For example, if $b(t):=\ell ^{\alpha }(t),\,t>0$, with $\alpha >0$, then $\lambda _{1r}(t)=\ell ^{-\alpha +\frac{1}{r}}(t),\,t\in (0,1)$, and $\lambda _{1r}\in {\mathcal {L}}_r^k$.

The next corollary is an analogue of Corollary 5.15 and concerns the case when $k=n$.

Corollary 5.18

Let $n\in {\mathbb {N}},\,r\in (0,+\infty ],\,\mu \in {\mathcal {L}}_r^n$ and let $b\in SV(0,+\infty )$ be equivalent to a non-increasing function.

• (i) If  $1\le r\le +\infty$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^nL_{1,1;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{n,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.50)
  $$\begin{aligned}&W^nL_{1,1;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{n,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.51)
  $$\begin{aligned}&\sup _{t\in (0,1)}\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)} (b(t^n))^{-1} <+\infty . \end{aligned}$$

  (5.52)

• (ii) If  $0<r<1$ and $\frac{1}{u}:=\frac{1}{r}-1$, then the following statements are equivalent:

  $$\begin{aligned}&\dot{W}^nL_{1,1;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{n,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.53)
  $$\begin{aligned}&W^nL_{1,1;b}({\mathbb {R}}^n)\hookrightarrow \dot{\Lambda }_{\infty ,r}^{n,\mu (\cdot )}({\mathbb {R}}^n),\end{aligned}$$

  (5.54)
  $$\begin{aligned}&\int \limits _0^1\left\| s^{-\frac{1}{r}}(\mu (s))^{-1}\right\| _{r;(t,1)}^{ur} (b(t^n))^{-u}\,(\mu (t))^{-r}\,\frac{\mathrm{d}t}{t} <+\infty . \end{aligned}$$

  (5.55)

Proof

(i-1) When $r=+\infty$, the result is a consequence of Theorem 5.7 (ii) with $q=1$ and $k=n$.

In this case, the result also follows from Theorem 5.1 with $X=L_{1,1;b}({\mathbb {R}}^n)$, where $b\in SV(0,+\infty )$ is equivalent to a non-increasing function. Note that (cf. [6, Theorem 9.1 (i)])

$$\begin{aligned} \Vert g\Vert _{X'}\approx \sup _{t>0}{g^{**}(t)(b(t))^{-1}}. \end{aligned}$$

(i-2) When $1\le r<+\infty$, the result is a consequence of Theorem 5.7 (i) with $q=1$ and $k=n$. Indeed, the second condition in Theorem 5.7 (i) is equivalent to (5.52) and, in our case (that is, when $\varphi (t):=b(t) ,\,t>0$), the second condition in Theorem 5.7 (i) implies the first one.

(ii) When $0<r<1$, the result is a consequence of Theorem 5.7 (iii) with $q=1$ and $k=n$. In fact, the second condition in Theorem 5.7 (iii) is equivalent to (5.55). Finally, in our case (that is, when $\varphi (t):=b(t) ,\,t>0$), the second condition in Theorem 5.7 (iii) implies the first one. $\square$

Remark 5.19

When $b(t)\rightarrow +\infty$ as $t\rightarrow 0_+$, then in Corollary 5.18 the embeddings are sharp if $r\ge 1$ and $\mu \in {\mathcal {L}}_r^k$ is given by $\mu :=\lambda _{1r}$, where

$$\begin{aligned} \lambda _{1r}(t)\!:&= \! \,\,\,t^{-1}\left( \int \limits _0^t(b(y^n))^{-1}\,\mathrm{d}y \right) ^{1+\frac{1}{r}} \nonumber \\&\quad \times \left( t (b(t^n))^{-1}-\int \limits _0^t(b(y^n))^{-1}\,\mathrm{d}y \right) ^{-\frac{1}{r}}, \quad t\in (0,1). \end{aligned}$$

(5.56)

Moreover, the embedding is optimal if $r=1$.

For example, if $b(t)=\ell ^{\alpha }(t),\,t\in (0,1]$, with $\alpha >0$, then $\lambda _{1r}(t)=\ell ^{-\alpha +\frac{1}{r}}(t),\,t\in (0,1)$, and $\lambda _{1r}\in {\mathcal {L}}_r^n$.