Some Embeddings of Morrey Spaces with Critical Smoothness

Abstract

We study embeddings of Besov–Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and of Triebel–Lizorkin–Morrey spaces ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ in the limiting cases when the smoothness s equals $s_o=d\max (1/u-p/u,0)$ or $s_{\infty }=d/u$, which is related to the embeddings in $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ or in $L_{\infty }({{{\mathbb {R}}}^d})$, respectively. When $s=s_o$ we characterise the embeddings in $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ and when $s=s_{\infty }$ we obtain embeddings into Orlicz–Morrey spaces of exponential type and into generalised Morrey spaces.

1 Introduction

In recent years function spaces built upon Morrey spaces ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$, $0< p \le u < \infty$, attracted some attention. They include in particular Besov–Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and Triebel–Lizorkin–Morrey spaces ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, $0< p \le u < \infty$, $0 < q \le \infty$, $s \in {{\mathbb {R}}}$. Some Besov–Morrey spaces were first introduced by Netrusov in [21] by means of differences. The attention paid to the spaces afterwards was motivated by possible applications. Using Fourier-analytical tools the Besov–Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ were introduced by Kozono and Yamazaki in [12] and used by them and later on by Mazzucato [14] in the study of Navier–Stokes equations. In [37] Tang and Xu introduced the corresponding Triebel–Lizorkin–Morrey spaces ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, thanks to establishing the Morrey version of the Fefferman–Stein vector-valued inequality. Some properties of these spaces including their atomic decompositions and wavelet characterisations were later described in the papers by Mazzucato [15], Sawano [25, 26], Sawano and Tanaka [28, 29] and Rosenthal [24]. Also embedding properties of these spaces were investigated in a series of papers [8, 9].

In our two recent papers [7, 10] we studied under which conditions the spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ contain only regular distributions, i.e., when

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\subset L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d}), \quad {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\subset L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d}) \end{aligned}$$

(1.1)

and when they consist of bounded functions

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\subset L_\infty ({{{\mathbb {R}}}^d}), \quad {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\subset L_\infty ({{{\mathbb {R}}}^d}). \end{aligned}$$

(1.2)

The embeddings (1.1) hold if the smoothness s is related to

$$\begin{aligned} s_o=\frac{p}{u}d \max \left( \frac{1}{p}-1,0\right) . \end{aligned}$$

Similarly the embeddings (1.2) are valid if s is related to $s_\infty =\frac{d}{u}$. Therefore we called $s_o$ and $s_\infty$ critical smoothnesses. In particular, the spaces contain a non-regular distribution if $s<s_o$ and consist of locally integrable functions if $s>s_o$. The behaviour in the critical case $s=s_o$ is a delicate question. The analogous situation occurs in the case of boundedness of the functions.

This paper is the direct continuation of the above mentioned recent articles. First in Sect. 3 we prove that the embeddings (1.1) with smoothness $s=s_o$ hold if and only if the spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ are embedded into some Morrey spaces ${\mathcal {M}}_{v,r}({{{\mathbb {R}}}^d})$ with properly defined indices v and r, cf. Theorem 3.2 and Theorem 3.4.

In Sect. 4 we investigate unboundedness properties of the functions belonging to smoothness spaces of Morrey type with $s=s_\infty$. First we prove that the spaces ${\mathcal {N}}^{d/u}_{u,p,q}({{{\mathbb {R}}}^d})$ and ${\mathcal {E}}^{d/u}_{u,p,q}({{{\mathbb {R}}}^d})$ can be embedded into Orlicz–Morrey spaces of exponential type, cf. Theorem 4.5 and Corollary 4.6. The idea that exponential Young functions can control the unboundedness of functions belonging to the Sobolev spaces of critical smoothness goes back to Trudinger [43], cf. also Strichartz [36]. There are several definitions of Orlicz–Morrey spaces. Here we follow the approach proposed by Nakai [19]. Afterwards we embed the above spaces of smoothness $\frac{d}{u}$ into the so-called generalised Morrey spaces, cf. Theorem 4.10 and Remark 4.12. The generalised Morrey spaces have been investigated by several authors. Our results here are close to that ones in papers by Sawano and Wadade [30], Nakamura et al. [20] and by Eridani et al. [5]. However in contrast to the above papers we study also the Besov–Morrey spaces. Moreover in the case of Sobolev–Morrey spaces we are able to prove the embeddings for a larger set of parameters.

In the very short last section, somewhat supplementary, we formulate some outcome concerning embeddings into spaces with smoothness 0, in particular to $\mathop {\mathrm {bmo}}({{{\mathbb {R}}}^d})$. In Sect. 2 we collect the definition and basic facts concerning the smoothness Morrey spaces that are needed in the next sections.

2 Smoothness Spaces of Morrey Type

First we fix some notation. By ${{\mathbb {N}}}$ we denote the set of natural numbers, by ${{\mathbb {N}}}_0$ the set ${{\mathbb {N}}}\cup \{0\}$, and by ${{\mathbb {Z}}}^d$ the set of all lattice points in${{{\mathbb {R}}}^d}$having integer components. Let ${{\mathbb {N}}}_0^d$, where $d\in {{\mathbb {N}}}$, be the set of all multi-indices, $\alpha = (\alpha _1, \ldots ,\alpha _d)$ with $\alpha _j\in {{\mathbb {N}}}_0$ and $|\alpha | := \sum _{j=1}^d \alpha _j$. If $x=(x_1,\ldots ,x_d)\in {{{\mathbb {R}}}^d}$ and $\alpha = (\alpha _1, \ldots ,\alpha _d)\in {{\mathbb {N}}}_0^d$, then we put $x^\alpha := x_1^{\alpha _1} \cdots x_d^{\alpha _d}$. For $a\in {{\mathbb {R}}}$, let $\left\lfloor a \right\rfloor :=\max \{k\in {{\mathbb {Z}}}: k\le a\}$ and $a_+:=\max (a,0)$. If $0<u\le \infty$, the number $u'$ is given by $\frac{1}{u'}=(1-\frac{1}{u})_+$, with the convention that $1/\infty =0$. All unimportant positive constants will be denoted by C, occasionally the same letter C is used to denote different constants in the same chain of inequalities. By the notation $A \lesssim B$, we mean that there exists a positive constant c such that $A \le c \,B$, whereas the symbol $A \sim B$ stands for $A \lesssim B \lesssim A$. We denote by $B(x,r) := \{y\in {{{\mathbb {R}}}^d}: |x-y|<r\}$ the ball centred at $x\in {{{\mathbb {R}}}^d}$ with radius $r>0$ and $|\cdot |$ denotes the Lebesgue measure when applied to measurable subsets of ${{{\mathbb {R}}}^d}$. Let ${\mathcal {Q}}$ be the collection of all dyadic cubes in ${{{\mathbb {R}}}^d}$, namely, ${\mathcal {Q}}:= \{Q_{j,k}:= 2^{-j}([0,1)^d+k):\ j\in {{\mathbb {Z}}},\ k\in {{\mathbb {Z}}}^d\}$.

Given two (quasi-)Banach spaces X and Y, we write $X\hookrightarrow Y$ if $X\subset Y$ and the natural embedding of X into Y is continuous.

We introduce smoothness spaces of Morrey type. We recall first the definition of Morrey spaces.

Definition 2.1

Let $0<p\le u<\infty$. The Morrey space${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$ is the set of all locally p-integrable functions $f\in L_p^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \Vert f \mid {{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})}\Vert :=\, \sup _{Q\in {\mathcal {Q}}} |Q|^{\frac{1}{u}-\frac{1}{p}} \biggl (\int _{Q} |f(y)|^p \;\mathrm {d}y \biggr )^{1/p}\, <\, \infty \, . \end{aligned}$$

(2.1)

Remark 2.2

The spaces ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$ are quasi-Banach spaces (Banach spaces for $p \ge 1$), that can be equivalently defined if the supremum in (2.1) is taken over all balls B(x, r) or over all cubes Q(x, r), with center $x\in {{{\mathbb {R}}}^d}$ and side length $r>0$, and sides parallel to the axes of coordinates.

These spaces originated from Morrey’s study on PDE (see [17]) and are part of the wider class of Morrey–Campanato spaces; cf. [23]. They can be considered as a complement to $L_p$ spaces, since ${{\mathcal {M}}}_{p,p}({{{\mathbb {R}}}^d}) = L_p({{{\mathbb {R}}}^d})$ with $p\in (0,\infty )$, extended by ${{\mathcal {M}}}_{\infty ,\infty }({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$. In a parallel way one can define the spaces ${{\mathcal {M}}}_{\infty ,p}({{{\mathbb {R}}}^d})$, $p\in (0, \infty )$, but using the Lebesgue differentiation theorem, one arrives at ${{\mathcal {M}}}_{\infty , p}({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$. Moreover, ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})=\{0\}$ for $u<p$, and for $0<p_2 \le p_1 \le u < \infty$,

$$\begin{aligned} L_u({{{\mathbb {R}}}^d})= {{\mathcal {M}}}_{u,u}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{u,p_1}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{u,p_2}({{{\mathbb {R}}}^d}). \end{aligned}$$

Let ${\mathcal {S}}({{{\mathbb {R}}}^d})$ be the set of all Schwartz functions on ${{{\mathbb {R}}}^d}$, endowed with the usual topology, and denote by ${\mathcal {S}}'({{{\mathbb {R}}}^d})$ its topological dual, namely, the space of all bounded linear functionals on ${\mathcal {S}}({{{\mathbb {R}}}^d})$ endowed with the weak $*$-topology. Let $\varphi _0=\varphi \in {\mathcal {S}}({{{\mathbb {R}}}^d})$ be such that

$$\begin{aligned} {\mathrm {supp}\,}{\varphi }\subset \{x\in {{{\mathbb {R}}}^d}:\,|x|\le 2\}\, \qquad \text {and}\qquad {\varphi }(x)=1 \quad \text {if}\quad |x|\le 1, \end{aligned}$$

(2.2)

and for each $j\in {{\mathbb {N}}}$ let $\varphi _j(x):=\varphi (2^{-j} x)-\varphi (2^{-j+1} x)$. Then $\{\varphi _j\}_{j=0}^{\infty }$ forms a smooth dyadic resolution of unity.

Definition 2.3

Let $0<p\le u<\infty$ or $p=u=\infty$. Let $q\in (0,\infty ]$, $s\in {{\mathbb {R}}}$ and $\{\varphi _j\}_{j=0}^{\infty }$ a smooth dyadic resolution of unity.

(i):

The Besov–Morrey space${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ is defined to be the set of all distributions $f\in {\mathcal {S}}'({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \big \Vert f\mid {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\big \Vert := \bigg (\sum _{j=0}^{\infty }2^{jsq}\big \Vert {{\mathcal {F}}}^{-1} (\varphi _j {{\mathcal {F}}}f)\vert {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\big \Vert ^q \bigg )^{1/q} < \infty \end{aligned}$$

with the usual modification made in case of $q=\infty$.

(ii):

Let $u\in (0,\infty )$. The Triebel–Lizorkin–Morrey space${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ is defined to be the set of all distributions $f\in {\mathcal {S}}'({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \big \Vert f \mid {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\big \Vert :=\bigg \Vert \bigg (\sum _{j=0}^{\infty }2^{jsq} | {{\mathcal {F}}}^{-1} (\varphi _j {{\mathcal {F}}}f)(\cdot )|^q\bigg )^{1/q} \mid {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\bigg \Vert <\infty \end{aligned}$$

with the usual modification made in case of $q=\infty$.

Remark 2.4

Occasionally we adopt the usual custom to write ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ instead of ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ or ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, when both scales are meant simultaneously in some context. The spaces ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ are independent of the particular dyadic partition of unity $\{\varphi _j\}_{j=0}^{\infty }$ appearing in their definitions. They are quasi-Banach spaces (Banach spaces for $p,\,q\ge 1$), and ${\mathcal {S}}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {\mathcal {S}}'({{{\mathbb {R}}}^d})$. Moreover, for $u=p$ we re-obtain the usual Besov and Triebel-Lizorkin spaces ${{{\mathcal {A}}}}^{s}_{p,p,q}({{{\mathbb {R}}}^d}) = A^s_{p,q}({{{\mathbb {R}}}^d})$. Besov–Morrey spaces were introduced by Kozono and Yamazaki in [12]. They studied semi-linear heat equations and Navier-Stokes equations with initial data belonging to Besov–Morrey spaces. The investigations were continued by Mazzucato [14, 15], where one can find the atomic decomposition of some spaces. The Triebel–Lizorkin–Morrey spaces were later introduced by Tang and Xu [37], we follow their approach. The ideas were further developed by Sawano and Tanaka [25, 26, 28, 29]. Closely related, alternative approaches can be found in the monographs [41, 42, 45] or in the survey papers by Sickel [33, 34].

We list some elementary embeddings within this scale of spaces. It holds

$$\begin{aligned} {{\mathcal {A}}}^{s+\varepsilon }_{u,p,r}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}), \quad \text {if} \quad \varepsilon >0, \quad r,q\in (0,\infty ], \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {A}}}^{s}_{u,p,q_1}({{{\mathbb {R}}}^d}) \hookrightarrow \mathcal{A}^{s}_{u,p,q_2}({{{\mathbb {R}}}^d}),\quad \quad \text {if} \quad q_1\le q_2. \end{aligned}$$

(2.3)

Sawano proved in [25] that, for $s\in {{\mathbb {R}}}$ and $0<p< u<\infty$,

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,\min (p,q)}({{{\mathbb {R}}}^d})\, \hookrightarrow \, {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\, \hookrightarrow \,{{\mathcal {N}}}^s_{u,p,\infty }({{{\mathbb {R}}}^d}), \end{aligned}$$

(2.4)

where, for the latter embedding, $r=\infty$ cannot be improved – unlike in case of $u=p$. More precisely, ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {N}}}^s_{u,p,r}({{{\mathbb {R}}}^d})$ if, and only if, $r=\infty$ or $u=p$ and $r\ge \max (p,q)$. On the other hand, Mazzucato has shown in [15, Prop. 4.1] that

$$\begin{aligned} {{\mathcal {E}}}^0_{u,p,2}({{{\mathbb {R}}}^d})={{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d}), \quad 1<p\le u<\infty , \end{aligned}$$

(2.5)

in particular,

$$\begin{aligned} {{\mathcal {E}}}^0_{p,p,2}({{{\mathbb {R}}}^d})=L_p({{{\mathbb {R}}}^d}), \quad 1<p<\infty . \end{aligned}$$

This is nothing else than the well known classical coincidence $F^0_{p,2}({{{\mathbb {R}}}^d})=L_p({{{\mathbb {R}}}^d})$, $1<p<\infty$, cf. [38, Thm. 2.5.6]. Further embedding results for the above scales of function spaces on ${{{\mathbb {R}}}^d}$ can be found in [8,9,10, 46].

The atomic decompositions

An important tool in our later considerations is the characterisation of the Besov–Morrey and Triebel–Lizorkin–Morrey spaces by means of atomic decompositions. We follow [24]; see also [28].

Definition 2.5

Let $0<p\le u< \infty$, $q\in (0,\infty ]$ and $s\in {{{\mathbb {R}}}}$. Let $K\in {{\mathbb {N}}}_0$ and $N\in \{-1\}\cup {{\mathbb {N}}}_0$. A collection of $L_{\infty }$-functions $a_{jm}:{{{\mathbb {R}}}^d}\rightarrow {{\mathbb {C}}}$, $j\in {{\mathbb {N}}}_0$, $m\in {{\mathbb {Z}}}^d$, is a family of (K, N)-atoms if, for some $c_1>1$ and $c_2>0$, it holds

(i):

${\mathrm {supp}\,}a_{jm} \subset c_1 Q_{jm}, \quad j\in {{\mathbb {N}}}_0,\; m\in {{\mathbb {Z}}}^d$,

(ii):

there exist all (classical derivatives) $\;\mathrm {D}^{\alpha }a_{jm}$ for $\alpha \in {{\mathbb {N}}}_0^n$ with $|\alpha |\le K$ and

$$\begin{aligned} \Vert \;\mathrm {D}^{\alpha }a_{jm} | L_{\infty }({{{\mathbb {R}}}^d})\Vert \le c_2 2^{j |\alpha |}, \quad j\in {{\mathbb {N}}}_0,\; m\in {{\mathbb {Z}}}^d, \end{aligned}$$

(2.6)

(iii):

if $\gamma \in {{\mathbb {N}}}_0^d$ with $|\gamma |\le N$, then

$$\begin{aligned} \int _{{{{\mathbb {R}}}^d}}x^{\gamma }a_{jm}(x) \;\mathrm {d}x=0, \quad j\in {{\mathbb {N}}}_0,\;\; m\in {{\mathbb {Z}}}^d. \end{aligned}$$

If $N=-1$, then this means that no moment condition is required.

For $0<u<\infty$, $j\in {{\mathbb {N}}}_0$ and $m\in {{\mathbb {Z}}}^d$ we denote by $\chi _{j,m}$ the characteristic function of the cube $Q_{j,m}$ and by $\chi _{j,m}^{(u)}:=2^{jd/u}\chi _{j,m}$ the u-normalised characteristic function of the same cube, i.e., such that $\Vert \chi _{j,m}^{(u)}\mid {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d}) \Vert =1$.

Definition 2.6

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

(i):

The sequence space$e^s_{u,p,q}({{{\mathbb {R}}}^d})$ is defined to be the set of all sequences $\lambda :=\{\lambda _{j,m}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}\subset {{\mathbb {C}}}$ such that

$$\begin{aligned} \Vert \lambda \mid e^s_{u,p,q}({{{\mathbb {R}}}^d})\Vert := \Big \Vert \big (\sum _{j=0}^\infty 2^{jq(s-\frac{d}{u})} \sum _{m\in {{\mathbb {Z}}}^d} |\lambda _{j,m}|^q \chi ^{(u)q}_{j, m}\big )^{1/q}| {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Big \Vert <\infty \end{aligned}$$

with the usual modification in case of $q=\infty$.

(ii):

The sequence space$n^s_{u,p,q}({{{\mathbb {R}}}^d})$ is defined to be the set of all sequences $\lambda :=\{\lambda _{j,m}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}\subset {{\mathbb {C}}}$ such that

$$\begin{aligned} \Vert \lambda \mid n^s_{u,p,q}({{{\mathbb {R}}}^d})\Vert := \Big (\sum _{j=0}^\infty 2^{jq(s-\frac{d}{u})} \big \Vert \sum _{m\in {{\mathbb {Z}}}^d} |\lambda _{j,m}|\chi ^{(u)}_{j, m}\,|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\big \Vert ^q\Big )^{1/q} <\infty \end{aligned}$$

with the usual modification in case of $q=\infty$.

Remark 2.7

It was proved in [8] that

$$\begin{aligned}&\Vert \lambda \mid n^s_{u,p,q} ({{{\mathbb {R}}}^d})\Vert \sim \Vert \lambda | { n}^{s}_{u,p,q}\Vert ^*: = \\&\quad = \Bigg (\sum _{j=0}^\infty 2^{qj(s-\frac{d}{u})} \sup _{\nu : \nu \le j; k\in {\mathbb {Z}}^d} 2^{qd(j-\nu )(\frac{1}{u} - \frac{1}{p} )}\Big (\sum _{m:Q_{j,m}\subset Q_{\nu ,k}}|\lambda _{j,m}|^p\Big )^{\frac{q}{p}}\Bigg )^{1/q} < \infty . \end{aligned}$$

A similar characterisation of a norm in discrete Besov-type spaces was used in the proof of Theorem 1.1 in [32].

For $p,q\in (0, \infty ]$, let

$$\begin{aligned} \sigma _p := d\left( \frac{1}{p} - 1\right) _{+} \quad \text{ and } \quad \sigma _{p,q} := d\left( \frac{1}{\min (p,q) }- 1\right) _{+} . \end{aligned}$$

(2.7)

According to [24, Thms. 2.30 and 2.36] (see also [28, Thm. 4.12]), we have the following atomic decomposition characterisation of ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, where we adopt the same custom to write $a^s_{u,p,q}({{{\mathbb {R}}}^d})$ instead of $n^s_{u,p,q}({{{\mathbb {R}}}^d})$ or $e^s_{u,p,q}({{{\mathbb {R}}}^d})$, for convenience.

Proposition 2.8

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

$$\begin{aligned} K \ge \max (\left\lfloor s+1 \right\rfloor ,0) \end{aligned}$$

and

$$\begin{aligned} N \ge \max (\left\lfloor \sigma _{p,q}-s \right\rfloor ,-1) \; ({{\mathcal {E}}}- \mathrm{case})\quad \text {or}\quad N\ge \max (\left\lfloor \sigma _{p}-s \right\rfloor ,-1) \; ({{\mathcal {N}}}- \mathrm{case}). \end{aligned}$$

Then for each $f\in {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, there exist a family $\{a_{jm}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}$ of (K, N)- atoms and a sequence $\lambda =\{\lambda _{jm}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}\in a^s_{u,p,q}({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} f=\sum _{j=0}^{\infty }\sum _{m\in {{\mathbb {Z}}}^d}\lambda _{jm} \,a_{jm} \quad \text{ in } \quad {\mathcal {S}}'({{{\mathbb {R}}}^d}) \end{aligned}$$

and

$$\begin{aligned} \Vert \lambda \mid a^s_{u,p,q}({{{\mathbb {R}}}^d})\Vert \le C\, \Vert f \mid {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\Vert , \end{aligned}$$

where C is a positive constant independent of $\lambda$ and f.

Conversely, there exists a positive constant C such that for all families $\{a_{jm}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}$ of (K, N)-atoms and $\lambda =\{\lambda _{jm}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}\in a^s_{u,p,q}({{{\mathbb {R}}}^d})$,

$$\begin{aligned} \Big \Vert \sum _{j=0}^{\infty }\sum _{m\in {{\mathbb {Z}}}^d}\lambda _{jm} \,a_{jm} \mid {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\Big \Vert \le C\, \Vert \lambda \mid a^s_{u,p,q}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

3 Embeddings with Smoothness $\frac{p}{u}\sigma _p$

We return to the remarkable coincidence (2.5) and consider the limiting situation when $p=1$. Recall that in case of $p=u=1$ it is well known that $F^0_{1,2}({{{\mathbb {R}}}^d}) \hookrightarrow L_1({{{\mathbb {R}}}^d})$ properly embedded, cf. [35]. Now we concentrate on the Morrey situation when $p=1<u$ and can prove some partial counterpart of (2.5).

Proposition 3.1

Let $1<u<\infty$. Then

$$\begin{aligned} {{\mathcal {E}}}^0_{u,1,2}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{u, 1}({{{\mathbb {R}}}^d}) . \end{aligned}$$

Proof

Step 1. We prove that there exists a positive constant $C>0$ such that

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{u,1}({{{\mathbb {R}}}^d})\Vert \le C\,\Vert f|{{\mathcal {E}}}^0_{u,1,2}({{{\mathbb {R}}}^d})\Vert \, . \end{aligned}$$

(3.1)

Let $f\in {{\mathcal {E}}}^0_{u,1,2}({{{\mathbb {R}}}^d})$. By the atomic decomposition theorem with atoms satisfying the conditions from Definition 2.5 with $N>\left\lfloor d(u-1) \right\rfloor$ and $c_1=3$, we have

$$\begin{aligned} f = \sum _{j=0}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} \lambda _{jm} a_{jm} \end{aligned}$$

with

$$\begin{aligned} \Big \Vert \Bigl (\sum _{j=0}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} |\lambda _{jm}|^2 \, \chi _{jm}(\cdot )\Bigr )^{\frac{1}{2}} \big | {{\mathcal {M}}}_{u,1}({{{\mathbb {R}}}^d})\Big \Vert \le c \Vert f|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

Let $Q=Q_{\nu ,k}$ be a dyadic cube and decompose f as follows

$$\begin{aligned} f = \sum _{j=0}^{\nu } \sum _{m\in {{{\mathbb {Z}}}^d}} \lambda _{jm} a_{jm} + \sum _{j=\nu +1}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} \lambda _{jm} a_{jm} =f_1+f_2. \end{aligned}$$

(3.2)

We remark that $f_i\in {{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})$ and $\Vert f_i|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert \le c \Vert f|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert$, $i=1,2$.

Step 2. We deal first with $f_1$. Recall $Q=Q_{\nu ,k}$. Let $\mu _{j,\nu }=\{m\in {{{\mathbb {Z}}}^d}: Q\cap {\mathrm {supp}\,}a_{j,m} \not = \emptyset \}$ and note that $\# \mu _{j,\nu }\le c$ if $j\le \nu$. Then we have

$$\begin{aligned}&|Q|^{\frac{1}{u}-1} \int _Q |f_1(x)| \;\mathrm {d}x \nonumber \\&\quad \le |Q|^{\frac{1}{u}-1} \sum _{j=0}^{\nu } \sum _{m\in \mu _{j,\nu }} |\lambda _{jm}| \int _Q |a_{jm}(x)| \;\mathrm {d}x \nonumber \\&\quad \le c |Q|^{\frac{1}{u}} \sum _{j=0}^{\nu } \sum _{m\in \mu _{j,\nu }} |\lambda _{jm}| \nonumber \\&\quad = c |Q|^{\frac{1}{u}} \sum _{j=0}^{\nu } \sum _{m\in \mu _{j,\nu }} |Q_{jm}|^{-1} \int _{Q_{jm}}\Bigl (|\lambda _{jm}|^2 \chi _{jm}(x) \Bigr )^{\frac{1}{2}} \;\mathrm {d}x \nonumber \\&\quad \le c \sum _{j=0}^{\nu } 2^{(j-\nu )\frac{d}{u}} \sum _{m\in \mu _{j,\nu }} |Q_{jm}|^{\frac{1}{u}-1} \int _{Q_{jm}}\Bigl (\sum _{\ell =1}^{\infty } \sum _{n\in {{{\mathbb {Z}}}^d}}|\lambda _{\ell n}|^2 \chi _{\ell n}(x) \Bigr )^{\frac{1}{2}} \;\mathrm {d}x \nonumber \\&\quad \le c \Bigl (\sum _{j=0}^{\nu } 2^{(j-\nu )\frac{d}{u}} \Bigr )\sup _{0\le j\le \nu } \sup _{m\in \mu _{j,\nu } }\Big \{ |Q_{jm}|^{\frac{1}{u}-1} \int _{Q_{jm}}\Bigl (\sum _{\ell =1}^{\infty } \sum _{n\in {{{\mathbb {Z}}}^d}}|\lambda _{\ell n}|^2 \chi _{\ell ,n}(x) \Bigr )^{\frac{1}{2}} \;\mathrm {d}x \Big \}\nonumber \\&\quad \le c \Big \Vert \Bigl (\sum _{j=0}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} |\lambda _{j m}|^2 \, \chi _{j,m}(\cdot )\Bigr )^{\frac{1}{2}} \big | {{\mathcal {M}}}_{u,1}({{{\mathbb {R}}}^d})\Big \Vert \le c \Vert f|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(3.3)

Step 3. Now we deal with $f_2$ for what we rely on the results in [11]. By checking the proof in [11, Thm. 4.3], we can see that $f_2 \in {{\mathcal {E}}}^0_{u,1,2}({{{\mathbb {R}}}^d})$ can be decomposed in terms of non-smooth atoms $b_{jm}$ supported in cubes $cQ_{jm}$ with side lengths less than $2^{-\nu }$, that is,

$$\begin{aligned} f_2=\sum _{j=\nu +1}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} \lambda _{jm} a_{jm} = \sum _{j=\nu +1}^\infty \sum _{m\in {{{\mathbb {Z}}}^d}} t_{jm} b_{jm} \end{aligned}$$

where

$$\begin{aligned} \Vert t\vert m_{u,1}\Vert :=\sup _{P\in \mathcal {Q}} |P|^{\frac{1}{u}-1} \Bigl (\sum _{Q_{\ell k}\subset P}|t_{\ell k}| \Bigr )\le c \Vert f_2|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert \end{aligned}$$

and $t=\{t_{jm}\}_{j\in {{\mathbb {N}}}_0,m\in {{\mathbb {Z}}}^d}$.

We remark that the non-smooth atoms from [11] differ from the smooth ones of Definition 2.5 in condition (2.6); in particular, we have

$$\begin{aligned} \Vert b_{jm}\vert L_u({{{\mathbb {R}}}^d})\Vert \le |Q_{jm}|^{\frac{1}{u}-1}, \quad j\in {{\mathbb {N}}}_0,\; m\in {{\mathbb {Z}}}^d. \end{aligned}$$

Then, using Hölder’s inequality and the properties of non-smooth atoms we obtain

$$\begin{aligned} |Q|^{\frac{1}{u}-1} \int _Q |f_2(x)| \;\mathrm {d}x&\le |Q|^{\frac{1}{u}-1} \sum _{j=\nu +1}^{\infty } \sum _{m\in {{{\mathbb {Z}}}^d}} |t_{jm}| \int _{3Q_{jm}\cap Q} |b_{jm}(x)| \;\mathrm {d}x \nonumber \\&\le c |Q|^{\frac{1}{u}-1} \sum _{j=\nu +1}^{\infty } \sum _{m\in {{{\mathbb {Z}}}^d}: Q_{jm}\subset cQ} |t_{jm}||Q_{jm}|^{1-\frac{1}{u}} \Vert b_{jm}\vert L_u({{{\mathbb {R}}}^d})\Vert \nonumber \\&\le c |Q|^{\frac{1}{u}-1} \sum _{j=\nu +1}^{\infty } \sum _{m\in {{{\mathbb {Z}}}^d}: Q_{jm}\subset c' Q} |t_{jm}| \le c \Vert (t_{jm})\vert m_{u,1}\Vert \nonumber \\&\le c \Vert f|{{\mathcal {E}}}_{u,1,2}^0({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(3.4)

The desired outcome is then a consequence of (3.2), (3.3) and (3.4). $\square$

We equip the space $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ with the metric

$$\begin{aligned} d(f,g)= \sum _{n=1}^\infty \frac{1}{2^n} \frac{\Vert f-g|L_1({\widetilde{Q}}_n)\Vert }{1+\Vert f-g|L_1({\widetilde{Q}}_n))\Vert }, \end{aligned}$$

where ${\widetilde{Q}}_n = [-n,n]^d$. The space $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ with this metric is a complete locally convex metric space, i.e., a Fréchet space, cf. [16, p. 40]. One can easily see that a Morrey space ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$ is continuously embedded into $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ if $p\ge 1$. Indeed we have

$$\begin{aligned} d(f,g)&\le C \sum _{n=1}^\infty \frac{1}{2^n} \frac{|{\widetilde{Q}}_n|^{1-\frac{1}{u}}}{1+ \Vert f-g|L_1 ({\widetilde{Q}}_n)\Vert }\Vert f-g|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert \\&\le C \sum _{n=1}^\infty \frac{|{\widetilde{Q}}_n|^{1-\frac{1}{u}}}{2^n }\Vert f-g|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert \le C \Vert f-g|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

Theorem 3.2

Let $0<p \le u <\infty$, $0<q\le \infty$, and $s=\frac{p}{u}\sigma _p$. The following assertions are equivalent:

(i):

${{\mathcal {E}}}^s_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$,

(ii):

${{\mathcal {E}}}^s_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{\min (p,1)},\max (p,1)}({{{\mathbb {R}}}^d})$,

(iii):

either $p\ge 1$ and $q\le 2$, or $0<p<1$.

Proof

Note that (ii) is clearly stronger than (i) since ${{\mathcal {M}}}_{\frac{u}{\min (p,1)},\max (p,1)}({{{\mathbb {R}}}^d})$ is embedded into $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$, and the part (i) $\Rightarrow$ (iii) is covered by [10, Thm. 3.4].

Step 1. Consider first the case $p=1$. The implication (iii) $\Rightarrow$ (ii) is a consequence of Proposition 3.1 and an elementary embedding,

$$\begin{aligned} {{\mathcal {E}}}^0_{u,1,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{u,1,2}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{u,1}({{{\mathbb {R}}}^d}) \quad \text {if} \quad 0<q\le 2 . \end{aligned}$$

Step 2. Let $0<p<1$. Then $s=\frac{p}{u}\sigma _p = \frac{d}{u}(1-p)$. The general properties of embeddings between Triebel–Lizorkin–Morrey spaces, cf. [9, Thm. 3.1], and the first step imply

$$\begin{aligned} {{\mathcal {E}}}^s_{u,p,\infty } ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{\frac{u}{p},1,1} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},1}({{{\mathbb {R}}}^d}), \end{aligned}$$

which shows that (iii) $\Rightarrow$ (ii), and there is nothing more to be proved in this case.

Step 3. Now assume $p>1$. The implication (iii) $\Rightarrow$ (ii) follows from

$$\begin{aligned} {{\mathcal {E}}}^0_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{u,p,2}({{{\mathbb {R}}}^d}) ={{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d}) \quad \text {if} \quad 0<q\le 2 . \end{aligned}$$

$\square$

Remark 3.3

The above theorem improves the statements of [10, Thm. 3.4] and extends Theorem 3.3.2(i) and Corollary 3.3.1 in [35] from classical Triebel–Lizorkin spaces to Triebel–Lizorkin–Morrey spaces.

Theorem 3.4

Let $0<p \le u <\infty$, $0<q\le \infty$, and $s=\frac{p}{u}\sigma _p$. The following assertions are equivalent:

(i):

${{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$,

(ii):

${{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{\min (p,1)},\max (p,1)}({{{\mathbb {R}}}^d})$,

(iii):

$0<q \le \min \bigl (\max (p,1),2 \bigr )$.

Proof

The case $p=u$ is well known, cf. [35, Thm. 3.3.2, Cor. 3.3.1], so we can restrict ourselves to the case $p<u$.

Step 1. We prove that (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i). The second implication has been already shown so it remains to prove the first one.

Let $0<p\le 1$. For $0<q\le 1$, by general properties of embeddings, in particular [8, Thm. 3.2], and Theorem 3.2, we have

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,q} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{\frac{u}{p},1,1} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{\frac{u}{p},1,1}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},1}({{{\mathbb {R}}}^d}) \end{aligned}$$

which proves the implication (iii) $\Rightarrow$ (ii).

Consider the case $p\ge 1$. If $0<q \le \min (p,2)$, then elementary embeddings, (2.5) and Theorem 3.2 yield

$$\begin{aligned} {{\mathcal {N}}}^0_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{u,p,2}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d}), \end{aligned}$$

which shows that (iii) $\Rightarrow$ (ii).

Step 2. We prove that the condition (i) implies (iii). For $0<p<1$ the implication was proved in [10], cf. Step 2 of the proof of Theorem 3.4. But the same argument works for $p=1$. Also the case $2\le p<\infty$ is covered by Theorem 3.4 in [10].

It remains to consider the case $1<p<2$. We assume that embedding (i) holds for some $q>p$. We choose a smooth function ${\tilde{a}}$ such that

$$\begin{aligned} {\mathrm {supp}\,}{\tilde{a}} \subset \left[ 0,\frac{1}{2}\right] ^d, \qquad 0\le {\tilde{a}} \le 1 \qquad \text {and} \qquad \left| \frac{\partial {\tilde{a}}}{\partial x_i}\right| \le 1, \quad i =1,\dots , d, \end{aligned}$$

(3.5)

and put

$$\begin{aligned} a(x)= {\tilde{a}}(x) - {\tilde{a}}(-x). \end{aligned}$$

(3.6)

Then a is an atom satisfying the first moment condition and supported in $[-\frac{1}{2},\frac{1}{2}]^d$. Moreover we consider the family of atoms $a_{j,m}$, $j=0,1,\ldots$ and $m\in {\mathbb {Z}}^d$ that are the dilations and translations of a, i.e.,

$$\begin{aligned} a_{j,m}(x)=a(2^{j}x-m-(1/2,\ldots ,1/2)). \end{aligned}$$

The function $a_{j,m}$ is an atom supported in $Q_{j,m}$.

Let us fix $n\in {\mathbb {N}}$. For any cube $Q_{j,m} \subset Q_{0,0}$, $1\le j\le n$, $m\in {{{\mathbb {Z}}}^d}$, we define a function

$$\begin{aligned} b_{j,m}(x) = 2^{j-n}\sum _{k: Q_{n,k}\subset Q_{j,m}} a_{n,k}(x). \end{aligned}$$

The function $b_{j,m}$ is an atom satisfying the first moment condition supported in $Q_{j,m}$. We define a smooth function $f_n$ by the following finite sum

$$\begin{aligned} f_n(x) = \sum _{j=1}^n j^{-\frac{1}{p}} \sum _{m: Q_{j,m} \subset Q_{0,0}}b_{j,m}(x). \end{aligned}$$

The functions $f_n$ belong to ${{\mathcal {N}}}^0_{u,p,q}({{{\mathbb {R}}}^d})$ and their norms are uniformly bounded since

$$\begin{aligned} \Vert f_n|{{\mathcal {N}}}^0_{u,p,q}({{{\mathbb {R}}}^d})\Vert&\le \left( \sum _{j=1}^n 2^{-jq\frac{d}{u}} \sup _{\nu :\nu \le j ,k\in {\mathbb {Z}}^d} 2^{dq(j-\nu )(\frac{1}{u}-\frac{1}{p})} \Big (\sum _{Q_{j,m}\subset Q_{\nu ,k}\subset Q_{0,0}} j^{-1} \Big )^\frac{q}{p} \right) ^\frac{1}{q} \\&\le \left( \sum _{j=1}^n j^{-\frac{q}{p}}2^{-jq\frac{d}{u}} \sup _{\nu :0\le \nu \le j ,k\in {\mathbb {Z}}^d} 2^{dq\frac{(j-\nu )}{u}}\right) ^\frac{1}{q} \\&\le \left( \sum _{j=1}^\infty j^{-\frac{q}{p}}\right) ^\frac{1}{q}= C < \infty , \end{aligned}$$

recall $q>p$. On the other hand, for $\nu \in {{\mathbb {N}}}$,

$$\begin{aligned} \Vert f_n|L_1({\widetilde{Q}}_\nu )\Vert&= \Vert f_n|L_1({\widetilde{Q}}_1)\Vert = \int _{{\widetilde{Q}}_1} \left| \sum _{j=1}^n j^{-\frac{1}{p}} \sum _{Q_{j,m} \subset Q_{0,0}} b_{j,m}(x)\right| \;\mathrm {d}x \\&= \int _{{\widetilde{Q}}_1} \left| \sum _{j=1}^n j^{-\frac{1}{p}} 2^{j-n}\sum _{Q_{j,m} \subset Q_{0,0}} \sum _{Q_{n,k}\subset Q_{j,m}} a_{n,k}(x)\right| \;\mathrm {d}x \\&= \int _{{\widetilde{Q}}_1} \left| \sum _{j=1}^n j^{-\frac{1}{p}} 2^{j-n} n \sum _{Q_{n,k} \subset Q_{0,0}} a_{n,k}(x)\right| \;\mathrm {d}x \\&= \int _{{\widetilde{Q}}_1} \left( \sum _{j=1}^n j^{-\frac{1}{p}} 2^{j-n} n\right) \left| \sum _{Q_{n,k} \subset Q_{0,0}} a_{n,k}(x)\right| \;\mathrm {d}x \\&= \sum _{j=1}^n j^{-\frac{1}{p}} 2^{j-n} n \sum _{Q_{n,k} \subset Q_{0,0}} \int _{{\widetilde{Q}}_1} | a_{n,k}(x)| \;\mathrm {d}x \\&= c n 2^{-n} \sum _{j=1}^n j^{-\frac{1}{p}} 2^j \ge c n^{1-\frac{1}{p}}. \end{aligned}$$

Thus $\Vert f_n|L_1({\widetilde{Q}}_\nu )\Vert = \Vert f_n|L_1({\widetilde{Q}}_1)\Vert \rightarrow \infty$ if $n\rightarrow \infty$ since $1<p$. The local base at zero in $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$ is given by the sets $V_{k,\varepsilon } = \{f: \Vert f| L_1({\widetilde{Q}}_\nu )\Vert <\varepsilon ,\quad \nu =1,\ldots , k\}$. So the sequence $f_n$ is not bounded in $L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$. This contradicts the continuity of the embedding (i). $\square$

Corollary 3.5

Let $0<p\le u< v<\infty$, $1\le q \le v$ and $s=\frac{d}{u}- \frac{d}{v}$. Then the following assertions are equivalent:

(i):

${{\mathcal {E}}}^s_{u,p,\infty }({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}}^d)$,

(ii):

${{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {M}}}_{v, q}({{{\mathbb {R}}}}^d)$,

(iii):

$q\le v\frac{p}{u}$.

Proof

Step 1. First we prove the sufficiency of the condition (iii). If $p= \frac{u}{v}$, then $\frac{p}{u}\sigma _p = \frac{d}{u}- \frac{d}{v}=s$. So (i) follows from Theorem 3.2. If $p > \frac{u}{v}$, then $s= \frac{d}{u}- \frac{d}{v}>0$ and $q_{\max }=\frac{pv}{u}>1$. In that case the embedding is a consequence of Sobolev embeddings [9, Thm. 3.2], the Paley–Littlewood formula (2.5) and the embeddings between Morrey spaces.

Similarly, the embedding (ii) in the case $p= \frac{u}{v}$ follows from Theorem 3.4 since $q= 1$. To prove (ii) for $p > \frac{u}{v}$ and $1<q\le p\frac{v}{u}$ one can use the Franke–Jawerth embeddings for smoothness Morrey spaces, cf. [9, Thm. 4.3]. Indeed, we have

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {E}}}^0_{v,q,2}({{{\mathbb {R}}}}^d) = {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}}^d). \end{aligned}$$

If $p>\frac{u}{v}$ and $q=1$, then

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {N}}}^s_{u,\frac{u}{v},q}({{{\mathbb {R}}}}^d) \hookrightarrow {{\mathcal {M}}}_{v,1}({{{\mathbb {R}}}}^d), \end{aligned}$$

the statement coincides with our previous statement, Theorem 3.4. We refer to [8] for a proof of the first embedding.

Step 2. Now we prove the necessity of the conditions.

First we assume that

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {M}}}_{v, q}({{{\mathbb {R}}}}^d). \end{aligned}$$

(3.7)

The last embeddings implies that the space ${{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}}^d)$ consists of locally integrable functions, so $s\ge \frac{p}{u}\sigma _p$, cf. Theorem 3.3. in [10]. This implies $p\ge \frac{u}{v}$. If $1<q$, then by (2.4) and (2.5) we get

$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{v, q}({{{\mathbb {R}}}}^d)= {{\mathcal {E}}}^0_{v, q,2}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{v, q,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$

But Theorem 3.3 of [8], implies $q\le \frac{pv}{u}$. If $q= 1$, then the condition $q\le \frac{pv}{u}$ is satisfied automatically since we have already proved that $p\ge \frac{u}{v}$.

Using the same argument as above we can prove that $q\le \frac{pv}{u}$ if

$$\begin{aligned} {{\mathcal {E}}}^s_{u,p,\infty }({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {M}}}_{v, q}({{{\mathbb {R}}}}^d) . \end{aligned}$$

$\square$

Remark 3.6

The embeddings (i) and (ii) of the last corollary hold also for $q< 1$ if $v\frac{p}{u}\ge 1$ since then we increase the target space whereas the source space is unchanged or smaller.

Remark 3.7

In [1], Adams proved that $I_s$, the Riesz potential of order $s \in (0,d)$, maps ${{\mathcal {M}}}_{u, p}({{{\mathbb {R}}}}^d)$ into ${{\mathcal {M}}}_{v, q}({{{\mathbb {R}}}}^d)$ if $1<p\le u<\infty$, $1<q\le v<\infty$,

$$\begin{aligned} \frac{p}{u}=\frac{q}{v} \quad \text {and} \quad s=\frac{d}{u}- \frac{d}{v}, \end{aligned}$$

cf. also [2, Theorem 7.1]. This implies the following embeddings of Triebel–Lizorkin–Morrey spaces ${{\mathcal {E}}}^s_{u,p,2}({{{\mathbb {R}}}}^d)\hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}}^d)$. So Corollary 3.5 gives the sharp version of the above statement.

Remark 3.8

Another class of generalisations of Besov and Triebel-Lizorkin spaces are Besov-type spaces ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ and Triebel-Lizorkin-type spaces ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, $0 < p,q \le \infty$ (with $p<\infty$ in case of ${F}_{p,q}^{s,\tau }$), $\tau \ge 0$, $s \in {{\mathbb {R}}}$ introduced in [45]. The spaces are strictly related to ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ spaces if $0\le \tau < \frac{1}{p}$. Namely

$$\begin{aligned} {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})= {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \qquad \text {if} \qquad 0\le \tau = \frac{1}{p}-\frac{1}{u}< \frac{1}{p}, \end{aligned}$$

(3.8)

and

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\qquad \text {if} \qquad 0\le \tau = \frac{1}{p}-\frac{1}{u}< \frac{1}{p}. \end{aligned}$$

(3.9)

The Besov–Morrey and Besov-type spaces coincide only if $\tau =0$ or $q=\infty$. In contrast to the spaces ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ these scales are embedded into each other like their classical counterparts, that is,

$$\begin{aligned} B^{s,\tau }_{p,\min (p,q)}({{{\mathbb {R}}}^d})\, \hookrightarrow \, {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\, \hookrightarrow \,B^{s,\tau }_{p,\max (p,q)}({{{\mathbb {R}}}^d}), \end{aligned}$$

(3.10)

whenever $0<p<\infty$, $0<q\le \infty$, $s\in {{\mathbb {R}}}$, $\tau \ge 0$. Similar assertions for homogeneous spaces can be found in [32, Thm. 1.1].

Using these relations we can easily transfer our results to the new class of function spaces if $0\le \tau <\frac{1}{p}$. In particular, if $0\le \tau <\frac{1}{p}$ and $s=(1-\tau p)\sigma _p$, then the following three conditions are equivalent:

(i):

${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \hookrightarrow L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})$,

(ii):

${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{v,\max (p,1)}({{{\mathbb {R}}}^d})$, where $v=\frac{p}{(1-\tau p)\min (p,1)}$,

(iii):

either $p\ge 1$ and $q\le 2$, or $0<p<1$.

The above statement improves Theorem 3.8 in [10] and corrects some misprint concerning non-embeddings of ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ spaces stated in formula (3.31) there.

Moreover we have the following counterpart of Corollary 3.5. If

$$\begin{aligned} 0\le \tau<\frac{1}{p}, \quad \frac{p}{1-p\tau }<v<\infty , \quad s=d\left( \frac{1}{p} - \tau \right) - \frac{d}{v}, \quad 1\le q \le v, \end{aligned}$$

(3.11)

then

$$\begin{aligned} F^{s,\tau }_{p,\infty }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}^d})\quad \iff \quad q\le v(1- p\tau ). \end{aligned}$$

(3.12)

This is a direct consequence of (3.8) and Corollary 3.5. In case of spaces ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ we have a partial counterpart only at the moment: assume (3.11) and $q \le p$. Then

$$\begin{aligned} B^{s,\tau }_{p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}^d}), \end{aligned}$$

(3.13)

as by assumption, $q\le p<v(1-p\tau )$; thus (3.12) together with (3.10) conclude the argument. In case of $q>p$ the situation is not yet complete: while the embedding (3.9) together with Corollary 3.5 always lead to $q\le v(1-p\tau )$ whenever ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}^d})$, the sufficiency is not clear in all cases: assume, in addition to (3.11) that $p<q\le \frac{v}{\tau v+1}$. Then using some Franke-Jawerth embedding, cf. [46, Thm. 3.10],

$$\begin{aligned} {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\hookrightarrow F^{\sigma ,\tau }_{q,\infty }({{{\mathbb {R}}}^d}),\quad \sigma = s-\frac{d}{p}+\frac{d}{q}<s, \end{aligned}$$

and, by (3.12) again,

$$\begin{aligned} F^{\sigma ,\tau }_{q,\infty }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{v,q}({{{\mathbb {R}}}^d})\quad \text {if}\quad q\le v(1-q\tau ), \end{aligned}$$

which is satisfied by our additional assumption on q, we get (3.13).

4 Embeddings with Smoothness $\frac{d}{u}$

In this section we are interested in embeddings of spaces with smoothness $s=\frac{d}{u}$. This is once more the borderline smoothness since if the smoothness is strictly bigger than $\frac{d}{u}$, then the space consists of bounded functions. We describe the properties of the functions belonging to the spaces with smoothness $s=\frac{d}{u}$ in terms of exponential Orlicz–Morrey spaces and some generalised Morrey spaces.

Our approach is based on the extrapolation argument that in the case of Besov and Triebel-Lizorkin spaces was elaborated by Triebel in [39]. We follow the general idea of his work. The extrapolation inequalities we need are formulated in the following lemma.

Lemma 4.1

Let $0< p\le u<\infty$, $0<q\le \infty$, $p<r<\infty$ and $v=\frac{u r}{p}$.

(i):

If $r\ge 1$, then there is a constant $C>0$ depending on d, p, u and q but independent of v, r such that for any $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})$ the following inequalities hold

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert&\le C v^{1-\frac{1}{q}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert \qquad \text {if}\quad q\ge 1, \end{aligned}$$

(4.1)

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert&\le C \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert \qquad \text {if}\quad q\le 1. \end{aligned}$$

(4.2)

(ii):

If $p<r< 1$, then there is a constant $C>0$ depending on d, p, u but independent of v and r such that for any $f\in {\mathcal {N}}^{\frac{d}{u}}_{u,p,\infty }({{{\mathbb {R}}}^d})$ the following inequality holds

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert \le C \Vert f|{\mathcal {N}}^{\frac{d}{u}}_{u,p,\infty }({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.3)

Remark 4.2

The constant C is independent of v in that sense that it depends on u and p but takes the same value as far as the the quotient $\frac{u}{p}$ is constant.

In [31] the authors proved the following inequality for the Riesz potential $I_s$ of order $0<s<1$,

$$\begin{aligned} \Vert I_s f|{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert \le C v \, \Vert f|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert , \qquad \frac{u}{p}=\frac{v}{r}<u\quad \text {and}\quad \frac{1}{v}=\frac{1}{u}-(1-s). \end{aligned}$$

This leads to an estimate similar to (4.1) if $d<u<v<ur$ and $q\le \min (p,2)$, but not as precise as (4.1).

Proof

First we prove that

$$\begin{aligned} |\psi (x)|\le C \Vert \psi |{{\mathcal {M}}}_{u,p}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

(4.4)

if ${\mathrm {supp}\,}{\mathcal {F}}\psi \subset B(0,2)$ and $0<p\le u<\infty$. The proof is standard and based on the Plancherel–Polya–Nikol’skii inequality

$$\begin{aligned} \sup _{y\in {{\mathbb {R}}}^d} \frac{|\psi (x-y)|}{(1+|y|)^{d/\delta }} \le C \Big (M(|\psi |^\delta )(x)\Big )^\frac{1}{\delta }, \qquad x\in {{\mathbb {R}}}^d,\; \delta >0, \end{aligned}$$

(4.5)

cf. [38, Theorem 1.3.1]. Here M stands for the Hardy-Littlewood maximal operator, as usual.

We repeat the argument for completeness, cf. also [20]. One can easily prove that if $|x-y|\le 2$, then

$$\begin{aligned} |\psi (x)| \le c \sup _{z\in {{\mathbb {R}}}^d} \frac{|\psi (z)|}{(1+|z-y|)^{d/\delta }} \le C \Big (M(|\psi |^\delta )(y)\Big )^\frac{1}{\delta }. \end{aligned}$$

(4.6)

So, if $\delta <p$, then the boundedness of the maximal function in Morrey spaces, cf. [3], gives us

$$\begin{aligned} |\psi (x)|&\le c \bigg ( |B(x,2)|^{-1} \int _{B(x,2)} \Big (M(|\psi |^\delta )(y)\Big )^\frac{p}{\delta } \;\mathrm {d}y\bigg )^{1/p} \\&\le C \left\| \Big (M(|\psi |^\delta )\Big )^\frac{1}{\delta } |{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\right\| = C \left\| M(|\psi |^\delta )| {{\mathcal {M}}}_{u/\delta ,p/\delta }({{\mathbb {R}}}^d)\right\| ^{\frac{1}{\delta }} \\&\le C \left\| |\psi |^\delta |{{\mathcal {M}}}_{u/\delta ,p/\delta }({{{\mathbb {R}}}^d}) \right\| ^{\frac{1}{\delta }} = C \Vert \psi |{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert , \qquad x\in {{{\mathbb {R}}}^d}. \end{aligned}$$

Thus, if $p<r$, then for any cube Q we have

$$\begin{aligned} \Big (\int _Q |\psi (x)|^r \;\mathrm {d}x\Big )^\frac{1}{r}&\le \sup _{x\in Q} |\psi (x)|^{1-\frac{p}{r}} \Big (\int _Q |\psi (x)|^p \;\mathrm {d}x \Big )^\frac{1}{r} \\&\le C |Q|^{\frac{1}{r}-\frac{p}{ru}} \Vert \psi |{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

So

$$\begin{aligned} \Vert \psi |{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert \le C \Vert \psi |{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.7)

The last inequality implies

$$\begin{aligned} \Vert {\mathcal {F}}^{-1}\big (D_{2^j}\varphi _j {\mathcal {F}}f\big )| {{\mathcal {M}}}_{v,r}({{\mathbb {R}}}^d)\Vert \le C \Vert {\mathcal {F}}^{-1}\big (D_{2^j}(\varphi _j {\mathcal {F}}f)\big )|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert \end{aligned}$$

(4.8)

since ${\mathrm {supp}\,}D_{2^j}(\varphi _j {\mathcal {F}}f) \subset B(0,2)$, where $D_\delta g(x) = g(\delta x)$. Now the formula for Fourier dilations and the relation between dilations and the Morrey norms give us

$$\begin{aligned} \Vert {\mathcal {F}}^{-1}\big (\varphi _j {\mathcal {F}}f\big )| {{\mathcal {M}}}_{v,r}({{\mathbb {R}}}^d)\Vert \le C 2^{jd(\frac{1}{u}-\frac{1}{v})} \Vert {\mathcal {F}}^{-1}\big (\varphi _j {\mathcal {F}}f\big )|{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.9)

In consequence, if $1\le r$ and $1<q\le \infty$ we have

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{\mathbb {R}}}^d)\Vert&\le \sum _{k=0}^\infty \Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert \\&\le C \sum _{k=0}^\infty 2^{kd \Big (\frac{1}{u}-\frac{1}{v}\Big )}\Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert \\&\le C \Big (\sum _{k=0}^\infty 2^{-kd\frac{q'}{v}}\Big )^\frac{1}{q'}\Big (\sum _{k=0}^\infty 2^{k\frac{d}{u}q}\Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{u,p}({{\mathbb {R}}}^d)\Vert ^q\Big )^\frac{1}{q} \\&\le C v^{1-\frac{1}{q}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert , \end{aligned}$$

with obvious changes if $q=\infty$; similarly,

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{\mathbb {R}}}^d)\Vert \le C \sum _{k=0}^\infty 2^{kd(\frac{1}{u}-\frac{1}{v})}\Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{u,p}({{\mathbb {R}}}^d)\Vert \le C \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert \end{aligned}$$

if $1\le r$ and $0<q\le 1$.

Now let $p< r< 1$ and $q=\infty$. We have

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert ^r&\le \sum _{k=0}^\infty \Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})\Vert ^r \\&\le C\sum _{k=0}^\infty 2^{kdr(\frac{1}{u}-\frac{1}{v})}\Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert ^r \\&\le C \Big (\sup _{k\in {{\mathbb {N}}}_0}2^{k\frac{d}{u}}\Vert {\mathcal {F}}^{-1}\varphi _k {\mathcal {F}}f| {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\Vert \Big )^r\ \sum _{k=0}^\infty 2^{-kd\frac{r}{v}} . \end{aligned}$$

But

$$\begin{aligned} \Big (\sum _{k=0}^\infty 2^{-kd\frac{r}{v}} \Big )^\frac{1}{r} \le C_{u,p} <\infty , \end{aligned}$$

so

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v,r}({{\mathbb {R}}}^d)\Vert \le C \Vert f|{\mathcal {N}}^{\frac{d}{u}}_{u,p,\infty }({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

This concludes the argument. $\square$

4.1 Embeddings in Orlicz–Morrey Spaces

The Orlicz–Morrey spaces considered below were introduced by Nakai [19]. They are a generalisation of both Morrey and Orlicz spaces. For different types of Orlicz–Morrey spaces we refer to e.g. [4, 6].

Definition 4.3

Let $\Phi :[0,\infty ) \rightarrow [0, \infty )$ be a Young function, i.e., a continuous convex function with $\Phi (0)=0$ and $\lim _{t\rightarrow \infty }\Phi (t)=\infty$. For $1\le r<\infty$ and a cube Q we put

$$\begin{aligned} \Vert f\Vert _{(r,\Phi );Q}:= \inf \left\{ \lambda >0: |Q|^{\frac{1}{r}-1}\int _Q\Phi \Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x\le 1 \right\} . \end{aligned}$$

The Orlicz–Morrey space ${{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})$ is the set of all measurable functions f such that

$$\begin{aligned} \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert := \sup _{Q\in {\mathcal {Q}}}\Vert f\Vert _{(r,\Phi );Q} < \infty . \end{aligned}$$

We consider also the following expression

$$\begin{aligned} \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert ^*:=\inf \left\{ \lambda >0: \sup _{Q\in {\mathcal {Q}}} |Q|^{\frac{1}{r}-1}\int _Q\Phi \Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x\le 1 \right\} . \end{aligned}$$

Since $\sup _Q\inf _\lambda \le \inf _\lambda \sup _Q$ we have $\Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert \le \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert ^*$. In the next lemma we show that if the Young function $\Phi$ is of exponential type, then both expressions are equivalent and the space ${{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})$ can be characterised by extrapolation.

Lemma 4.4

Let $0< p\le u<\infty$, $0<q < \infty$, and $\Phi _{p,q}(t):=t^p \exp ( t^q)$. Then f belongs to the Orlicz–Morrey space ${{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})$ if, and only if,

$$\begin{aligned} \sup _{j\ge 1}j^{-1/q}\, \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert <\infty , \end{aligned}$$

where $p(j)=p+jq$ and $v(j)=\frac{u}{p}p(j)$. Moreover,

$$\begin{aligned} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert \sim \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert ^* \sim \sup _{j\ge 1}j^{-1/q}\, \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

Proof

Step 1. It is sufficient to prove that there are constants $c,C>0$ such that for any measurable function f we have

$$\begin{aligned} c \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert ^*\le & {} \sup _{j\ge 1}j^{-1/q}\, \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \nonumber \\\le & {} C \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.10)

Consider a dyadic cube Q and $\lambda >0$. Using the Taylor expansion of $\Phi _{p,q}$ and the Stirling’s formula, we have

$$\begin{aligned} \int _Q\Phi _{p,q} \Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x&=\sum _{j=1}^{\infty } \frac{1}{j !} \int _Q \frac{|f(x)|^{p+jq}}{\lambda ^{p+jq}} \;\mathrm {d}x \\&= \sum _{j=1}^{\infty } j^{-j} e^j (2\pi j)^{-1/2} \lambda ^{-(p+jq)}\int _Q |f(x)|^{p+jq} \;\mathrm {d}x . \end{aligned}$$

Let $\kappa \in {{\mathbb {R}}}$ (be at our disposal) and let

$$\begin{aligned} \lambda _j:=(2\pi )^{\frac{1}{2(p+qj)}} e^{-\frac{j}{p+qj}} j^{\left( \kappa -\frac{p}{q}+\frac{1}{2}\right) /(p+qj)}, \quad j\in {{\mathbb {N}}}. \end{aligned}$$

It can be easily seen that the sequence $(\lambda _j)_j$ converges to $e^{-1/q}$, thus there are positive constants $c_0,c_1$ such that $0<c_0<\lambda _j<c_1<\infty$ for any $j\in {{\mathbb {N}}}$. Therefore,

$$\begin{aligned} \int _Q\Phi _{p,q}\Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x \sim \sum _{j=1}^{\infty } j^{\kappa -\frac{p}{q}-j} \lambda ^{-(p+jq)}\int _Q |f(x)|^{p+jq} \;\mathrm {d}x . \end{aligned}$$

(4.11)

Step 2. We prove the left-hand side inequality in (4.10). Assume that

$$\begin{aligned} \sup _{j\ge 1}j^{-1/q}\, \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \le \lambda . \end{aligned}$$

Then, for any dyadic cube Q and any $j\in {{\mathbb {N}}}$, it holds

$$\begin{aligned} \lambda ^{-(p+qj)} j^{-\frac{p+qj}{q}} |Q|^{\frac{p}{u}-1 }\int _Q |f(x)|^{p(j)} \;\mathrm {d}x \le 1. \end{aligned}$$

For this $\lambda$, inserting the above inequality in (4.11) entails

$$\begin{aligned} \int _Q\Phi _{p,q}\Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x \le c \sum _{j=1}^{\infty } j^{\kappa } |Q|^{1-\frac{p}{u}}. \end{aligned}$$

By choosing $\kappa <-1$, we conclude that

$$\begin{aligned} \sup _{Q\in {{{\mathcal {Q}}}}} |Q|^{\frac{p}{u}-1} \int _Q\Phi _{p,q}\Bigl ( \frac{|f(x)|}{\lambda }\Bigr )\;\mathrm {d}x \le c. \end{aligned}$$

Step 3. It remains to show the right-hand side inequality in (4.10). Let now

$$\begin{aligned} \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert := \sup _{Q\in {\mathcal {Q}}}\Vert f\Vert _{(r,\Phi );Q} \le 1 \end{aligned}$$

and let $\varepsilon >0$. Then for any dyadic cube Q there is $\lambda _Q$, $0<\lambda _Q\le \Vert f\Vert _{(r,\Phi );Q}+\varepsilon$, such that

$$\begin{aligned} |Q|^{\frac{p}{u}-1} \int _Q\Phi _{p,q}\Bigl ( \frac{|f(x)|}{\lambda _Q}\Bigr )\;\mathrm {d}x \le 1 . \end{aligned}$$

Then, by (4.11),

$$\begin{aligned} \sum _{j=1}^{\infty } j^{\kappa -\frac{p}{q}-j} \lambda _Q^{-(p+jq)} |Q|^{\frac{p(j)}{v(j)}-1} \int _Q |f(x)|^{p+jq} \;\mathrm {d}x \le c \end{aligned}$$

for all dyadic cubes Q and for some positive constant c independent of Q. Hence, for any $j\in {{\mathbb {N}}}$ and any dyadic cube Q, it holds

$$\begin{aligned} j^{-\frac{p}{q}-j} |Q|^{\frac{p(j)}{v(j)}-1} \int _Q |f(x)|^{p+jq} \;\mathrm {d}x \le c j^{-\kappa } \lambda _Q^{p+jq}\le c j^{-\kappa } (\Vert f\Vert _{(r,\Phi );Q}+\varepsilon )^{p+jq}. \end{aligned}$$

Taking the infimum over $\varepsilon$ we get

$$\begin{aligned} j^{-\frac{p}{q}-j} |Q|^{\frac{p(j)}{v(j)}-1} \int _Q |f(x)|^{p+jq} \;\mathrm {d}x \le c j^{-\kappa } \Vert f\Vert _{(r,\Phi );Q}^{p+jq} \end{aligned}$$

and afterwards taking the supremum over all dyadic cubes gives

$$\begin{aligned} j^{-1/q} \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \le c^{1/p(j)} j^{-\kappa /p(j)} \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

Now we take the supremum over all j. The expression on the right-hand side is of the size $j^{1/j}$ so it can be bounded by a positive constant. This yields

$$\begin{aligned} \sup _{j\ge 1} j^{-1/q} \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \le c \Vert f\vert {{\mathcal {M}}}_{r,\Phi }({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

$\square$

Theorem 4.5

Let $0< p\le u<\infty$ and $1<q \le \infty$. Then

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q'}}({{{\mathbb {R}}}^d}), \end{aligned}$$

where $q'$ is the conjugate exponent of q.

Proof

For each $j\in {{\mathbb {N}}}$, by Lemma 4.1 with $r=p(j)=p+qj$ and $v(j)=\frac{u}{p}p(j)$,

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \le c \{v(j)\}^{1/q'} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert , \end{aligned}$$

where c is a positive constant independent of v(j) and p(j), and thus of j. Since $\{v(j)\}^{1/q'} \sim j^{1/q'}$, using also Lemma 4.4, we get

$$\begin{aligned} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q'}}({{{\mathbb {R}}}^d})\Vert \sim \sup _{j\ge 1}j^{-1/q'}\, \Vert f\vert {{\mathcal {M}}}_{v(j),p(j)}({{{\mathbb {R}}}^d})\Vert \le c \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

$\square$

Corollary 4.6

Let $0< p\le u<\infty$ and $0<q\le \infty$. Then

$$\begin{aligned} {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,1}}({{{\mathbb {R}}}^d}). \end{aligned}$$

Proof

The result follows from the above theorem and elementary embeddings:

$$\begin{aligned} {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,\infty }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,\infty }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,1}}({{{\mathbb {R}}}^d}). \end{aligned}$$

$\square$

Remark 4.7

According to [30, Cor. 1.5] it holds

$$\begin{aligned} {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,2}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p}}({{{\mathbb {R}}}^d}), \end{aligned}$$

where $1<p\le u<\infty$ and

$$\begin{aligned} \Phi _p(t):=\sum _{j=j_p}^{\infty }\frac{t^j}{j!}, \quad t\ge 0, \end{aligned}$$

with $j_p:=\min \{j\in {{\mathbb {N}}}: j\ge p\}$. Since there exists a constant $c>0$ such that

$$\begin{aligned} \Phi _{p,1}(t) \ge c \Phi _p(t) \quad \text {for all} \quad t\ge 0, \end{aligned}$$

it turns out that

$$\begin{aligned} {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,1}}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p}}({{{\mathbb {R}}}^d}). \end{aligned}$$

Hence Corollary 4.6 does not only extend [30, Cor. 1.5] from $q=2$ to any $1<q\le \infty$, it improves it.

4.2 Embeddings in Generalised Morrey Spaces

Now we turn to the generalised Morrey spaces. The spaces were extensively studied, cf. Nakai [18], Nakamura et al. [20], Sawano and Wadade [30] and the references given there. Let $0< r < \infty$ and let $\varphi :[0,\infty )\rightarrow [0,\infty )$ be a suitable function. For a locally r-integrable function f we put

$$\begin{aligned} \Vert f|{{\mathcal {M}}}_r^{\varphi }\Vert :=\, \sup _{Q\in {\mathcal {Q}}} \varphi (|Q|) \biggl ( |Q|^{-1}\int _{Q} |f(y)|^r \;\mathrm {d}y \biggr )^{1/r}\, . \end{aligned}$$

(4.12)

The space ${{\mathcal {M}}}_r^{\varphi }({{{\mathbb {R}}}^d})$ is the set of all measurable functions f for which the quasi-norm (4.12) is finite. If $\varphi (t)=t^{1/u}$, $0<r\le u$, then the definition coincides with the definition of the Morrey space ${{\mathcal {M}}}_{u,r}({{{\mathbb {R}}}^d})$. Since we will work with the given examples of functions $\varphi$ we avoid the discussions which functions $\varphi$ define the reasonable spaces, and we refer the interested reader to the above mentioned papers.

We start with a proposition that somehow compares the Orlicz–Morrey spaces with generalised Morrey spaces we will use.

Proposition 4.8

Let $0< p\le u<\infty$, $1\le q < \infty$ and $r\ge p+q$. Then there is a positive constant C depending on u, p, q and r such that the following inequality

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}\le c \,(1+|Q|)^{-\frac{p}{ru}} \left( \log \left( e+|Q|^{-1}\right) \right) ^{\frac{1}{q}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert \end{aligned}$$

holds for all dyadic cubes Q and all $f\in {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})$.

Proof

It should be clear that it is sufficient to consider the case $r=p+j_0q$ for some $j_0\in {{\mathbb {N}}}$. Note that this refers to $r=p(j_0)$ in the notation of Lemma 4.4.

Let Q be a dyadic cube with side length $2^{-j}$ and $|Q|=2^{-jd}$, $j\in {{\mathbb {Z}}}$. Assume first $j\le 0$. Then by Lemma  4.4 we get

$$\begin{aligned} \left( \frac{1}{|Q|} \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}&\le C \,j_0^{1/q} |Q|^{-\frac{p}{ur}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert \nonumber \\&\le C (1+|Q|)^{-\frac{p}{ur}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.13)

Next we assume that $j\ge j_0$. Then Hölder’s inequality and Lemma 4.4 imply

$$\begin{aligned} \left( \frac{1}{|Q|} \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}&\le \left( \frac{1}{|Q|} \int _Q|f(x)|^{p(j)}\;\mathrm {d}x\right) ^{\frac{1}{p(j)}} \nonumber \\&\le C \,j^{1/q} |Q|^{-\frac{p}{u p(j)}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert \nonumber \\&\le C \left( \log (e+|Q|^{-1})\right) ^{\frac{1}{q}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert \end{aligned}$$

(4.14)

since

$$\begin{aligned} \log (e+|Q|^{-1}) \sim j \quad \text {and}\quad |Q|^{-\frac{p}{u p(j)}}= 2^{\frac{jdp}{u (p+ jq)}} \le 2^{\frac{dp}{u q}} . \end{aligned}$$

At the end we consider the cubes with $0<j < j_0$. We have

$$\begin{aligned}&2^{-\frac{j_0 d}{v(j_0)}} \sup _{Q:\; |Q|=2^{-jd},\;0<j < j_0}\left( \frac{1}{|Q|} \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \\&\quad \le \ \sup _{Q} |Q|^{\frac{1}{v(j_0)}} \left( \frac{1}{|Q|} \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \le C \,j_0^{\frac{1}{q}}\ \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert , \end{aligned}$$

by Lemma 4.4, recall $r=p(j_0)$. Thus

$$\begin{aligned} \left( \frac{1}{|Q|} \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \le C \left( \log (e+|Q|^{-1})\right) ^{\frac{1}{q}} \Vert f\vert {{\mathcal {M}}}_{\frac{u}{p},\Phi _{p,q}}({{{\mathbb {R}}}^d})\Vert . \end{aligned}$$

(4.15)

Consequently, the inequalities (4.13)-(4.15) prove the proposition. $\square$

The next corollary is an immediate consequence of Theorem 4.5, Corollary 4.6 and Proposition 4.8.

Corollary 4.9

Let $0< p\le u<\infty$ and $0< q \le \infty$.

(i):

If $1< q \le \infty$ and $r\ge p+q'$ . Then there is a positive constant C such that the inequality

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \nonumber \\&\quad \le c \,(1+|Q|)^{-\frac{p}{ru}} \left( \log \left( e+|Q|^{-1}\right) \right) ^{\frac{1}{q'}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

(4.16)

holds for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})$.

(ii):

If $0 < q \le \infty$ and $r\ge p + 1$ . Then there is a positive constant C such that the inequality

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \nonumber \\&\quad \le c \,(1+|Q|)^{-\frac{p}{ru}} \left( \log \left( e+|Q|^{-1}\right) \right) \Vert f|{{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

(4.17)

holds for all dyadic cubes Q and all $f\in {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})$.

The inequalities (4.16) and (4.17) can be extended to the smallest values of r and q. Recall that $q'$ is defined by $q'=\frac{q}{q-1}$ if $1<q<\infty$ and $q'=\infty$ if $0<q\le 1$, where the usual convention $1/\infty =0$ is assumed.

Theorem 4.10

Let $0< p\le u<\infty$ and $0<q\le \infty$. If $1\le r<\infty$, then there exists a positive constant c such that

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}\\&\quad \le c \,(1+|Q|)^{-\frac{\min \left( 1,\frac{p}{r}\right) }{u }} \left( \log \left( e+|Q|^{-1}\right) \right) ^{\frac{1}{q'}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

holds for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d})$.

Proof

Step 1. Given $r\ge 1$, let $r_0$ be such that $r_0>\max (p,r)$. By Lemma 4.1, there exists a positive constant c, not depending on $r_0$ and $v_0$, with $v_0=\frac{ur_0}{p}$, such that the inequality

$$\begin{aligned} |Q|^{\frac{1}{v_0}-\frac{1}{r_0}}\left( \int _Q|f(x)|^{r_0}\;\mathrm {d}x\right) ^{\frac{1}{r_0}}\le c\, v_0^{\frac{1}{q'}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

(4.18)

holds for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$. The Hölder inequality and (4.18) yield

$$\begin{aligned} \left( \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}&\le |Q|^{\frac{1}{r}-\frac{1}{r_0}} \left( \int _Q|f(x)|^{r_0}\;\mathrm {d}x\right) ^{\frac{1}{r_0}} \nonumber \\&\le c\, |Q|^{\frac{1}{r}-\frac{1}{v_0}} \, v_0^{\frac{1}{q'}} \, \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

(4.19)

for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$.

For convenience we deal with the case $q\ge 1$, the other case is even easier. Assume first that the cubes are small, that is, they satisfy $|Q|< e^{-u\max (1,\frac{r}{p})}$. Then $v_0=\log (|Q|^{-1})$ and $r_0=\frac{p}{u} v_0$ satisfy the above assumptions. Hence (4.19) leads to

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}\le c\, \left( \log \left( e+ |Q|^{-1}\right) \right) ^{\frac{1}{q'}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert , \end{aligned}$$

(4.20)

for any small enough cube Q with $|Q|< e^{-u\max (1,\frac{r}{p})}$, and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$. It remains to deal with the bigger cubes.

Step 2. Let $r\ge p$. Elementary embeddings and [8, Thm. 3.3] yield

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{\frac{ur}{p},r,1} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{\frac{ur}{p},r,2} ({{{\mathbb {R}}}^d}) ={{\mathcal {M}}}_{\frac{ur}{p},r} ({{{\mathbb {R}}}^d}), \quad \text {if}\quad r>1, \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{\frac{u}{p},1,1} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{\frac{u}{p},1} ({{{\mathbb {R}}}^d}), \quad \text {if}\quad r=1, \end{aligned}$$

where the last embedding is due to Theorem 3.4. Therefore, for $r\ge 1$ and $r\ge p$, we have

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}\le c\, |Q|^{-\frac{p}{ur}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert , \end{aligned}$$

(4.21)

for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$. Together with Step 1 this completes the argument in case of $r\ge p$.

Step 3. Let $1\le r<p$. Using Hölder’s inequality we obtain

$$\begin{aligned} \left( \int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}&\le |Q|^{\frac{1}{r}-\frac{1}{p}} \left( \int _Q|f(x)|^{p}\;\mathrm {d}x\right) ^{\frac{1}{p}} \le |Q|^{\frac{1}{r}-\frac{1}{u}} \Vert f|{{\mathcal {M}}}_{u,p}({{\mathbb {R}}}^d)\Vert \nonumber \\&\le c |Q|^{\frac{1}{r}-\frac{1}{u}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert , \end{aligned}$$

(4.22)

for all dyadic cubes Q and all $f\in {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$, where in the last step we used the fact that

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,\infty } ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{u,p,1} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {E}}}^0_{u,p,2} ({{{\mathbb {R}}}^d}) ={{\mathcal {M}}}_{u,p} ({{{\mathbb {R}}}^d}). \end{aligned}$$

Again the final outcome in this case follows from Step 1 and (4.22). $\square$

Remark 4.11

The logarithmic term in Theorem 4.10 can not be removed if $1<q\le \infty$. One can consider the following example. Let $\psi \in C_0^\infty ({{{\mathbb {R}}}^d})$, $0\le \psi \le 1$, ${\mathrm {supp}\,}\psi \subset [-3/4, 3/4]^d$ and $\psi (x)=1$ if $x\in [-1/2, 1/2]^d$. Then the function

$$\begin{aligned} f(x) = \sum _{j=1}^\infty j^{-\frac{1}{r}}\psi (2^j x), \qquad 1<r<q, \end{aligned}$$

is an element of the space ${{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$ and by the atomic decomposition theorem $\Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \le (\sum _{j={1}}^\infty j^{-\frac{q}{r}})^\frac{1}{q}<\infty$. On the other hand, if $x\in [-2^{-\ell },2^{-\ell }]^d\setminus [-2^{-\ell -1},2^{-\ell -1}]^d$, $\ell \in {\mathbb {N}}$, then $f(x)\ge \sum _{j=1}^\ell j^{-\frac{1}{r}} \sim \ell ^{1-\frac{1}{r}}$. So if we take $Q=[-2^{-\ell },2^{-\ell }]^d$, then

$$\begin{aligned} \left( \log \left( e+|Q|^{-1}\right) \right) ^{\frac{1}{r'}} \sim \ell ^{\frac{1}{r'}}&\le \left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}} \\&\le C \left( \log \left( e+|Q|^{-1}\right) \right) ^{\frac{1}{q'}} \Vert f|{{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert . \end{aligned}$$

Remark 4.12

(i) In terms of embeddings in generalised Morrey spaces, what has been proved could be stated as follows. Let $0< p\le u<\infty$, $0<q\le \infty$, and $1\le r<\infty$. Then

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_r^{\varphi _{r,q}}({{{\mathbb {R}}}^d}) \end{aligned}$$

where

$$\begin{aligned} \varphi _{r,q}(t)= {\left\{ \begin{array}{ll} \big (\log (t^{-1})\big )^{-\frac{1}{q'}} &{} \text {if} \quad 0<t< e^{-\frac{u}{d}\max \big (1,\frac{r}{p}\big )}, \\ t^{\frac{d}{u}\min \big (1,\frac{p}{r}\big )} &{} \text {if} \quad t \ge e^{-\frac{u}{d}\max \big (1,\frac{r}{p}\big )} . \end{array}\right. } \end{aligned}$$

(4.23)

(ii) If we would consider local generalised Morrey spaces ${\mathcal {L}}{{\mathcal {M}}}_p^{\varphi }({{{\mathbb {R}}}^d})$, where the supremum taken in the definition of the norm is restricted to cubes with volume less or equal than 1 (cf. [42, p. 7]), then we can state that

$$\begin{aligned} {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,q} ({{{\mathbb {R}}}^d}) \hookrightarrow {\mathcal {L}}{{\mathcal {M}}}_r^{\varphi }({{{\mathbb {R}}}^d}), \qquad \varphi (t)=|\log (t)|^{-\frac{1}{q'}}, \quad 1\le r<\infty . \end{aligned}$$

Corollary 4.13

Let $0< p\le u<\infty$ and $0<q\le \infty$. If $1\le r<\infty$, then there exists a positive constant c such that

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q|f(x)|^{r}\;\mathrm {d}x\right) ^{\frac{1}{r}}\le c \,(1+|Q|)^{-\frac{\min \left( 1,\frac{p}{r}\right) }{u}} \log \left( e+|Q|^{-1}\right) \Vert f|{{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

holds for all dyadic cubes Q and all $f\in {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{\mathbb {R}}}^d)$.

Proof

The outcome is a direct consequence of Theorem 4.10 taking into account the embedding

$$\begin{aligned} {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^{\frac{d}{u}}_{u,p,\infty } ({{{\mathbb {R}}}^d}). \end{aligned}$$

$\square$

Remark 4.14

• (i) As in Remark 4.12, we can state the following: Let $0< p\le u<\infty$, $0<q\le \infty$, and $1\le r<\infty$. Then

  $$\begin{aligned} {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q} ({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_r^{\varphi _r}({{{\mathbb {R}}}^d}) \qquad \text {and} \qquad {{\mathcal {E}}}^{\frac{d}{u}}_{u,p,q} ({{{\mathbb {R}}}^d}) \hookrightarrow {\mathcal {L}}{{\mathcal {M}}}_r^{\varphi }({{{\mathbb {R}}}^d}) \end{aligned}$$

  with $\varphi _r=\varphi _{r,\infty }$ given by (4.23) and $\varphi (t)=|\log (t)|^{-1}$.

• (ii) In the particular case of $p>1$, $q=2$ and $r=1$, the result in the above corollary coincides with Theorem 5.1 of [30].

• (iii) In the particular case of $p=1$, $q=2$ and $r=1$, the result in the above corollary is comparable with Proposition 1.7 of [5].

5 Further Embeddings into Spaces with Smoothness $s=0$

In the preceding subsection we dealt with (limiting) embeddings of spaces ${{\mathcal {A}}}^{s}_{u,p,q}$ when $s=\frac{d}{u}$, into spaces of Orlicz–Morrey type or generalised Morrey type. In Corollary 3.5 the parallel setting was studied for embeddings into Morrey spaces ${\mathcal {M}}_{v,q}({{{\mathbb {R}}}^d})$ when $s=\frac{d}{u}-\frac{d}{v}$. For convenience we briefly recall the forerunners, that is, when ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ is embedded into classical Lebesgue spaces $L_r({{{\mathbb {R}}}^d})$, and into the space $C({{{\mathbb {R}}}^d})$ of bounded, uniformly continuous functions. We also consider the target space $\mathop {\mathrm {bmo}}({{\mathbb {R}}}^d)$, i.e., the local (non-homogeneous) space of functions of bounded mean oscillation, consisting of all locally integrable functions $\ f\in L_1^{\mathrm {loc}}({{\mathbb {R}}}^d)$ satisfying that

$$\begin{aligned} \left\| f \right\| _{\mathop {\mathrm {bmo}}({{{\mathbb {R}}}^d})}:= \sup _{|Q|\le 1}\; \frac{1}{|Q|} \int \limits _Q |f(x)-f_Q| \;\mathrm {d}x + \sup _{|Q|> 1}\; \frac{1}{|Q|} \int \limits _Q |f(x)| \;\mathrm {d}x<\infty , \end{aligned}$$

where Q appearing in the above definition runs over all cubes in ${{\mathbb {R}}}^d$, and $f_Q$ denotes the mean value of f with respect to Q, namely, $f_Q := \frac{1}{|Q|} \;\int _Q f(x)\;\mathrm {d}x$.

Most of these results have been obtained in different papers before, we recall it for completeness and to simplify the comparison with our new findings presented above.

Corollary 5.1

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

(i):

Then

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow C({{{\mathbb {R}}}^d})\quad \iff \quad {\left\{ \begin{array}{ll} s>\frac{d}{u}, &{} \text {or}\\ s=\frac{d}{u}&{} \text {and}\quad q\le 1, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow C({{{\mathbb {R}}}^d})\quad \iff \quad {\left\{ \begin{array}{ll} s>\frac{d}{u}, &{} \text {or}\\ s=\frac{d}{u}&{} \text {and}\quad u=p\le 1. \end{array}\right. } \end{aligned}$$

Here $C({{{\mathbb {R}}}^d})$ can be replaced by $L_\infty ({{{\mathbb {R}}}^d})$.

(ii):

Then

$$\begin{aligned} {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow \mathop {\mathrm {bmo}}({{{\mathbb {R}}}^d})\qquad \text {if and only if} \qquad s\ge \frac{d}{u}. \end{aligned}$$

(5.1)

(iii):

Let $1\le r<\infty$. If $p<u$, then ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ is never embedded into $L_r({{{\mathbb {R}}}^d})$. If $p=u$, then

$$\begin{aligned} {\mathcal {N}}^s_{u,u,q}({{{\mathbb {R}}}^d})\hookrightarrow L_r({{{\mathbb {R}}}^d}) \iff r\ge u\;\,\text {and}\;\, {\left\{ \begin{array}{ll} s>\frac{d}{u}-\frac{d}{r},&{} \text {or}\\ s=\frac{d}{u}-\frac{d}{r}&{} \text {and}\quad q\le r, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\mathcal {E}}^s_{u,u,q}({{{\mathbb {R}}}^d}) \hookrightarrow L_r({{{\mathbb {R}}}^d}) \iff r\ge u\;\,\text {and}\;\, {\left\{ \begin{array}{ll} s\ge \frac{d}{u}-\frac{d}{r}&{} \text {and}\;\, s>0,\ \text {or}\\ s=\frac{d}{u}-\frac{d}{r}=0 &{} \text {and}\;\, q\le 2. \end{array}\right. } \end{aligned}$$

Proof

Step 1.  Part (i) is well known, we refer to [8, Prop. 5.5] and [9, Prop. 3.8] for the Morrey situation when $p<u$, while the classical setting $p=u$ can be found in [40, Theorem 11.4].

Step 2.  We prove (ii). In case of ${{\mathcal {A}}}^{s}_{u,p,q}={{\mathcal {E}}}^{s}_{u,p,q}$, this follows from the analogous statement for $F^{s,\tau }_{p,q}({{\mathbb {R}}}^d)$ spaces and the coincidence ${{\mathcal {E}}}^{s}_{u,p,q}({{\mathbb {R}}}^d)=F^{s,\tau }_{p,q}({{\mathbb {R}}}^d)$ if $\tau =\frac{1}{p}-\frac{1}{u}$, cf. [47, Props. 5.13, 5.14] and (3.8). The similar statement for ${{\mathcal {N}}}^{s}_{u,p,q}({{\mathbb {R}}}^d)$ spaces

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{\mathbb {R}}}^d)\hookrightarrow \mathop {\mathrm {bmo}}({{\mathbb {R}}}^d)\qquad \text {if, and only if,}\qquad s\ge \frac{d}{u} \end{aligned}$$

(5.2)

can be proved analogously. Let $\tau =\frac{1}{p}-\frac{1}{u}$. Then, in view of (3.9) and the subsequent remark,

$$\begin{aligned} {\mathcal {N}}^s_{u,p,\infty }({{\mathbb {R}}}^d) = B^{s,\tau }_{p,\infty }({{\mathbb {R}}}^d)\hookrightarrow \mathop {\mathrm {bmo}}({{\mathbb {R}}}^d)= B^{0,\frac{1}{2}}_{2,2}({{\mathbb {R}}}^d) \end{aligned}$$

if $\frac{1}{p}-\frac{1}{u}- \frac{1}{2}\not = 0$, cf. [47, Prop. 5.10, Theorem 2.5]. If $\frac{1}{p}-\frac{1}{u}- \frac{1}{2} = 0$, then we can choose r such that $u<r<p$ and $\frac{1}{r}-\frac{1}{u}- \frac{1}{2} > 0$. Hence

$$\begin{aligned} {\mathcal {N}}^s_{u,p,\infty }({{\mathbb {R}}}^d) \hookrightarrow {\mathcal {N}}^s_{u,r,\infty }({{\mathbb {R}}}^d) \hookrightarrow \mathop {\mathrm {bmo}}({{\mathbb {R}}}^d). \end{aligned}$$

This proves the sufficiency of the conditions.

To prove necessity let us take $s<\frac{d}{u}$. If ${\mathcal {N}}^s_{u,p,\infty }({{\mathbb {R}}}^d) \hookrightarrow \mathop {\mathrm {bmo}}({{\mathbb {R}}}^d)$, then ${{\mathcal {E}}}^{s}_{u,p,q}({{\mathbb {R}}}^d)\hookrightarrow \mathop {\mathrm {bmo}}({{\mathbb {R}}}^d)$. This contradicts (5.1).

Step 3. Part (iii) in case of $p<u$ was proved in [8, 9] with a forerunner in [22], whereas the classical results for $p=u$ are well known. $\square$

Remark 5.2

A partial forerunner of (i) can be found in [27, Prop. 1.11] dealing with the sufficiency part; see also [34]. In some sense the embeddings into $C({{{\mathbb {R}}}^d})$ and $\mathop {\mathrm {bmo}}({{{\mathbb {R}}}^d})$ can be understood as limiting cases of Corollary 3.5 when $v\rightarrow \infty$, whereas the embedding into $L_r({{{\mathbb {R}}}^d})$ refers to the situation of $r=v=q$ in Corollary 3.5.

Remark 5.3

Based on arguments on the known properties of Triebel-Lizorkin type spaces one can easily strengthen Remark 3.6, in particular (3.12) with (3.11), as follows. Let $0<p<\infty$, $0<q\le \infty$, $1\le r\le v<\infty$, $s\in {{\mathbb {R}}}$, $\tau \ge 0$. Then

$$\begin{aligned} {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d}) \end{aligned}$$

(5.3)

if, and only if,

$$\begin{aligned} r\le v(1-p\tau ),\quad \text {and}\quad {\left\{ \begin{array}{ll} s>d \left( \frac{1}{p}-\tau -\frac{1}{v}\right) \ge 0, &{}\text {or}\\ s=d \left( \frac{1}{p}-\tau -\frac{1}{v}\right) >0, &{}\text {or}\\ s=d \left( \frac{1}{p}-\tau -\frac{1}{v}\right) =0,&{}\text {and}\quad q\le 2. \end{array}\right. } \end{aligned}$$

The case $0\le \tau <\frac{1}{p}$ is covered by Remark 3.6 together with the usual monotonicity for spaces ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ and ${{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})$, see also the forerunner [9, Thm. 3.1]. Thus it remains to disprove any embedding of type (5.3) whenever $\tau \ge \frac{1}{p}$. Assume first $\tau > \frac{1}{p}$ or $\tau = \frac{1}{p}$ and $q=\infty$. Thus the coincidence ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})=B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$, cf. [44], together with (5.3) would imply

$$\begin{aligned} B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d})={F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{v,r,\infty }({{{\mathbb {R}}}^d}), \end{aligned}$$

and hence $v=\infty$ in view of [8, Thm. 3.3]. But this contradicts our general assumption. Otherwise, if $\tau = \frac{1}{p}$ and $q<\infty$, then we may use [45, Prop. 2.6] which states, that ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \hookrightarrow B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$. We choose a number $\varrho$ such that $v<\varrho <\infty$, and apply an embedding proved by Marschall, cf. [13], to obtain

$$\begin{aligned} B^{s+\frac{d}{\varrho }}_{\varrho ,\infty }({{{\mathbb {R}}}^d})\hookrightarrow F^s_{\infty ,q}({{{\mathbb {R}}}^d})=F^{s,\frac{1}{p}}_{p,q}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {N}}}^0_{v,r,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$

Again the embedding (5.3) would thus lead to $\varrho \le v$ in view of [8, Thm. 3.3], i.e., to a contradiction by our choice of $\varrho$. Here we also used the identification $F^s_{\infty ,q}({{{\mathbb {R}}}^d})=F^{s,\frac{1}{p}}_{p,q}({{{\mathbb {R}}}^d})$, see [33, Props. 3.4 and 3.5] and [34, Rem. 10]. In a completely parallel way one can show that an embedding ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{v,r}({{{\mathbb {R}}}^d})$ is never possible when $\tau \ge \frac{1}{p}$.

Note that the limiting case $v=\infty$ is covered by [47, Prop. 5.4], [46, Prop. 4.1], in view of ${{\mathcal {M}}}_{\infty ,r}({{{\mathbb {R}}}^d})=L_\infty ({{{\mathbb {R}}}^d})$. In particular, if $0<p<\infty$, $0<q\le \infty$, $s\in {{\mathbb {R}}}$ and $\tau > 0$, then

$$\begin{aligned} {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\hookrightarrow L_\infty ({{{\mathbb {R}}}^d}) \quad \text {if, and only if,}\quad s>d\left( \frac{1}{p}-\tau \right) . \end{aligned}$$

The result for ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ looks alike.