Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator

Abstract

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$ and $\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$, where $\Omega \subset {{{\mathbb {R}}}^d}$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $\text {id}_\tau$. This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces.

1 Introduction

Besov-type spaces ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ and Triebel-Lizorkin-type spaces ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, $0< p <\infty$ (or $p=\infty$ in the B-case), $0 < q \le \infty$, $\tau \ge 0$, $s \in {{\mathbb {R}}}$, are part of a class of function spaces built upon Morrey spaces ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$, $0< p \le u < \infty$. They are regularly called in the literature as smoothness spaces of Morrey type or, shortly, smoothness Morrey spaces, and they have been increasingly studied in the last decades, motivated firstly by possible applications.

The classical Morrey spaces ${{\mathcal {M}}}_{u,p}$, $0< p \le u < \infty$, were introduced by Morrey in [19] and are part of a wider class of Morrey-Campanato spaces, cf. [21]. They can be seen as a complement to $L_p$ spaces, since ${{\mathcal {M}}}_{p,p}({{{\mathbb {R}}}^d})= L_p({{{\mathbb {R}}}^d})$.

The (inhomogeneous) Besov-type and Triebel-Lizorkin-type spaces we work here with were introduced and intensively studied in [47] by Yuan, Sickel and Yang. Their homogeneous versions were previously investigated by El Baraka in [3,4,5], and also by Yuan and Yang [41, 42]. Considering $\tau =0$, one recovers the classical Besov and Triebel-Lizorkin spaces. Moreover, they are also closely connected with Besov-Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and Triebel-Lizorkin-Morrey spaces ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, $0< p \le u < \infty$, $0 < q \le \infty$, $s \in {{\mathbb {R}}}$, which are also included in the class of smoothness Morrey spaces. Namely, when $p\le u$, $\tau =\frac{1}{p}-\frac{1}{u}$, then the Triebel-Lizorkin-type space ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ coincides with ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$, and it is also known that the Besov-Morrey space ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ is a proper subspace of ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ with $\tau =\frac{1}{p}-\frac{1}{u}$, $p<u$ and $q<\infty$. The Besov-Morrey spaces were introduced by Kozono and Yamazaki in [15] and used by them and later on by Mazzucato [17] in the study of Navier–Stokes equations. In [33] Tang and Xu introduced the corresponding Triebel-Lizorkin-Morrey spaces, thanks to establishing the Morrey version of the Fefferman-Stein vector-valued inequality. Some properties of these spaces including their wavelet characterisations were later described in the papers by Sawano [24, 25], Sawano and Tanaka [28, 29] and Rosenthal [22]. The surveys [31, 32] by Sickel are also worth of being consulted when studying these scales. Recently, some limiting embedding properties of these spaces were investigated in a series of papers [10,11,12,13,14]. As for the Besov-type and Triebel-Lizorkin-type spaces, also embedding properties have been recently studied in [8, 45, 46].

Undoubtedly the question of necessary and sufficient conditions for continuous embeddings of certain function spaces is a natural and classical one. Beyond that, this paper should essentially be understood as the continuation of our earlier studies in [8]. Proceeding contrary to the usual, there we started by studying compactness of the embeddings of Besov-type ($A=B$) and Triebel-Lizorkin-type ($A=F$) spaces,

$$\begin{aligned}\text {id}_\tau ~:~{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )~\hookrightarrow ~{A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ),\end{aligned}$$

where $\Omega \subset {{{\mathbb {R}}}^d}$ is a bounded domain. Now we finally deal with the continuity of such embeddings, obtaining sufficient and necessary conditions on the parameters under which they hold true. According to the results obtained in [8], the embedding $\text {id}_\tau$ is compact if, and only if,

$$\begin{aligned} \frac{s_1-s_2}{d} > \max \left\{ \left( \tau _2-\frac{1}{p_2} \right) _+- \left( \tau _1-\frac{1}{p_1} \right) _+, \frac{1}{p_1}-\tau _1 -\min \left\{ \frac{1}{p_2}-\tau _2, \frac{1}{p_2}(1-p_1\tau _1)_+\right\} \right\} =: \gamma , \end{aligned}$$

with $a_+{:=} \max \{a,0\}$, and there is no continuous embedding $\text {id}_\tau$ when $s_1-s_2<d \, \gamma$. Consequently, only the case when

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma \end{aligned}$$

is of interest to us here. In what follows, we call this setting ‘limiting situation’, giving meaning to the expression ‘limiting embedding’. In that way we complement earlier results in [11, 12, 46] in related settings. Our main results are stated in Theorem 4.9, Propositions 4.3 and 4.6. In the propositions we consider the situation when one of the spaces, the source one or the target one, coincides with some classical Besov space $B^\sigma _{\infty ,\infty }(\Omega )$. The outcome for the spaces that do not satisfy this assumption can be found in the theorem. In almost all cases we prove the sharp sufficient and necessary conditions. Only in one case we have a small gap between them, cf. Remark 4.10.

Of independent interest is also the extension theorem we are able to prove for the spaces under consideration. The first extension operators for Besov-type and Triebel-Lizorkin-type spaces were constructed by Sickel, Yang and Yuan in [47], cf. Theorem 6.11 and 6.13 ibidem. However it is assumed there that domains are $C^\infty$ smooth and the extension operators were not universal. Another not universal construction for the smooth domains was given by Moura, Neves and Schneider in [20]. The Rychkov universal extension operator for the Triebel-Lizorkin-type spaces defined on Lipschitz domains was recently constructed by Zhou, Hovemann and Sickel in [52] with additional assumptions $p,q \in [1, \infty )$. Here we considered all admissible parameters p and q. We concentrate on the Besov-type spaces, that are not a real interpolation space of Triebel-Lizorkin-type spaces, in contrast to the classical case, cf. [48]. The extension theorem will not only help us to obtain the results about the continuity, but also will allow us to improve some necessary conditions of such embeddings on ${{{\mathbb {R}}}^d}$. We follow Rychkov’s approach from [23] and construct such an operator, for all possible values of $p,q\in (0, \infty ]$. We learned only recently that in [51, 52] the authors followed a similar approach to construct such an extension operator adapted to their purposes, that is, for $p,q\ge 1$. For the convenience of the reader we keep our argument for the full range of parameters here.

This paper is organized as follows. In Sect. 2 we recall the definition, on ${{{\mathbb {R}}}^d}$ and on bounded domains $\Omega \subset {{{\mathbb {R}}}^d}$, of the spaces considered in the paper and collect some basic properties, among them the wavelet characterisations. In Sect. 3 we deal with the construction of a universal linear bounded extension operator for the spaces ${A}_{p,q}^{s,\tau }$. Section 4 is, finally, devoted to the study of continuity properties of limiting embeddings of the spaces ${A}_{p,q}^{s,\tau }(\Omega )$. Moreover, we make use of these results and the extension theorem from Section 3 to improve prior results on the continuity of embeddings of the corresponding spaces on ${{{\mathbb {R}}}^d}$, cf. [46].

2 Preliminaries

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, $\alpha =(\alpha _1,..., \alpha _d)$ with $\alpha _j \in {{\mathbb {N}}}_0$ and $|\alpha |{:=}\sum _{j=1}^d\alpha _j$. If $x=(x_1,..., x_d) \in {{{\mathbb {R}}}^d}$ and $\alpha =(\alpha _1,..., \alpha _d) \in {{\mathbb {N}}}_0^d$, then we put $x^\alpha =x_1^{\alpha _1} \cdot \cdot \cdot x_d^{\alpha _d}$. For $a\in {{\mathbb {R}}}$, let $\left\lfloor a \right\rfloor {:=}\max \{k\in {{\mathbb {Z}}}: k\le a\}$ and $a_+{:=}\max \{a,0\}$. All unimportant positive constants will be denoted by C, occasionally with subscripts. 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 ${{{\mathbb {R}}}^d}$.

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.

2.1 Smoothness spaces of Morrey type on ${{{\mathbb {R}}}^d}$

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. For all $f\in {\mathcal {S}}({{{\mathbb {R}}}^d})$ or ${\mathcal {S}}'({{{\mathbb {R}}}^d})$, we use ${\widehat{f}}$ to denote its Fourier transform, and $f^\vee$ for its inverse. Let $\mathcal {Q}$ be the collection of all dyadic cubes in ${{{\mathbb {R}}}^d}$, namely, $\mathcal {Q}{:=} \{Q_{j,k}{:=} 2^{-j}([0,1)^d+k):\ j\in {{\mathbb {Z}}},\ k\in {{{\mathbb {Z}}}^d}\}.$ The symbol $\ell (Q)$ denotes the side-length of the cube Q and $j_Q{:=}-\log _2\ell (Q)$. Moreover, we denote by $\chi _{Q_{j,m}}$ the characteristic function of the cube $Q_{j,m}$.

Let $\varphi _0,$ $\varphi \in \mathcal {S}({{{\mathbb {R}}}^d})$. We say that $(\varphi _0, \varphi )$ is an admissible pair if

$$\begin{aligned} {\mathop {\text {supp}}\nolimits }\,\widehat{\varphi _0}\subset \{\xi \in {{{\mathbb {R}}}^d}:\,|\xi |\le 2\}\,, \qquad |\widehat{\varphi _0}(\xi )|\ge C\ \text{ if }\ |\xi |\le 5/3, \end{aligned}$$

(2.1)

and

$$\begin{aligned} {\mathop {\text {supp}}\nolimits }\,\widehat{\varphi }\subset \{\xi \in {{{\mathbb {R}}}^d}: 1/2\le |\xi |\le 2\}\quad \text{ and }\quad |\widehat{\varphi }(\xi )|\ge C\ \text{ if }\ 3/5\le |\xi |\le 5/3, \end{aligned}$$

(2.2)

where C is a positive constant. In what follows, for all $\varphi \in {\mathcal {S}}({{{\mathbb {R}}}^d})$ and $j\in {{\mathbb {N}}}$, $\varphi _j(\cdot ){:=}2^{jd}\varphi (2^j\cdot )$.

Definition 2.1

Let $s\in {{{\mathbb {R}}}}$, $\tau \in [0,\infty )$, $q \in (0,\infty ]$ and $(\varphi _0,\varphi )$ be an admissible pair.

(i):

Let $p\in (0,\infty ]$. The Besov-type space ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ is defined to be the collection of all $f\in \mathcal {S}'({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \Vert f \mid {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty \!\! 2^{js q}\left[ \int \limits _P |\varphi _j*f(x)|^p\;\textrm{d}x\right] ^{\frac{q}{p}}\right\} ^{\frac{1}{q}}<\infty \end{aligned}$$

with the usual modifications made in case of $p=\infty$ and/or $q=\infty$.

(ii):

Let $p\in (0,\infty )$. The Triebel-Lizorkin-type space ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ is defined to be the collection of all $f\in \mathcal {S}'({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \Vert f \mid {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \int \limits _P\left[ \sum _{j=\max \{j_P,0\}}^\infty \!\! 2^{js q} |\varphi _j*f(x)|^q\right] ^{\frac{p}{q}}\;\textrm{d}x\right\} ^{\frac{1}{p}}<\infty \end{aligned}$$

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

Remark 2.2

These spaces were introduced in [47] and proved therein to be quasi-Banach spaces. In the Banach case the scale of Nikol’skij-Besov type spaces ${{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}$ had already been introduced and investigated in [3,4,5]. It is easy to see that, when $\tau =0$, then ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ and ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ coincide with the classical Besov space $B^s_{p,q}({{{\mathbb {R}}}^d})$ and Triebel-Lizorkin space $F^s_{p,q}({{{\mathbb {R}}}^d})$, respectively. In case of $\tau <0$ the spaces are trivial, ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})={F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})=\{0\}$, $\tau <0$. There exists extensive literature on such spaces; we refer, in particular, to the series of monographs [34,35,36, 39] for a comprehensive treatment.

Convention. We adopt the nowadays usual custom to write $A^s_{p,q}({{{\mathbb {R}}}^d})$ instead of $B^s_{p,q}({{{\mathbb {R}}}^d})$ or $F^s_{p,q}({{{\mathbb {R}}}^d})$, and ${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ instead of ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ or ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, respectively, when both scales of spaces are meant simultaneously in some context.

We have elementary embeddings within this scale of spaces (see [47, Proposition 2.1]),

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

(2.3)

and

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

(2.4)

as well as

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

(2.5)

which directly extends the well-known classical case from $\tau =0$ to $\tau \in [0,\infty )$, $p\in (0,\infty )$, $q\in (0,\infty ]$ and $s\in {{\mathbb {R}}}$.

It is also known from [47, Proposition 2.6] that

$$\begin{aligned} A^{s,\tau }_{p,q}({{{\mathbb {R}}}^d}) \hookrightarrow B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$

(2.6)

The following remarkable feature was proved in [44].

Proposition 2.3

Let $s\in {{{\mathbb {R}}}}$, $\tau \in [0,\infty )$ and $p,\,q\in (0,\infty ]$ (with $p<\infty$ in the F-case). If either $\tau >\frac{1}{p}$ or $\tau =\frac{1}{p}$ and $q=\infty$, then $A^{s,\tau }_{p,q}({{{\mathbb {R}}}^d}) = B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$.

Now we come to smoothness spaces of Morrey type ${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ and ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$. Recall first that the Morrey space ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$, $0<p\le u<\infty$, is defined to be the set of all locally p-integrable functions $f\in L_p^{\textrm{loc}}({{{\mathbb {R}}}^d})$ such that

$$\begin{aligned} \Vert f \mid {{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})}\Vert {:=}\, \sup _{x\in {{{\mathbb {R}}}^d}, R>0} R^{\frac{d}{u}-\frac{d}{p}} \left[ \int _{B(x,R)} |f(y)|^p \;\textrm{d}y \right] ^{\frac{1}{p}}\, <\, \infty \,. \end{aligned}$$

Remark 2.4

The spaces ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$ are quasi-Banach spaces (Banach spaces for $p \ge 1$). They originated from Morrey’s study on PDE (see [19]) and are part of the wider class of Morrey-Campanato spaces; cf. [21]. They can be considered as a complement to $L_p$ spaces. As a matter of fact, ${{\mathcal {M}}}_{p,p}({{{\mathbb {R}}}^d}) = L_p({{{\mathbb {R}}}^d})$ with $p\in (0,\infty )$. To extend this relation, we put ${{\mathcal {M}}}_{\infty ,\infty }({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$. One can easily see that ${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})=\{0\}$ for $u<p$, and that for $0<p_2 \le p_1 \le u < \infty$,

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

(2.7)

In an analogous way, one can define the spaces ${{\mathcal {M}}}_{\infty ,p}({{{\mathbb {R}}}^d})$, $p\in (0, \infty )$, but using the Lebesgue differentiation theorem, one can easily prove that ${{\mathcal {M}}}_{\infty , p}({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$.

Definition 2.5

Let $0<p\le u<\infty$ or $p=u=\infty$. Let $q\in (0,\infty ]$, $s\in {{\mathbb {R}}}$ and $\varphi _0$, $\varphi \in {\mathcal {S}}({{{\mathbb {R}}}^d})$ be as in (2.1) and (2.2), respectively.

(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

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

(2.8)

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

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

(2.9)

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

Convention. Again we adopt the usual custom to write ${{\mathcal {A}}}^{s}_{u,p,q}$ instead of ${{\mathcal {N}}}^{s}_{u,p,q}$ or ${{\mathcal {E}}}^{s}_{u,p,q}$, when both scales of spaces are meant simultaneously in some context.

Remark 2.6

The spaces ${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ are independent of the particular choices of $\varphi _0$, $\varphi$ 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,

$$\begin{aligned} {{\mathcal {A}}}^{s}_{p,p,q}({{{\mathbb {R}}}^d}) = A^s_{p,q}({{{\mathbb {R}}}^d}) = A^{s,0}_{p,q}({{{\mathbb {R}}}^d}). \end{aligned}$$

(2.10)

Besov-Morrey spaces were introduced by Kozono and Yamazaki in [15]. They studied semi-linear heat equations and Navier–Stokes equations with initial data belonging to Besov-Morrey spaces. The investigations were continued by Mazzucato [17], where one can find the atomic decomposition of some spaces. The Triebel-Lizorkin-Morrey spaces were later introduced by Tang and Xu [33]. We follow the ideas of Tang and Xu [33], where a somewhat different definition is proposed. The ideas were further developed by Sawano and Tanaka [24, 25, 28, 29]. The most systematic and general approach to the spaces of this type can be found in the monograph [47] or in the survey papers by Sickel [31, 32], which we also recommend for further up-to-date references on this subject. We refer to the recent monographs [26, 27] for applications.

It turned out that many of the results from the classical situation have their counterparts for the spaces $\mathcal {A}^s_{u,p,q}({{{\mathbb {R}}}^d})$, e. g.,

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

(2.11)

and ${{\mathcal {A}}}^{s}_{u,p,q_1}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q_2}({{{\mathbb {R}}}^d})$ if $q_1\le q_2$. However, there also exist some differences. Sawano proved in [24] that, for $s\in {{\mathbb {R}}}$ and $0<p< u<\infty$,

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

(2.12)

where, for the latter embedding, $r=\infty$ cannot be improved – unlike in case of $u=p$ (see (2.5) with $\tau =0$). More precisely,

$$\begin{aligned} {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {N}}}^s_{u,p,r}({{{\mathbb {R}}}^d})\quad \text {if, and only if,}\quad r=\infty ~~ \text {or} ~~ u=p\ \text {and}\ r\ge \max \{p,\,q\}. \end{aligned}$$

On the other hand, Mazzucato has shown in [17, Proposition 4.1] that

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

in particular,

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

(2.13)

Remark 2.7

Let s, u, p and q be as in Definition 2.5 and $\tau \in [0,\infty )$. It is known that

$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \qquad \text {with}\qquad \tau ={1}/{p}- {1}/{u}, \end{aligned}$$

(2.14)

cf. [47, Corollary 3.3]. Moreover, the above embedding is proper if $\tau >0$ and $q<\infty$. If $\tau =0$ or $q=\infty$, then both spaces coincide with each other, in particular,

$$\begin{aligned} \mathcal {N}^{s}_{u,p,\infty }({{{\mathbb {R}}}^d}) = B^{s,\frac{1}{p}- \frac{1}{u}}_{p,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$

(2.15)

As for the F-spaces, if $0\le \tau <{1}/{p}$, then

$$\begin{aligned} {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\, = \, {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\quad \text {with }\quad \tau = {1}/{p}-{1}/{u}\,,\quad 0< p\le u < \infty \,; \end{aligned}$$

(2.16)

cf. [47, Corollary 3.3]. Moreover, if $p\in (0,\infty )$ and $q\in (0,\infty )$, then

$$\begin{aligned} F^{s,\, \frac{1}{p} }_{p\,,\,q}({{{\mathbb {R}}}^d}) \, = \, F^{s}_{\infty ,\,q}({{{\mathbb {R}}}^d})\, = \, B^{s,\, \frac{1}{q} }_{q\,,\,q}({{{\mathbb {R}}}^d}) \,; \end{aligned}$$

(2.17)

cf. [31, Propositions 3.4, 3.5] and [32, Remark 10].

Remark 2.8

Recall that the space $\textrm{bmo}({{{\mathbb {R}}}^d})$ is covered by the above scale. More precisely, consider the local (non-homogeneous) space of functions of bounded mean oscillation, $\textrm{bmo}({{{\mathbb {R}}}^d})$, consisting of all locally integrable functions $\ f\in L_1^{\textrm{loc}}({{{\mathbb {R}}}^d})$ satisfying that

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

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, $f_Q {:=} \frac{1}{|Q|} \;\int _Q f(x)\;\textrm{d}x$, cf. [34, 2.2.2 (viii)]. The space $\textrm{bmo}({{{\mathbb {R}}}^d})$ coincides with $F^{0}_{\infty , 2}({{{\mathbb {R}}}^d})$, cf. [34, Theorem 2.5.8/2]. Hence the above result (2.17) implies, in particular,

$$\begin{aligned} \textrm{bmo}({{{\mathbb {R}}}^d})= F^{0}_{\infty ,2}({{{\mathbb {R}}}^d})= F^{0, 1/p}_{p, 2}({{{\mathbb {R}}}^d})= {B^{0, 1/2}_{2, 2}({{{\mathbb {R}}}^d})}, \quad 0<p<\infty . \end{aligned}$$

(2.18)

Remark 2.9

In contrast to this approach, Triebel followed the original Morrey-Campanato ideas to develop local spaces $\mathcal {L}^rA^s_{p,q}({{{\mathbb {R}}}^d})$ in [37], and so-called ‘hybrid’ spaces $L^rA^s_{p,q}({{{\mathbb {R}}}^d})$ in [38], where $0<p<\infty$, $0<q\le \infty$, $s\in {{\mathbb {R}}}$, and $-\frac{d}{p}\le r<\infty$. This construction is based on wavelet decompositions and also combines local and global elements as in Definitions 2.1 and 2.5. However, Triebel proved in [38, Theorem 3.38] that

$$\begin{aligned} L^rA^s_{p,q}({{{\mathbb {R}}}^d}) = {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}), \qquad \tau =\frac{1}{p}+\frac{r}{d}, \end{aligned}$$

(2.19)

in all admitted cases. We return to this coincidence below.

As mentioned previously, in this paper we are interested in studying embeddings of type

$$\begin{aligned} \text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}\end{aligned}$$

on domains. To do so, we will strongly rely on the corresponding results for these spaces on ${{{\mathbb {R}}}^d}$, obtained in [46]. In order to make the reading easier, we recall those results here. We begin with the situation of Besov-type spaces where the results can be found in [46, Theorems 2.4, 2.5].

Theorem 2.10

( [46]) Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i\le \infty$ and $\tau _i\ge 0$, $i=1,2$.

(i):

Let $\tau _2 > \frac{1}{p_2}$ or $\tau _2=\frac{1}{p_2}$ and $q_2=\infty$. Then the embedding

$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$

(2.20)

holds if, and only if, $\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$.

(ii):

Let $\tau _1 > \frac{1}{p_1}$ or $\tau _1=\frac{1}{p_1}$ and $q_1=\infty$. Then the embedding (2.20) holds if, and only if,

$$\begin{aligned}&\qquad \frac{s_1-s_2}{d}> \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 \ge \frac{1}{p_2}\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 > \frac{1}{p_2}\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 = \frac{1}{p_2} \quad \text{ and } \quad q_2=\infty . \end{aligned}$$

(iii):

Assume that $\tau _i < \frac{1}{p_i}$ or $\tau _i=\frac{1}{p_i}$ and $q_i<\infty$, $i=1,2$.

• (a) The embedding (2.20) holds true if

  $$\begin{aligned} \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0, \qquad \frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1} \end{aligned}$$

  and

  $$\begin{aligned}&\qquad \frac{s_1-s_2}{d}> \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \\&\qquad \text{ with } \\&\qquad \qquad \quad (s_1-s_2)(\tau _1-\tau _2)\ne 0, \quad \frac{\tau _1}{p_2}< \frac{\tau _2}{p_1}\\&\qquad \text{ or } \qquad (s_1-s_2)(\tau _1-\tau _2)\ne 0, \quad \frac{\tau _1}{p_2}= \frac{\tau _2}{p_1}, \quad \frac{\tau _1}{q_2}\le \frac{\tau _2}{q_1}\\&\qquad \text{ or } \qquad (s_1-s_2)(\tau _1-\tau _2)=0, \quad q_1 \le q_2,\\&\qquad \qquad \quad \text{ and } \quad p_1\ge p_2 \quad \text{ if } \quad s_1=s_2 \quad \text{ and } \quad \tau _1 p_1 = \tau _2 p_2 =1. \end{aligned}$$

• (b) The conditions $\frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0$ and $\frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1}$, and $s_1-s_2\ge \frac{d}{p_1}- d\tau _1 - \frac{d}{p_2}+d\tau _2$ as well as $q_1 \le q_2$ if $s_1=s_2$ are also necessary for the embedding (2.20).

The counterpart for F-spaces reads as follows, we refer to [46, Corollaries 5.8, 5.9] for details.

Theorem 2.11

( [46]) Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i< \infty$ and $\tau _i\ge 0$, $i=1,2$.

(i):

Let $\tau _i > \frac{1}{p_i}$ or $\tau _i=\frac{1}{p_i}$ and $q_i=\infty$, $i=1,2$. Then the embedding

$$\begin{aligned} {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$

(2.21)

holds if, and only if, $\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$.

(ii):

Assume that $\tau _i < \frac{1}{p_i}$, $i=1,2$. Then the embedding (2.21) holds if, and only if,

$$\begin{aligned} \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0, \qquad \frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1} \end{aligned}$$

and

$$\begin{aligned}&\qquad \frac{s_1-s_2}{d}>\frac{1}{p_1} - \tau _1 - \frac{1}{p_2}+\tau _2\\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \\ \text{ or }&\qquad s_1=s_2 \quad \text{ and } \quad q_1 \le q_2. \end{aligned}$$

2.2 Spaces on domains

Let $\Omega$ denote an open, nontrivial subset of ${{{\mathbb {R}}}^d}$. We consider smoothness Morrey spaces on $\Omega$ defined by restriction. Let ${{\mathcal {D}}}(\Omega )$ be the set of all infinitely differentiable functions supported in $\Omega$ and denote by ${{\mathcal {D}}}'(\Omega )$ its dual. Since we are able to define the extension operator $\textrm{ext}: {{\mathcal {D}}}(\Omega ) \rightarrow \mathcal {S}({{{\mathbb {R}}}^d})$, cf. [30], the restriction operator $\textrm{re}: \mathcal {S}'({{{\mathbb {R}}}^d})\rightarrow {{\mathcal {D}}}'(\Omega )$ can be defined naturally as an adjoint operator

$$\begin{aligned} \langle \textrm{re}(f), \varphi \rangle = \langle f, \textrm{ext}(\varphi )\rangle , \quad f\in \mathcal {S}'({{{\mathbb {R}}}^d}), \end{aligned}$$

where $\varphi \in \mathcal {D}(\Omega )$. We will write $f\vert _{\Omega }=\textrm{re } (f)$.

Definition 2.12

Let $s\in {{\mathbb {R}}}$, $\tau \in [0,\infty )$, $q\in (0,\infty ]$ and $p\in (0,\infty ]$ (with $p<\infty$ in the case of ${A}_{p,q}^{s,\tau }={F}_{p,q}^{s,\tau }$). Then ${A}_{p,q}^{s,\tau }(\Omega )$ is defined by

$$\begin{aligned} {A}_{p,q}^{s,\tau }(\Omega ){:=}\big \{f\in {{\mathcal {D}}}'(\Omega ): f=g\vert _{\Omega } \text { for some } g\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\big \} \end{aligned}$$

endowed with the quasi-norm

$$\begin{aligned} \big \Vert f\mid {A}_{p,q}^{s,\tau }(\Omega )\big \Vert {:=} \inf \big \{ \Vert g\mid {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Vert : f=g\vert _{\Omega }, \; g\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\big \}. \end{aligned}$$

Remark 2.13

The spaces ${A}_{p,q}^{s,\tau }(\Omega )$ are quasi-Banach spaces (Banach spaces for $p,q\ge 1$). When $\tau =0$ we re-obtain the usual Besov and Triebel-Lizorkin spaces defined on domains. For the particular case of $\Omega$ being a bounded $C^{\infty }$ domain in ${{{\mathbb {R}}}^d}$, some properties were studied in [47, Section 6.4.2]. In particular, according to [47, Theorem 6.13], for such a domain $\Omega$, there exists a linear and bounded extension operator

$$\begin{aligned} \textrm{ext}: {A}_{p,q}^{s,\tau }(\Omega )\rightarrow {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}),\quad \text {where}\quad 1\le p<\infty , 0<q\le \infty , s\in {{\mathbb {R}}}, \tau \ge 0, \end{aligned}$$

(2.22)

such that

$$\begin{aligned} \textrm{re} \circ \textrm{ext}= \text {id}\quad \text {in}\quad {A}_{p,q}^{s,\tau }(\Omega ), \end{aligned}$$

(2.23)

where $\textrm{re}: {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\rightarrow {A}_{p,q}^{s,\tau }(\Omega )$ is the restriction operator as above.

Moreover, in [9] we studied the question under what assumptions these spaces consist of regular distributions only.

Remark 2.14

Let us mention that we have the counterparts of many continuous embeddings stated in the previous subsection for spaces on ${{{\mathbb {R}}}^d}$ when dealing with spaces restricted to bounded domains. We recall them in further detail if appropriate and necessary for our arguments below.

Later we shall mainly deal with Lipschitz domains. Therefore we recall the concept for convenience. By a Lipschitz domain we mean either a special or bounded Lipschitz domain. A special Lipschitz domain is defined as an open set $\Omega \subset {{{\mathbb {R}}}^d}$ lying above a graph of a Lipschitz function $\omega : {{\mathbb {R}}}^{d-1} \rightarrow {{\mathbb {R}}}$. More precisely,

$$\begin{aligned} \Omega = \{(x', x_d) \in {{{\mathbb {R}}}^d}: x_d > \omega (x') \}, \end{aligned}$$

where

$$\begin{aligned} |\omega (x')-\omega (y')|\le A \, |x'-y'|, \quad x', y' \in {{\mathbb {R}}}^{d-1}. \end{aligned}$$

(2.24)

A bounded Lipschitz domain is a bounded domain $\Omega$ whose boundary $\partial \Omega$ can be covered by a finite number of open balls $B_k$ so that, possibly after an appropriate rotation, $\partial \Omega \cap B_k$ for each k is a part of the graph of a Lipschitz function. Let $\Omega$ be a special Lipschitz domain defined by a Lipschitz function $\omega$ that satisfies the condition (2.24). We put

$$\begin{aligned} K{:=} \{ (x', x_n) \in {{{\mathbb {R}}}^d}: |x'|<A^{-1} x_n\} \end{aligned}$$

and $-K {:=}\{-x: x\in K\}$. Then K has the property that $x+K \subset \Omega$ for any $x \in \Omega$. Moreover let $\mathcal {S}'(\Omega )$ denote the subspace of $\mathcal {D}'(\Omega )$ consisting of all distributions of finite order and of at most polynomial growth at infinity, that is, $f\in \mathcal {S}'(\Omega )$ if, and only if, the estimate

$$\begin{aligned} |\langle f,\eta \rangle | \le c \sup _{x\in \Omega , |\alpha |\le M}|\;\textrm{D}^\alpha \eta (x)|(1+|x|)^M,\qquad \eta \in \mathcal {D}(\Omega ), \end{aligned}$$

holds with some constants c and M depending on f.

Remark 2.15

As already mentioned, we shall study continuous embeddings of function spaces on domains. For convenience let us recall what is well known for the classical function spaces $A^s_{p,q}(\Omega )$. Let $s_i\in {{\mathbb {R}}}$, $0<p_i, q_i\le \infty$, $i=1,2$, and $\Omega \subset {{{\mathbb {R}}}^d}$ a bounded Lipschitz domain. Then

$$\begin{aligned} \text {id}^B_\Omega : {B}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2}(\Omega ) \end{aligned}$$

is continuous if, and only if,

$$\begin{aligned} \text {either}\qquad&\frac{s_1-s_2}{d} > \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} \nonumber \\ \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} \quad \text {and}\quad q_1\le q_2. \end{aligned}$$

(2.25)

Assume, in addition, $p_i<\infty$, $i=1,2$. Then

$$\begin{aligned} \text {id}^F_\Omega : {F}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2}(\Omega ) \end{aligned}$$

is continuous if, and only if, either (2.25) is satisfied,

$$\begin{aligned} \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2},0\right\} =\frac{1}{p_1}-\frac{1}{p_2}>0,\\ \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} =0 \quad \text {and}\quad q_1\le q_2. \end{aligned}$$

For (partial) results we refer to [2, Section 2.5.1], [36, p. 60], and, quite recently, the extension to spaces $F^s_{\infty ,q}(\Omega )$ in [39, Section 2.6.5]; cf. also our results in [11, Theorem 3.1] and [12, Theorem 5.2]. However, in most of the above cases, the domain $\Omega$ is there assumed to be smooth.

2.3 Wavelet decomposition in Besov-type and Triebel-Lizorkin-type spaces

In this section we describe the wavelet characterisation of Besov-type and Triebel-Lizorkin-type spaces proved in [16]. This is a key tool when studying embedding properties of function spaces, since it allows one to transfer the problem to the corresponding sequence spaces.

Let $\widetilde{\phi }$ be a scaling function on ${{\mathbb {R}}}$ with compact support and of sufficiently high regularity, and $\widetilde{\psi }$ the corresponding orthonormal wavelet. Then we can extend these wavelets from ${{\mathbb {R}}}$ to ${{{\mathbb {R}}}^d}$ by the usual tensor procedure, cf. [40, Section 5.1]. This yields a scaling function $\phi$ and associated wavelets $\psi _1, \dots , \psi _{2^d-1}$, all defined on ${{{\mathbb {R}}}^d}$; see, e.g. [40, Proposition 5.2]. We suppose that

$$\begin{aligned} \phi , \psi _i \in C^{N_1}({{{\mathbb {R}}}^d}) \qquad \text { and } \qquad {\mathop {\text {supp}}\nolimits }\,\phi , {\mathop {\text {supp}}\nolimits }\,\psi _i \subset [-N_2, N_2]^d,\quad i=1, \dots , 2^d-1, \end{aligned}$$

for some $N_1, N_2 \in {{\mathbb {N}}}$.

For $k\in {{{\mathbb {Z}}}^d}$, $j\in {{\mathbb {N}}}_0$ and $i\in \{1,\ldots ,2^d-1\}$, define

$$\begin{aligned} \phi _{j,k}(x){:=} 2^{jd/2} \phi (2^jx-k){}\;{\text {and}}\;{} \psi _{i,j,k}(x){:=} 2^{jd/2} \psi _i(2^jx-k),x\in {{{\mathbb {R}}}^d}. \end{aligned}$$

It is well known that

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^d}\psi _{i,j,k}(x)\, x^\gamma \;\textrm{d}x = 0 \qquad \text{ if } \qquad |\gamma |\le N_1 \end{aligned}$$

(see [40, Proposition 3.1]), and

$$\begin{aligned} \{\phi _{0,k}: \ k\in {{{\mathbb {Z}}}^d}\}\, \cup \, \{\psi _{i,j,k}:\ k\in {{{\mathbb {Z}}}^d},\ j\in {{\mathbb {N}}}_0,\ i\in \{1,\ldots ,2^d-1\}\} \end{aligned}$$

forms an orthonormal basis of $L_2({{{\mathbb {R}}}^d})$ (see, e. g., [18, Section 3.9] or [36, Section 3.1]). Hence

$$\begin{aligned} f=\sum _{k\in {{{\mathbb {Z}}}^d}}\, \lambda _k \, \phi _{0,k}+\sum _{i=1}^{2^d-1} \sum _{j=0}^\infty \sum _{k\in {{{\mathbb {Z}}}^d}}\, \lambda _{i,j,k}\, \psi _{i,j,k}\, \end{aligned}$$

(2.26)

in $L_2({{{\mathbb {R}}}^d})$, where $\lambda _k {:=} \langle f,\,\phi _{0,k}\rangle$ and $\lambda _{i,j,k}{:=} \langle f,\,\psi _{i,j,k}\rangle$. We will denote by $\lambda (f)$ the following sequence:

$$\begin{aligned}\lambda (f) {:=} \left( \lambda _k,\lambda _{i,j,k}\right) = \big ( \langle f,\,\phi _{0,k}\rangle , \langle f,\,\psi _{i,j,k}\rangle \big ). \end{aligned}$$

Definition 2.16

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

(i):

Let $p \in (0, \infty ]$. The sequence space ${{b}_{p,q}^{s,\tau }}{:=}{{b}_{p,q}^{s,\tau }}({{{\mathbb {R}}}^d})$ is defined to be the space of all complex-valued sequences $t{:=}\{t_{i,j,m}:\ i\in \{1,\ldots ,2^d-1\}, j\in {{\mathbb {N}}}_0, m\in {{{\mathbb {Z}}}^d}\}$ such that $\Vert t\mid {{b}_{p,q}^{s,\tau }}\Vert <\infty$, where

$$\begin{aligned} \Vert t\mid {{b}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2}-\frac{d}{p})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^p\right] ^{\frac{q}{p}}\right\} ^{\frac{1}{q}}, \end{aligned}$$

with the usual modification when $p=\infty$ or $q=\infty$.

(ii):

Let $p \in (0, \infty )$. The sequence space ${{f}_{p,q}^{s,\tau }}{:=}{{f}_{p,q}^{s,\tau }}({{{\mathbb {R}}}^d})$ is defined to be the space of all complex-valued sequences $t{:=}\{t_{i,j,m}:\ i\in \{1,\ldots ,2^d-1\}, j\in {{\mathbb {N}}}_0, m\in {{{\mathbb {Z}}}^d}\}$ such that $\Vert t\mid {{f}_{p,q}^{s,\tau }}\Vert <\infty$, where

$$\begin{aligned} \Vert t\mid {{f}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \int _P \left[ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2})q} \sum _{i=1}^{2^d-1} \sum _{m\in {{{\mathbb {Z}}}^d}} |t_{i,j,m}|^q \, \chi _{Q_{j,m}}(x)\right] ^{\frac{p}{q}} \;\textrm{d}x \right\} ^{\frac{1}{p}}, \end{aligned}$$

with the usual modification when $q=\infty$.

As a special case of [16, Theorem 4.12], we have the following wavelet characterisation of ${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$.

Proposition 2.17

( [16]) Let $s\in {{{\mathbb {R}}}}$, $\tau \in [0,\infty )$, $q\in (0,\infty ]$. Moreover, let $N_1 \in {{\mathbb {N}}}_0$ be such that, when $A=B$ and $p\in (0,\infty ]$,

$$\begin{aligned} N_1+1>\max \left\{ d+\frac{d}{p}-d\tau -s, \frac{2 d}{\min \{p,1\}}+d\tau +1, d+\frac{d}{p}+\frac{d}{2}, d+s, \frac{d}{p} -s, {s+d\tau }\right\} , \end{aligned}$$

and when $A=F$ and $p\in (0,\infty )$,

$$\begin{aligned} {N_1+1>\max \left\{ d+\frac{d}{p}-d\tau -s, \frac{2 d}{\min \{p,{q},1\}}+d\tau +1, d+\frac{d}{p}+\frac{d}{2}, d+s, \frac{d}{p} -s, {s+d\tau }\right\} .} \end{aligned}$$

Let $f\in {\mathcal {S}}'({{{\mathbb {R}}}^d})$. Then $f\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ if, and only if, f can be represented as (2.26) in ${\mathcal {S}}'({{{\mathbb {R}}}^d})$ and

$$\begin{aligned} \Vert \lambda (f)\mid {a}_{p,q}^{s,\tau }\Vert ^*{:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }} \left\{ \sum _{m:\ Q_{0,m}\subset P} | \langle f,\,\phi _{0,m}\rangle |^p\right\} ^{\frac{1}{p}} + \Vert \{\langle f,\,\psi _{i,j,m}\rangle \}_{i,j,m}\mid {a}_{p,q}^{s,\tau }\Vert <\infty . \end{aligned}$$

Moreover, $\Vert f \mid {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Vert$ is equivalent to $\Vert \lambda (f)\mid {a}_{p,q}^{s,\tau }\Vert ^*.$

3 Extension Operator

As mentioned in Remark 2.13 above, if $\Omega$ is a bounded $C^{\infty }$ domain in ${{{\mathbb {R}}}^d}$, an extension theorem for the spaces ${A}_{p,q}^{s,\tau }(\Omega )$ was stated in [47], but with the assumption that $p \in [1, \infty )$. It is our aim in this section to establish an extended result, which holds for all $p \in (0, \infty ]$ ($p\in (0,\infty )$ in the F-case). Similarly to what was done in [48, Proposition 4.13] for Besov-Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}$, we will follow Rychkov [23, Theorem 2.2] in the construction of such an operator. We start with some preparation.

Let $q \in (0, \infty ]$ and $\tau \in [0, \infty )$. Denote by $\ell _q(L_p^\tau )$ the set of all sequences $\{g_j\}_{j \in {{\mathbb {N}}}_0}$ of measurable functions on ${{{\mathbb {R}}}^d}$ such that

$$\begin{aligned} \Vert \{g_j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert {:=} \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j = \max \{j_P, 0\}}^{\infty } \Vert g_j \mid L_p(P)\Vert ^q\right\} ^{1/q} < \infty . \end{aligned}$$

(3.1)

Similarly, $L_p^\tau (\ell _q)$ with $p\in (0, \infty )$ denotes the set of all sequences $\{g_j\}_{j \in {{\mathbb {N}}}_0}$ of measurable functions on ${{{\mathbb {R}}}^d}$ such that

$$\begin{aligned} \Vert \{g_j\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert {:=} \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\| \left\{ \sum _{j = \max \{j_P, 0\}}^{\infty } |g_j|^q\right\} ^{1/q} \mid L_p(P)\right\| < \infty . \end{aligned}$$

(3.2)

The first auxiliary lemma can be seen as a particular case of [16, Lemma 2.9], where the authors considered more general quasi-norms.

Lemma 3.1

Let $q\in (0, \infty ]$, $\tau \in [0, \infty )$. Let $D_1, D_2 \in (0, \infty )$ with $D_2 >d \tau$. For any sequence $\{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}$ of measurable functions on ${{{\mathbb {R}}}^d}$, consider

$$\begin{aligned} G_j(x){:=} \sum _{\nu =0}^j 2^{-(j -\nu )D_2} g_\nu (x) + \sum _{\nu =j+1}^\infty 2^{-(\nu -j)D_1}g_\nu (x), \quad j \in {{\mathbb {N}}}_0, \quad x \in {{{\mathbb {R}}}^d}. \end{aligned}$$

Then there exists a positive constant C, independent of $\{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}$, such that, for all $p \in (0, \infty ]$,

$$\begin{aligned} \Vert \{ G_j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert \le C\, \Vert \{g_\nu \}_{\nu \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert , \end{aligned}$$

and, for all $p \in (0, \infty )$,

$$\begin{aligned} \Vert \{ G_j\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert \le C\, \Vert \{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert . \end{aligned}$$

Let $(\varphi _0,\varphi )$ be an admissible pair and $f \in \mathcal {S}'({{{\mathbb {R}}}^d})$. For all $j \in {{\mathbb {N}}}_0$, $a \in (0, \infty )$ and $x \in {{{\mathbb {R}}}^d}$, we denote

$$\begin{aligned} \varphi _j^{*, a}f(x) {:=} \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\varphi _j *f(y)|}{(1+2^j|x-y|)^a}. \end{aligned}$$

Another tool we will need later is the characterisation of the spaces via Peetre maximal functions. For the homogeneous version of the spaces, this result was proved in [43, Theorem 1.1]. Here we use the results proved in [50, Theorem 5.1] and [49, Theorem 3.11] for the more general scale of Besov-type spaces and Triebel-Lizorkin-type spaces with variable exponents, respectively, which in particular cover our spaces. Adapted to our case, the results read as follows.

Theorem 3.2

Let $s \in {{\mathbb {R}}}$, $\tau \in [0, \infty )$, $q \in (0, \infty ]$ and $(\varphi _0,\varphi )$ be an admissible pair.

(i):

Let $p\in (0, \infty ]$ and $a > \displaystyle \frac{d}{p}+d\tau$. Then

$$\begin{aligned} \Vert f \mid {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}\varphi _j^{*, a}f\}_{j \in {{\mathbb {N}}}_0}\mid \ell _q(L_p^\tau )\Vert . \end{aligned}$$

(ii):

Let $p\in (0, \infty )$ and $a > \displaystyle \frac{d}{\min \{p,q\}}+d\tau$. Then

$$\begin{aligned} \Vert f \mid {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}\varphi _j^{*, a}f\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert . \end{aligned}$$

Remark 3.3

Note that in the above theorem the sequence $\{\varphi _j\}_{j \in {{\mathbb {N}}}_0}$ is built upon an admissible pair $(\varphi _0, \varphi )$, as in Definition 2.1. Hence $\Vert f | {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}(\varphi _j *f)\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert \$ and $\Vert f | {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}(\varphi _j *f)\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert$. However, in [6] and [7] the authors proved that, for Besov-type and Triebel-Lizorkin-type spaces with variable exponents, one can consider more general pairs of functions not only in this result, but also in the definition of the spaces.

Following Rychkov [23], we will prove the existence of a universal linear bounded extension operator if $\Omega$ is a Lipschitz domain, recall the explanations given at the end of Sect. 2.2. We first recall two results of Rychkov we shall need in our argument below.

Lemma 3.4

([23, Proposition 3.1]) Let $\Omega$ be a special Lipschitz domain. The distribution $f\in \mathcal {D}'(\Omega )$ belongs to $\mathcal {S}'(\Omega )$ if, and only if, there exist $g\in \mathcal {S}'({{{\mathbb {R}}}^d})$ such that $f=g|_\Omega$.

Lemma 3.5

([23, Theorem 4.1 (a)]) Let $\Omega$ be a special Lipschitz domain and K its associated cone. Let $-K {:=}\{-x: x\in K\}$ be a ‘reflected’ cone. Then there exist functions $\phi _0,\phi , \psi _0, \psi \in \mathcal {S}({{{\mathbb {R}}}^d})$ supported in $-K$ such that

$$\begin{aligned} \int _{{{{\mathbb {R}}}^d}}x^\alpha \phi (x) \;\textrm{d}x&= \int _{{{{\mathbb {R}}}^d}}x^\alpha \psi (x) \;\textrm{d}x = 0 \quad \text {for all multi-indices}\;\alpha , \end{aligned}$$

(3.3)

and

$$\begin{aligned} f&= \sum _{j \in {{\mathbb {N}}}_0} \psi _j *\phi _j *f \quad \text{ in } \quad {{\mathcal {D}}}'(\Omega ), \quad \text {for any} \; f\in \mathcal {S}'(\Omega ), \end{aligned}$$

(3.4)

where $\phi _j(\cdot )= 2^{jd}\phi (2^j\cdot )$ and $\psi _j(\cdot )= 2^{jd}\psi (2^j\cdot )$, $j\in {{\mathbb {N}}}$.

We can now state the main result of this section.

Theorem 3.6

Let $\Omega \subset {{{\mathbb {R}}}^d}$ be a special or bounded Lipschitz domain if $d\ge 2$, or an interval if $d=1$. Then there exists a linear bounded operator $\textrm{Ext}$ which maps ${A}_{p,q}^{s,\tau }(\Omega )$ into ${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ for all $s \in {{\mathbb {R}}}$, $\tau \in [0, \infty )$, $p \in (0, \infty ]$ ($p<\infty$ in the F-case) and $q \in (0, \infty ]$, such that, for all $f \in {{\mathcal {D}}}'(\Omega )$, $\textrm{Ext}f|_{\Omega }=f$ in ${{\mathcal {D}}}'(\Omega )$.

Proof

We concentrate now on Besov-type spaces with $d\ge 2$ and give some details on the Triebel-Lizorkin scale at the end of the proof.

We apply the extension operator constructed by V. Rychkov in [23] and follow the main ideas of his proof. By a standard procedure (see [23, Subsection 1.2]), we only need to consider the case when $\Omega$ is a special Lipschitz domain.

Let $\Omega$ be a special Lipschitz domain. The spaces ${B}_{p,q}^{s,\tau }(\Omega )$ are defined by restriction therefore it follows from Lemma 3.4 and Lemma 3.5 that any distribution $f\in {B}_{p,q}^{s,\tau }(\Omega )$ can be represented in the form (3.4) with the functions $\phi$ and $\psi$ satisfying (3.3).

For any distribution $f \in {{\mathcal {S}}}'(\Omega )$, we define the mapping

$$\begin{aligned} \text {ext}\,f \,{:=} \sum _{j \in {{\mathbb {N}}}_0} \psi _j *(\phi _j *f)_\Omega . \end{aligned}$$

(3.5)

Here we use the notation $g_\Omega$ to denote the extension of a function $g:\Omega \rightarrow {{\mathbb {R}}}$ from $\Omega$ to ${{{\mathbb {R}}}^d}$ by setting

$$\begin{aligned} g_\Omega (x) = {\left\{ \begin{array}{ll} g(x), &{} \text{ if } x \in \Omega , \\ 0, &{} \text{ if } x \in {{{\mathbb {R}}}^d}\setminus \Omega . \end{array}\right. } \end{aligned}$$

Step 1. Let $\{g_j\}_{j \in {{\mathbb {N}}}_0}$ be a sequence of measurable functions. Moreover, let $\mathcal {M}_{N}^{g_j}(x)$ denote the Peetre maximal function of $g_j$, namely,

$$\begin{aligned} \mathcal {M}_{N}^{g_j}(x){:=} \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|g_j(y)|}{(1+2^j|x-y|)^N} \end{aligned}$$

for all $x \in {{{\mathbb {R}}}^d}$ and $N \in {{\mathbb {N}}}\cap \left( \frac{d}{\min \{1,p\}}, \infty \right)$. We prove that if $\{2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \in \ell _q(L_p^\tau )$, then the series $\sum _{j \in {{\mathbb {N}}}_0} \psi _j *g_j$ converges in $\mathcal {S}'({{{\mathbb {R}}}^d})$ and the ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ norm of its sum can be estimated by $\Vert \{2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert$.

We start with the following elementary inequality

$$\begin{aligned} |\phi _\ell *\psi _j *g_j(x)| \le \mathcal {M}_{N}^{g_j}(x) \int |\phi _\ell *\psi _j(y)|(1+2^j|y|)^N \;\textrm{d}y\; . \end{aligned}$$

(3.6)

Bui, Paluszyński and Taibleson proved in [1] that

$$\begin{aligned} \int |\phi _\ell *\psi _j(y)|(1+2^j|y|)^N \;\textrm{d}y \le C_{M,N} 2^{-|\ell -j|M}\qquad \text {for all} \; M>0, \end{aligned}$$

(3.7)

cf. [1, Lemma 2.1] or [23, proof of Theorem 4.1]. We take $M>|s|-d\tau$ and put $\sigma =M- |s|$. Then

$$\begin{aligned} 2^{\ell s}|\phi _\ell *\psi _j *g_j(x)| \lesssim 2^{-|\ell -j|\sigma }2^{js} \mathcal {M}^{g_j}_{N}(x), \quad x \in {{{\mathbb {R}}}^d}, \quad \ell \in {{\mathbb {N}}}_0. \end{aligned}$$

If the sequence $\{g_j\}_{j \in {{\mathbb {N}}}_0}$ is such that $\Vert \{2^{js}\,\mathcal {M}^{g_j}_{N}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert <\infty$, then there exists a constant c such that for any dyadic cube P we have $\Vert \mathcal {M}^{g_j}_{N}|L_p(P)\Vert \le c |P|^\tau$. In consequence, any function $g_j$ is a tempered distribution and $\psi _j *g_j\in \mathcal {S}'({{{\mathbb {R}}}^d})$, $j \in {{\mathbb {N}}}_0$.

It holds true that

$$\begin{aligned} \big \Vert \psi _j * g_j \mid B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})\big \Vert&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } 2^{-(\ell 2 \sigma + |\ell -j|\sigma )q}\right\} ^{1/q} \bigg \Vert 2^{js} \mathcal {M}_N^{g_j}\mid L_p(P)\bigg \Vert \nonumber \\&\lesssim 2^{-j \sigma }\, \left\| \{2^{ks}\, \mathcal {M}_N^{g_k}\}_{k \in {{\mathbb {N}}}_0}\mid \ell _q(L_p^\tau )\right\| , \end{aligned}$$

(3.8)

using that $|\ell -j|\ge j-\ell$. Therefore $\sum _{j \in {{\mathbb {N}}}_0} \psi _j *g_j$ converges in $B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})$ and hence in $\mathcal {S}'({{{\mathbb {R}}}^d})$, since $B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d}) \hookrightarrow \mathcal {S}'({{{\mathbb {R}}}^d})$. In this way, we further have

$$\begin{aligned} 2^{\ell s}\left| \phi _\ell *\left( \sum _{j=0}^\infty \psi _j *g_j \right) (x)\right| \lesssim \sum _{j=0}^\infty 2^{-|\ell -j|\sigma } 2^{js}\mathcal {M}_{N}^{g_j}(x), \quad x \in {{{\mathbb {R}}}^d}, \quad \ell \in {{\mathbb {N}}}_0. \end{aligned}$$

Applying this, we see that

$$\begin{aligned}&\bigg \Vert \sum _{j=0}^\infty \psi _j *g_j \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\bigg \Vert \\&\quad = \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } 2^{\ell sq} \left[ \int _P \Big | \phi _\ell *\left( \sum _{j=0}^\infty \psi _j *g_j \right) (x)\Big |^p {\;\textrm{d}x} \right] ^{q/p} \right\} ^{1/q}\\&\quad \lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } \bigg \Vert \sum _{j=0}^\infty 2^{-|\ell -j|\sigma } \, 2^{js}\,\mathcal {M}^{g_j}_{N} \mid L_p(P)\bigg \Vert ^{q} \right\} ^{1/q}. \end{aligned}$$

Now we can apply Lemma 3.1 since $\sigma >d\tau$. We get

$$\begin{aligned} \Big \Vert \sum _{j=0}^\infty \psi _j *g_j \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Big \Vert&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } \big \Vert 2^{js}\,\mathcal {M}^{g_j}_{N} \mid L_p(P) \big \Vert ^{q} \right\} ^{1/q}\nonumber \\&= \big \Vert \{ 2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\big \Vert . \end{aligned}$$

(3.9)

Step 2. Let $f \in {B}_{p,q}^{s,\tau }(\Omega )$. Then, for any $\varepsilon \in (0, \infty )$, there exists an $h \in {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ such that $h|_{\Omega }=f$ in ${{\mathcal {D}}}'(\Omega )$ and

$$\begin{aligned} \left\| h \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \le \left\| f \mid {B}_{p,q}^{s,\tau }(\Omega )\right\| + \varepsilon . \end{aligned}$$

(3.10)

We have $\phi _j *f(y)= \phi _j *h(y)$ if $y\in \Omega$ since ${\mathop {\textrm{supp}}\nolimits }\phi _j\subset -K$ and $y+K\subset \Omega$ for any point $y\in \Omega$. In consequence

$$\begin{aligned} \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|x-y|)^N}\, \lesssim \displaystyle \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\phi _j *h(y)|}{(1+2^j|{\tilde{x}}-y|)^N}, \quad&x \not \in \overline{\Omega }, \end{aligned}$$

(3.11)

where ${\tilde{x}}{:=}(x', 2\omega (x')-x_n) \in \Omega$ is the symmetric point to $x=(x',x_n) \not \in \overline{\Omega }$ with respect to $\partial \Omega$.

Let $g_j {:=} (\phi _j *f)_{\Omega }$ for all $j \in {{\mathbb {N}}}_0$. It was proved in [23, p. 248] that

$$\begin{aligned} \sup _{y \in \Omega } \frac{| g_j(y)|}{(1+2^j|x-y|)^N}\,&{\left\{ \begin{array}{ll}\displaystyle \,= \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|x-y|)^N}, &{} x \in \Omega , \\ \, \lesssim \displaystyle \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|{\tilde{x}}-y|)^N}, &{} x \not \in \overline{\Omega }. \end{array}\right. } \end{aligned}$$

(3.12)

Now, we conclude from (3.9), (3.11) and (3.12) that

$$\begin{aligned} \left\| \textrm{ext}f\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\|&= \Big \Vert \sum _{j=0}^{\infty } \psi _j *(\phi _j *f)_\Omega \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Big \Vert \lesssim \left\| \{2^{js}\,\mathcal {M}_N^{g_j} \}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\right\| \\&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } 2^{jsq} \Big \Vert \sup _{y \in \Omega } \frac{|\phi _j *h(y)|}{(1+2^j|\cdot -y|)^N} \mid L_p(P)\Big \Vert ^q\right\} ^{1/q}\\&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } 2^{jsq} \Big \Vert \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\phi _j *h(y)|}{(1+2^j|\cdot -y|)^N} \mid L_p(P)\Big \Vert ^q\right\} ^{1/q} . \end{aligned}$$

Thus the last inequalities, the characterisation of ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ via the Peetre maximal functions stated in Theorem 3.2 (i) and the choice of g imply that

$$\begin{aligned} \left\| \textrm{ext}f\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \lesssim \left\| h\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \lesssim \left\| f\mid {B}_{p,q}^{s,\tau }(\Omega )\right\| +\varepsilon . \end{aligned}$$

(3.13)

Letting $\varepsilon \rightarrow 0$, we then know that $\textrm{ext}$ is a bounded linear operator from ${B}_{p,q}^{s,\tau }(\Omega )$ into ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$.

Finally, since the supports of $\psi _0$ and $\psi$ lie in $-K$, it follows that

$$\begin{aligned} \textrm{ext}f|_\Omega = \sum _{j=0}^{\infty } \psi _j *\phi _j *f =f\quad \text{ in } {{\mathcal {D}}}'(\Omega ). \end{aligned}$$

Therefore, $\textrm{ext}$ is the desired extension operator from ${B}_{p,q}^{s,\tau }(\Omega )$ into ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, which concludes the proof for the Besov-type spaces.

Step 3. The proof for the Triebel-Lizorkin-type spaces follows similarly. We can define the extension operator by the formula (3.5) and use once more the Peetre maximal function $\mathcal {M}^{g_j}_N$ as the main tool for estimations. Therefore, we point out the differences without giving the details. The estimation of the norm $\big \Vert \psi _j *g_j \mid F^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})\big \Vert$ is the first difference, but it can be done similarly as in (3.8), now with the $L_p^\tau (\ell _q)$ norm instead of $\ell _q(L_p^\tau )$ norm on the right hand side of the inequality. Afterwards, the counterpart of (3.9) can be obtained using again Lemma 3.1, but now the estimate related to the spaces $L_p^\tau (\ell _q)$. Finally, the characterisation of ${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ via the Peetre maximal functions stated in Theorem 3.2 (ii) leads us to obtain a similar estimate as (3.13). $\square$

Remark 3.7

As mentioned above essentially the same proof for Triebel-Lizorkin-type spaces can be found in [52] with additional restrictions $p,q\ge 1$. Note that, in view of the coincidence (2.16), also the Triebel-Lizorkin-Morrey spaces ${{\mathcal {E}}}^{s}_{u,p,q}$ are covered by our theorem. This complements the corresponding result obtained in [48, Proposition 4.13] for the class of Besov-Morrey spaces ${{\mathcal {N}}}^{s}_{u,p,q}$. Very recently similar arguments were used in [51, 52] for the construction of the extension operator, but restricted to the case $p,q\ge 1$. For the sake of completeness, we briefly sketched our proof here.

Corollary 3.8

Let $\Omega \subset {{{\mathbb {R}}}^d}$ be an interval if $d=1$ or a Lipschitz domain if $d\ge 2$. Then there exists a linear bounded operator $\textrm{ext}$ which maps ${{\mathcal {E}}}^{s}_{u,p,q}(\Omega )$ into ${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$ for all $s \in {{\mathbb {R}}}$, $q \in (0, \infty ]$ and $0<p\le u< \infty$, such that, for all $f \in {{\mathcal {D}}}'(\Omega )$, $\textrm{ext}f|_{\Omega }=f$ in ${{\mathcal {D}}}'(\Omega )$.

Proof

This follows immediately from Theorem 3.6 and (2.16). $\square$

Using Theorem 3.6 and the wavelet decomposition of the spaces ${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, cf. Proposition 2.17, we can now prove a result on the monotonicity of the spaces ${A}_{p,q}^{s,\tau }(\Omega )$ regarding the parameter $\tau$, which in fact does not hold when considering the spaces on ${{{\mathbb {R}}}^d}$.

Proposition 3.9

Let $0< p\le \infty$ ($p<\infty$ in the F-case), $s\in {{\mathbb {R}}}$, $0<q\le \infty$, $0\le \tau _2\le \tau _1$. Let $\Omega \subset {{{\mathbb {R}}}^d}$ be a bounded interval if $d=1$ or a bounded Lipschitz domain if $d\ge 2$. Then

$$\begin{aligned} \text {id}_{\tau }: A^{s,\tau _1}_{p,q}(\Omega )\hookrightarrow A^{s,\tau _2}_{p,q}(\Omega ). \end{aligned}$$

Proof

Let $\widetilde{Q}_0$ be a dyadic cube that contains $\overline{\Omega }$ in its interior and let $\widetilde{Q}$ be a (fixed) dyadic cube that contains the supports of all the functions $\psi _{i,j,k}$ and $\phi _{0,k}$ with non-empty intersection with $\widetilde{Q}_0$. Let $f\in A^{s,\tau _1}_{p,q}(\Omega )$. The compactly supported smooth functions are pointwise multipliers in ${A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d)$, cf. [31] or [47, Theorem 6.1] for $\tau \le \frac{1}{p}$ and Proposition 2.3 for $\tau >\frac{1}{p}$, therefore

$$\begin{aligned} \Vert f \mid {A}_{p,q}^{s,\tau }(\Omega )\Vert \sim \inf \{ \Vert g \mid {A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d )\Vert :\quad g\in {A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d )\quad \text {and}\quad {\mathop {\textrm{supp}}\nolimits }g\subset \widetilde{Q}_0\}. \end{aligned}$$

(3.14)

There exists a dyadic cube ${\widetilde{Q}}_1$ such that $\langle g,\psi _{i,j,m}\rangle = \langle g,\phi _{0,m}\rangle =0$ for any $g\in A^{s,\tau _1}_{p,q}({{\mathbb {R}}}^d)$ with ${\mathop {\textrm{supp}}\nolimits }g\subset \widetilde{Q}_0$ and $\phi _{0,\ell }$, $\psi _{i,j,m}$ such that ${\mathop {\textrm{supp}}\nolimits }\phi _{0,\ell }\nsubseteq {\widetilde{Q}}_1$, ${\mathop {\textrm{supp}}\nolimits }\psi _{i,j,m}\nsubseteq {\widetilde{Q}}_1$. Moreover there exists a positive constant C such that

$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}} \le C\, \frac{ 1}{|P|^{\tau _1}}\quad \text {for }\quad 0\le \tau _2 \le \tau _1\ \end{aligned}$$

if $P\subset \widetilde{Q}_1$. Therefore

$$\begin{aligned} \Vert \lambda (g)\mid&a^{s,\tau _2}_{p,q}\Vert ^*\le c \Vert \lambda (g)\mid a^{s,\tau _1}_{p,q}\Vert ^*. \end{aligned}$$

(3.15)

By Theorem 3.6 there exists a linear and bounded extension operator $\textrm{ext}$ from $A^{s, \tau _1}_{p,q}(\Omega )$ into $A^{s, \tau _1}_{p,q}({{\mathbb {R}}}^d)$. So, using also the wavelet decomposition of these spaces, if $\varphi \in C^\infty _0({{\mathbb {R}}}^d)$ is supported in $\widetilde{Q}_0$ and equals 1 on $\Omega$, then by (3.15)

$$\begin{aligned} \Vert f \mid A^{s, \tau _2}_{p,q}(\Omega )\Vert&\le C\, \Vert \lambda (\varphi \textrm{ext}(f))\mid a^{s, \tau _2}_{p,q}\Vert ^*\le C\, \Vert \lambda (\varphi \textrm{ext}(f))\mid a^{s, \tau _1}_{p,q}\Vert ^*\nonumber \\&\le C\, \Vert \varphi \textrm{ext}(f)\mid A^{s, \tau _1}_{p,q}({{\mathbb {R}}}^d)\Vert \le C \, \Vert f \mid A^{s, \tau _1}_{p,q}(\Omega )\Vert . \end{aligned}$$

(3.16)

The proof is complete. $\square$

4 Limiting embeddings

We shall always assume in the sequel that $\Omega$ is a bounded Lipschitz domain in ${{{\mathbb {R}}}^d}$. As already mentioned, we shall deal – differently from the standard approach – with continuous embeddings of type

$$\begin{aligned} \text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

only after we studied their compactness in [8].

But it will turn out that only the limiting cases are of particular interest. So we collect first what is more or less obvious. Note that we use the above notation always with the understanding that either both, source and target space, are of Besov-type ($A=B$), or both are of Triebel-Lizorkin-type ($A=F$).

For convenience we use the following abbreviation:

$$\begin{aligned} \gamma (\tau _1,\tau _2,p_1,p_2)&:= \max \left\{ \left( \tau _2-\frac{1}{p_2}\right) _+ -\left( \tau _1-\frac{1}{p_1}\right) _+, \frac{1}{p_1} -\tau _1 - \frac{1}{p_2}+\tau _2,\right. \nonumber \\&\left. \frac{1}{p_1}-\tau _1 - \min \left\{ \frac{1}{p_2}-\tau _2, \frac{1}{p_2}(1-p_1\tau _1)_+\right\} \right\} \nonumber \\&\quad = {\left\{ \begin{array}{ll} \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2, &{} \text {if}\quad \tau _2\ge \frac{1}{p_2}, \\ \frac{1}{p_1}-\tau _1, &{}\text {if}\quad \tau _1\ge \frac{1}{p_1}, \ \tau _2< \frac{1}{p_2}, \\ \max \{0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \{\tau _2,\frac{p_1}{p_2}\tau _1\}\}, &{}\text {if}\quad \tau _1< \frac{1}{p_1}, \ \tau _2< \frac{1}{p_2}. \end{array}\right. } \end{aligned}$$

(4.1)

Here and in the sequel we put $p_i\tau _i=1$ in case of $p_i=\infty$ and $\tau _i=0$. Similarly we shall understand $\frac{p_i}{p_k}=1$ if $p_i=p_k=\infty$.

Theorem 4.1

([8]) Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i\le \infty$ (with $p_i<\infty$ in case of $A=F$), $\tau _i\ge 0$, $i=1,2$.

(i):

The embedding

$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

is compact if, and only if,

$$\begin{aligned} \frac{s_1-s_2}{d} > \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$

(ii):

There is no continuous embedding

$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

if

$$\begin{aligned} \frac{s_1-s_2}{d} < \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$

(4.2)

This result was proved in [8] and shows us that we are indeed left to deal with the limiting case

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$

First we prove the following lemma that extends [11, Theorem 3.1] to the case $u=p=\infty$. We recall that $\mathcal {N}^{s}_{\infty ,\infty ,q}({{{\mathbb {R}}}^d})= B^{s}_{\infty ,q}({{{\mathbb {R}}}^d})$.

Lemma 4.2

Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i\le u_i< \infty$ or $p_i=u_i=\infty$, $i=1,2$. Then

$$\begin{aligned}&{{\mathcal {N}}}^{s_1}_{u_1,p_1,q_1}(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{\infty ,\infty ,q_2}(\Omega ) \quad \text{ if, } \text{ and } \text{ only } \text{ if, }\nonumber \\&\frac{s_1-s_2}{d}> \frac{1}{u_1}\quad \text {or}\quad \frac{s_1-s_2}{d}= \frac{1}{u_1}\quad \text {and}\quad q_1\le q_2 , \end{aligned}$$

(4.3)

and

$$\begin{aligned} \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega )&\quad \text{ if, } \text{ and } \text{ only } \text{ if, } \quad s_1> s_2 \quad \text {or}\quad s_1=s_2 \quad \text {and}\quad q_1\le q_2 . \end{aligned}$$

(4.4)

Proof

The necessity of the conditions in (4.3) follows easily by the following chain of embeddings

$$\begin{aligned} B^{s_1}_{u_1,q_1}(\Omega )\hookrightarrow {{\mathcal {N}}}^{s_1}_{u_1,p_1,q_1}(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{\infty ,\infty ,q_2}(\Omega ) = B^{s_2}_{\infty ,q_2}(\Omega ) \end{aligned}$$

and the properties of embeddings of classical Besov spaces. Whereas the sufficiency can be proved in the same way as in the proof of [11, Theorem 3.1].

To prove the second embedding it is sufficient to note that

$$\begin{aligned} \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) = B^{s_1}_{\infty ,q_1}(\Omega ) \hookrightarrow B^{s_2}_{u_2,q_2}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega ). \end{aligned}$$

On the other hand it follows from (4.3) that

$$\begin{aligned} B^{s_1+\frac{d}{u_1}}_{u_1,q_1}(\Omega )\hookrightarrow \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega ).\end{aligned}$$

So if the last embedding holds, then it follows from [11, Corollary 3.7] that $s_1>s_2$, or $s_1=s_2$ and $q_1\le q_2$. $\square$

Proposition 4.3

Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i\le \infty$ (with $p_i<\infty$ in case of $A=F$), $\tau _i\ge 0$, $i=1,2$. Assume

$$\begin{aligned} \tau _2\ge \frac{1}{p_2}\quad \text {with}\quad q_2=\infty \quad \text {if}\quad \tau _2=\frac{1}{p_2}. \end{aligned}$$

Then the embedding

$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

is continuous if, and only if,

$$\begin{aligned} \frac{s_1-s_2}{d} \ge \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$

Proof

Note that by Theorem 4.1 we are left to deal with the limiting case $s_1-s_2 = d \gamma (\tau _1,\tau _2,p_1,p_2)$ only. In view of Proposition 2.3 we always have ${A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega )$ now. Assume first $\tau _1\ge \frac{1}{p_1}$ with $q_1=\infty$ if $\tau _1=\frac{1}{p_1}$, then by the same result also ${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )$ such that $\text {id}_\tau$ is continuous if, and only if,

$$\begin{aligned} \text {id}:B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega ) \hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ). \end{aligned}$$

But this is always true if $\frac{s_1-s_2}{d}=\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2$, recall Remark 2.15.

Assume next $0\le \tau _1<\frac{1}{p_1}$. We put $\frac{1}{u_1}=\frac{1}{p_1}-\tau _1$. We first show the sufficiency of $s_1-s_2= d\gamma (\tau _1,\tau _2,p_1,p_2)$ for the continuity of $\text {id}_\tau$. We use (2.15), (2.5), Proposition 2.3 and Lemma 4.2 to obtain

$$\begin{aligned}{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow B^{s_1,\tau _1}_{p_1,\infty } (\Omega )=\mathcal {N}^{s_1}_{u_1,p_1,\infty }(\Omega )\hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega )={A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ). \end{aligned}$$

On the other hand, for the necessity,

$$\begin{aligned}\mathcal {N}^{s_1}_{u_1,p_1,\min \{p_1,q_1\}}(\Omega )\hookrightarrow {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ) \end{aligned}$$

and (4.3) implies $s_1-s_2 \ = \ d\gamma (\tau _1,\tau _2,p_1,p_2)$ if $\text {id}_\tau$ is continuous.

It remains to consider the case $\tau _1=\frac{1}{p_1}$, $q_1<\infty$. Now we benefit from the following chains of embeddings

$$\begin{aligned} {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ) = {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

(4.5)

for the sufficiency of the condition $s_1-s_2= d\gamma (\tau _1,\tau _2,p_1,p_2)$, and

$$\begin{aligned} B^{s_1}_{\infty ,\min (p_1,q_1)}(\Omega ) \hookrightarrow {{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )} = B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ), \end{aligned}$$

(4.6)

for its necessity, cf. (2.5) and [47, Proposition 2.4]. $\square$

Remark 4.4

Note that the above result is the direct counterpart of our result for spaces on ${{{\mathbb {R}}}^d}$ obtained in Theorem 2.10 (i), since $\gamma (\tau _1,\tau _2,p_1,p_2)=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2$ in the above setting.

Remark 4.5

Recall the definition of the spaces $\textrm{bmo}({{{\mathbb {R}}}^d})$ in Remark 2.8 and define $\textrm{bmo}(\Omega )$ by restriction, that is, in analogy to Definition 2.12. Then

$$\begin{aligned} \textrm{bmo}(\Omega )=F^{0,1/p}_{p,2}(\Omega ) = {B^{0,1/2}_{2,2}(\Omega ), \quad 0<p<\infty }, \end{aligned}$$

(4.7)

extending (2.18) and (2.17) to domains $\Omega$. Taking $\textrm{bmo}(\Omega )$ as the source space, that is, $\tau _1=\frac{1}{p_1}$, $s_1=0$, then Proposition 4.3 implies that for $s\in {{\mathbb {R}}}$, $0<p,q\le \infty$, and $\tau \ge \frac{1}{p}$ with $q=\infty$ if $\tau =\frac{1}{p}$, then

$$\begin{aligned} \text {id}_\tau : \textrm{bmo}(\Omega ) \hookrightarrow {A}_{p,q}^{s,\tau }(\Omega ) \end{aligned}$$

is continuous if, and only if, $s\le -d (\tau -\frac{1}{p})\le 0$. If $\textrm{bmo}(\Omega )$ was the target space, then Proposition 4.3 cannot be applied since $q_2=2<\infty$.

In view of Theorem 4.1 and Proposition 4.3 we are left to study the situation

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)\quad \text {and}\quad \tau _2\le \frac{1}{p_2}\quad \text {with}\quad q_2<\infty \quad \text {if}\quad \tau _2=\frac{1}{p_2} \end{aligned}$$

(4.8)

in the sequel. Next we give some counterpart of Proposition 4.3 dealing with the case when in the source space the parameter $\tau _1$ is large.

Proposition 4.6

Let $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0<p_i\le \infty$, $\tau _i\ge 0$, $i=1,2$. Assume that (4.8) is satisfied and

$$\begin{aligned} \tau _1\ge \frac{1}{p_1}\quad \text {with}\quad q_1=\infty \quad \text {if}\quad \tau _1=\frac{1}{p_1}. \end{aligned}$$

Then the embedding

$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

is continuous if, and only if,

$$\begin{aligned} q_2=\infty . \end{aligned}$$

Proof

First note that $\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1$ by (4.1) and thus ${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )=B^{s_2}_{\infty ,\infty }(\Omega )$ in view of Proposition 2.3.

We first deal with the case $A=B$ and start with the sufficiency of $q_2=\infty$. Then Proposition 4.3 covers the case $\tau _2 =\frac{1}{p_2}$ and we may assume $\tau _2<\frac{1}{p_2}$, recall (4.8). But in view of (2.6) and (2.15) we get

$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{u_2,p_2,\infty }(\Omega )= B^{s_2,\tau _2}_{p_2,\infty }(\Omega ). \end{aligned}$$

Now assume $A=F$ and again $\tau _2<\frac{1}{p_2}$. If $f\in {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) =B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )$, recall Proposition 2.3, then there exists some $g\in B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$ such that $f=g|_\Omega$ We can choose g such that it can be represented as in (2.26) with the summation over $k\in {{{\mathbb {Z}}}^d}$ restricted to the indices k such that $|k|\le K$ for some fixed K since $\Omega$ is a bounded domain. Moreover we can choose g in such a way that

$$\begin{aligned} \Vert \lambda _k \mid \ell _\infty ({{{\mathbb {Z}}}^d})\Vert + \sup _{j\in {{\mathbb {N}}}} 2^{j(s_1+d(\tau _1-\frac{1}{p_1})+\frac{d}{2})} \sup _{i=1,\ldots ,2^d-1;\;k\in {{{\mathbb {Z}}}^d}} |\lambda _{i,j,k}| \le C \Vert f\mid B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\Vert ,\end{aligned}$$

for some constant $C>0$ independent of f. We have to show that $f\in F^{s_2,\tau _2}_{p_2,\infty }(\Omega )$ and $\Vert f \mid F^{s_2,\tau _2}_{p_2,\infty }(\Omega ) \Vert \le c \Vert f\mid B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\Vert$. It is sufficient to note that for any $i=1,\ldots , 2^d-1$,

$$\begin{aligned}&\left\| \sup _{j,k} 2^{j(s_2-\frac{d}{u_2}+\frac{d}{2})} |\lambda _{i,j,k}|{2^{j\frac{d}{u_2}}\chi _{j,k}(\cdot )}\mid {\mathcal {M}_{u_2,p_2}}{({{{\mathbb {R}}}^d})}\right\| \nonumber \\&\quad \le C \sup _{j\in {{\mathbb {N}}}} 2^{j(s_1+d(\tau _1-\frac{1}{p_1})+\frac{d}{2})} \sup _{i=1,\ldots ,2^d-1;\;k\in {{{\mathbb {Z}}}^d}} |\lambda _{i,j,k}| . \end{aligned}$$

(4.9)

The rest follows from the wavelet characterisation of $\mathcal {E}^{s_2}_{u_2,p_2,\infty }({{{\mathbb {R}}}^d})={F^{s_2,\tau _2}_{p_2,\infty }}({{{\mathbb {R}}}^d})$, $\frac{1}{u_2}=\frac{1}{p_2}-\tau _2$, cf. [16]. But (4.9) follows easily from the identities $s_2=s_1+d(\tau _1-\frac{1}{p_1})$ and $\Vert {2^{j\frac{d}{u_2}}\chi _{j,k}(\cdot )}\mid \mathcal {M}_{u_2,p_2}{({{{\mathbb {R}}}^d})}\Vert =1$.

Now we prove the necessity and assume that $\text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$ is continuous. We start with the case $A=B$. If $p_2=\infty$, then $\tau =0$ by assumption (4.8) and $B^{s_2,\tau _2}_{\infty ,q_2}(\Omega )= B^{s_2}_{\infty ,q_2}(\Omega )$. So both the source and the target space are classical Besov spaces and it is well-known that, in that case, $q_2=\infty$, recall Remark 2.15. So it remains to consider the case $p_2<\infty$. To simplify the notation we assume that the support of the wavelets $\psi _{i,j,m}$ such that $Q_{j,m}\subset Q_{0,0}$ are contained in $\Omega$. If it is not true one can easily rescale the argument.

We take a sequence $\lambda =(\lambda _{i,j,m}){_{i,j,m}, i=1,...,2^d-1, j\in {{\mathbb {N}}}_0,m\in {{{\mathbb {Z}}}^d},}$ defined by the formula

$$\begin{aligned} \lambda _{i,j,m}= {\left\{ \begin{array}{ll} 2^{-j(s_2+\frac{d}{2})} &{} \textrm{if }\quad i=1 \quad \textrm{and}\quad Q_{j,m}\subset Q_{0,0},\\ 0 &{} \textrm{otherwise}. \end{array}\right. } \end{aligned}$$

Then, using the sequence space version of Proposition 2.3,

$$\begin{aligned} \Vert \lambda \mid {b}_{p_1,q_1}^{s_1,\tau _1}\Vert \sim \Vert \lambda \mid b^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty } \Vert = 1 \end{aligned}$$

since $s_2= s_1+d(\tau _1-\frac{1}{p_1})$. Here we used the notation $b^\sigma _{p,q} = b^{\sigma ,0}_{p,q}$. On the other hand, for any dyadic cube $P\subset Q_{0,0}$ and any $j\ge j_P$, we have

$$\begin{aligned} 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})}\left( \sum _{Q_{j,m}\subset P}|\lambda _{i,j,m}|^{p_2}\right) ^{\frac{1}{p_2}} = 2^{-j_P\frac{1}{p_2}}. \end{aligned}$$

So if $q_2<\infty$, then

$$\begin{aligned} \Vert \lambda \mid b^{s_2,\tau _2}_{p_2,q_2} \Vert = \infty . \end{aligned}$$

Therefore, if $q_2<\infty$, the function

$$\begin{aligned} f=\sum _{i,j,m} \lambda _{i,j,m} \psi _{i,j,m} \end{aligned}$$

belongs to $B^{s_1,\tau _1}_{p_1,q_1}(\Omega )$ but not to $B^{s_2,\tau _2}_{p_2,q_2}(\Omega )$, which contradicts our assumption and thus finishes the proof of the necessity for the B-case.

The case $A=F$ follows by (2.5) and by what we just proved for the Besov-type spaces. Note that, for the F-spaces, we always have $p<\infty$. Therefore, by the following chain of embeddings

$$\begin{aligned} B^{s_1, \tau _1}_{p_1, \min \{p_1, q_1\}}(\Omega ) \hookrightarrow {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \hookrightarrow B^{s_2, \tau _2}_{p_2, \max \{p_2,q_2\}}(\Omega ), \end{aligned}$$

we obtain the necessity of the condition $\max \{p_2,q_2\}=\infty$, which here reads as $q_2=\infty$. $\square$

Remark 4.7

Note that the above result differs from its ${{{\mathbb {R}}}^d}$-counterpart in Theorem 2.10 (ii). In that case, there is never a continuous embedding in the setting of Proposition 4.6, that is, when conditions (4.8) and $\tau _1\ge \frac{1}{p_1}$ with $q_1=\infty$ when $\tau _1=\frac{1}{p_1}$ are satisfied.

Remark 4.8

Again we return to the special case when the source or target space of $\text {id}_\tau$ coincides with $\textrm{bmo}(\Omega )$. Parallel to Remark 4.5 we cannot apply Proposition 4.6 in case of ${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=\textrm{bmo}(\Omega )$. Otherwise, if ${A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=\textrm{bmo}(\Omega )$, then Proposition 4.6 implies that there is never a continuous embedding of type

$$\begin{aligned}\text {id}_\tau : {A}_{p,q}^{s,\tau }(\Omega ) \hookrightarrow \textrm{bmo}(\Omega )\end{aligned}$$

in the limiting case (4.8) which reads here as $0<p\le \infty$ (with $p<\infty$ in case of $A=F$), $0<q\le \infty$, $\tau \ge 0$ and $s = d(\frac{1}{p}-\tau )$. Moreover, there is no such continuous embedding whenever $\tau _2=\frac{1}{p_2}$ and $q_2<\infty$ in the limiting case (4.8).

For the rest of this section we shall now assume that

$$\begin{aligned} 0\le \tau _i\le \frac{1}{p_i}\quad \text {with}\quad q_i<\infty \quad \text {if}\quad \tau _i=\frac{1}{p_i}, \quad i=1,2, \quad \textrm{and}\quad \tau _1+\tau _2>0, \end{aligned}$$

(4.10)

and thus

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)= \max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$

(4.11)

In view of the embeddings and coincidences (2.14), (2.15) and (2.16), together with our previous findings for the spaces ${{\mathcal {A}}}^{s}_{u,p,q}(\Omega )$ in [11, 12] (as well as some ${{{\mathbb {R}}}^d}$-counterparts of ${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$ in Theorems 2.10 and 2.11), we expect some q-dependence now. For the moment, we restrict ourselves to the case of ${B}_{p,q}^{s,\tau }$ spaces.

Theorem 4.9

Let $0< p_1,p_2\le \infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0\le \tau _i\le \frac{1}{p_i}$, $i=1,2$. Assume that the conditions (4.10) hold and that

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$

(i):

The embedding

$$\begin{aligned} \text {id}_{\tau }: {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

(4.12)

is continuous if one of the following conditions holds:

$$\begin{aligned}&\frac{s_1-s_2}{d} = \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0, \quad \text {and}\quad p_1\tau _1<p_2\tau _2, \end{aligned}$$

(4.13)

$$\begin{aligned} \text{ or } \qquad&\frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\frac{p_1}{p_2}\tau _1>0 \quad \text {and}\quad q_1\le \frac{p_1}{p_2}q_2 , \end{aligned}$$

(4.14)

$$\begin{aligned} \text{ or } \qquad&s_1=s_2 \quad \text {and}\quad q_1\le \min \left\{ 1, \frac{p_1}{p_2}\right\} q_2. \end{aligned}$$

(4.15)

(ii):

If the embedding (4.12) is continuous and one of the following conditions holds

$$\begin{aligned}&\gamma (\tau _1,\tau _2,p_1,p_2)= 0, \end{aligned}$$

(4.16)

$$\begin{aligned} \text{ or }\qquad&\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\frac{p_1}{p_2}\tau _1>0 , \end{aligned}$$

(4.17)

then $q_1\le q_2$. If the embedding (4.12) is continuous for any $q_1$ and $q_2$ with fixed $s_1,s_2,p_1,p_2,\tau _1,\tau _2$, then (4.13) holds.

Remark 4.10

As mentioned above, we are left to consider the embedding in the limiting case (4.11) when (4.10) is satisfied. However, in case of $\tau _i=\frac{1}{p_i}$, $q_i=\infty$, for $i=1$ or $i=2$, the above Theorem 4.9 coincides with Propositions 4.3 or 4.6, respectively. So in fact situation (4.10) is the only interesting one now.

We have always $p_1<p_2$ in (4.14) so we have a small gap between sufficient and necessary conditions on $q_i$ here. We meet a similar situation if $s_1=s_2$, $p_1<p_2$, $\tau _1=\frac{1}{p_1}$ and $q_1<\infty$ in (4.15). In all other cases the result is sharp.

Proof (of Theorem 4.9)

Step 1. We start by proving part (i). For this, we use an argument similar to the one used in the proof of Proposition 3.9, based on the extension operator from Theorem 3.6 and the wavelet decomposition of the spaces ${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$, cf. Proposition 2.17. We use the same notation as there. Let us denote by $\widetilde{{b}_{p,q}^{s,\tau }}(\widetilde{Q}_0)$ the sequence space defined by

$$\begin{aligned} \widetilde{{b}_{p,q}^{s,\tau }}(\widetilde{Q}_0){:=}&\left\{ t= \{t_{i,j,m}\}_{i,j,m}: t_{i,j,m} \in {{\mathbb {C}}},\, j \in {{\mathbb {N}}}_0, i=1, \dots , 2^d-1, \right. \\&\qquad \qquad \left. m\in {{{\mathbb {Z}}}^d}, Q_{j,m} \subset \widetilde{Q}_0,\, \Vert t \mid \widetilde{{b}_{p,q}^{s,\tau }}\Vert <\infty \right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}_0} \frac{ 1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2}-\frac{d}{p})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p}\right] ^{\frac{ q}{p}}\right\} ^{\frac{1}{q}}.\nonumber \\ \end{aligned}$$

(4.18)

Then, we have to prove that for some $C>0$

$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \le C \, \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert \end{aligned}$$

(4.19)

holds true for all $t\in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$. Here and in the sequel we assume for convenience that $p_i, q_i<\infty$, otherwise the modifications are obvious. Please note, once more, that the assumption $P\subset \widetilde{Q}_{0}$ implies that

$$\begin{aligned} \frac{ 1}{|P|^{a}} \le C \frac{ 1}{|P|^{b}}\quad \text {if }\quad a\le b \quad \text {and}\quad \#\{m:\;Q_{j,m}\subset P \} \sim 2^{jd}\min \{1, 2^{-j_P d}\}\quad \text {if} \quad j\ge j_P.\nonumber \\ \end{aligned}$$

(4.20)

Moreover, if $1\le |P|\le | \widetilde{Q}_0|$, then

$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}} \sim \frac{ 1}{|P|^{\tau _1}} \end{aligned}$$

(4.21)

for any $\tau _1$ and $\tau _2$. To shorten the notation we put $\gamma = \gamma (\tau _1,\tau _2,p_1,p_2)$.

Substep 1.1. If $\gamma =\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0$, then $\frac{p_1}{p_2}\tau _1\le \tau _2$. In this case the statement follows from Theorem 3.6 and Theorem 2.10 (iii) if $\tau _1\not = \tau _2$.

Indeed Theorem 3.6 and Remark 3.7 imply that there exists a common bounded extension operator $\textrm{ext}$ for the spaces ${B}_{p_1,q_1}^{s_1,\tau _1}(\Omega )$ and ${B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$ and we thus have the following commutative diagram

Now the case (4.13) follows from Theorem 2.10 (iii), as well as the situation when

$$\begin{aligned} \frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0,\quad \tau _1\not = \tau _2,\quad \frac{p_1}{p_2}=\frac{\tau _2}{\tau _1}\quad \text {and}\quad q_1\le \frac{p_1}{p_2}q_2. \end{aligned}$$

(4.22)

Substep 1.2. Let $\gamma =\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0$ and $\tau _1=\tau _2$. In that case $p_1<p_2$ and $s_1-\frac{d}{p_1}=s_2-\frac{d}{p_2}$. Let $t = \{t_{i,j,m}\}_{i,j,m}\in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$ and let $\Vert t\mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert =1$. To simplify the notation we put $\lambda _{i,j,m}= 2^{-j(s_1+\frac{d}{2}-\frac{d}{p_1})}t_{i,j,m}$ and $\tau =\tau _1=\tau _2$.

For any i, j and m we have

$$\begin{aligned} |\lambda _{i,j,m}|\le 2^{-jd\tau }, \end{aligned}$$

and in consequence

$$\begin{aligned} |\lambda _{i,j,m}|^{p_2}\le |\lambda _{i,j,m}|^{p_1} 2^{-jd\tau (p_2-p_1)}. \end{aligned}$$

(4.23)

In a parallel way, for any dyadic cube $P\subset \widetilde{Q}_0$ we have

$$\begin{aligned} \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_1} \le 2^{-j_Pd\tau p_1}, \end{aligned}$$

(4.24)

so in consequence

$$\begin{aligned} \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_2}\right) ^{\frac{q_2}{p_2}}&\le \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_1}\right) ^{\frac{q_2}{p_2}} 2^{-jd\tau (p_2-p_1)\frac{q_2}{p_2}} \nonumber \\&\le \ c\ 2^{-j_Pd\tau \frac{p_1}{p_2}q_2} 2^{-jd\tau (1-\frac{p_1}{p_2})q_2} \end{aligned}$$

(4.25)

for any $i=1,\dots , 2^d-1$. Summing up over j we get

$$\begin{aligned} \sum _{j=j_P}^\infty \sum _{i=1}^{2^d-1} \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_2}\right) ^{\frac{q_2}{p_2}} \le c\ 2^d 2^{-j_Pd\tau \frac{p_1}{p_2}q_2} \sum _{j=j_P}^\infty 2^{-jd\tau (1-\frac{p_1}{p_2})q_2} = C 2^{-j_Pd\tau q_2}. \end{aligned}$$

(4.26)

This proves that $t = \{t_{i,j,m}\}_{i,j,m}\in \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}$ and $\Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \le C$.

Substep 1.3. Let $\gamma =0$, i.e., $s_1=s_2$. Then $p_2\le p_1$ and $\frac{1}{p_1}-\frac{1}{p_2} \le \tau _1-\tau _2$ or $p_2 > p_1$ and $\tau _1=\frac{1}{p_1}$. First we assume that $p_2\le p_1$ and $\frac{1}{p_1}-\frac{1}{p_2} \le \tau _1-\tau _2$. We conclude by Hölder’s inequality for any $i=1, \dots , 2^d-1$, that

$$\begin{aligned} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{1}{p_2}} \le 2^{d(j-j_P)(\frac{1}{p_2}-\frac{1}{p_1})} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{1}{p_1}} \, . \end{aligned}$$

(4.27)

In consequence, for any $q\in (0,\infty ]$,

$$\begin{aligned} \left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{{p_2}}\right] ^{\frac{q}{p_2}} \right\} ^{\frac{1}{q}} \,\le 2^{j_P d(\frac{1}{p_1}-\frac{1}{p_2})} \nonumber \\ \times \left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q} 2^{j(s_2-s_1)q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{q}{p_1}} \right\} ^{\frac{1}{q}}\, . \end{aligned}$$

(4.28)

If $\gamma (\tau _1,\tau _2,p_1,p_2)=0$, then $s_1=s_2$ and $a=\frac{1}{p_1}-\frac{1}{p_2}+\tau _2\le \tau _1=b$. So (4.19) follows from (4.28) and (4.20)-(4.21) for any $q_1\le q_2$.

Now let $\gamma =0$, $p_2 > p_1$ and $\tau _1=\frac{1}{p_1}$. First we consider the case $\tau _2=\frac{1}{p_2}$. Let

$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert =1, \end{aligned}$$

(4.29)

which implies that, for every cube $P \in \mathcal {Q}, P \subset \widetilde{Q}_{0}$ and for every $j \ge \max \{j_P,0\}$, we have

$$\begin{aligned} \frac{ 1}{|P|^{\tau _1 q_1}} \, 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{q_1}{p_1}} \le 1. \end{aligned}$$

In particular, we know that for every cube $Q_{\nu ,m} \in \mathcal {Q}, Q_{\nu ,m} \subset \widetilde{Q}_{0}$, with $\nu \ge 0$ and for every $i=1,..., 2^d-1$, we have

$$\begin{aligned} 2^{\nu (s_1 + \frac{d}{2}- \frac{d}{p_1}+d \tau _1)}|t_{i,\nu ,m}|= 2^{\nu (s_1 + \frac{d}{2})}|t_{i,\nu ,m}| \le 1. \end{aligned}$$

So the condition $p_1< p_2$ implies

$$\begin{aligned} 2^{\nu ( s_1 + \frac{d}{2})p_2}|t_{i,\nu ,m}|^{p_2} \le 2^{\nu ( s_1 + \frac{d}{2})p_1}|t_{i,\nu ,m}|^{p_1}. \end{aligned}$$

(4.30)

We have to prove that $\Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \lesssim 1$. Let us fix a cube $P\in \mathcal {Q}, P \subset \widetilde{Q}_{0}$. Thus, by the inequality (4.30), it follows that for $q_1=\frac{p_1}{p_2}q_2$ we have

$$\begin{aligned}&\sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\\&\quad = \sum _{j=\max \{j_P,0\}}^\infty 2^{-j \frac{d}{p_2}q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P}2^{j(s_1+\frac{d}{2})p_2} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\\&\quad \le \sum _{j=\max \{j_P,0\}}^\infty 2^{-j\frac{d}{p_1} q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P}2^{j(s_1+\frac{d}{2})p_1} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\\&\quad = \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\\&\quad \le \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert ^{q_1}\, |P|^{\tau _1 q_1}. \end{aligned}$$

In consequence for any cube $P\subset \widetilde{Q}_{0}$ we have

$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}}&\left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\right) ^\frac{1}{q_2}\le |P|^{\frac{\tau _1 q_1}{q_2}-\tau _2 } \le 1\\ \end{aligned}$$

since $\tau _1 q_1 -\tau _2q_2 = 0$. So by monotonicity if $\tau _2=\frac{1}{p_2}$, then for any $q_1$ and $q_2$ such that $q_1 \le \frac{p_1}{p_2}q_2$,

$$\begin{aligned} \Big \Vert t \mid {\widetilde{b^{s_2,\frac{1}{p_2}}_{p_2,q_2}}}\Big \Vert \le \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert . \end{aligned}$$

If $\gamma =0$ and $\tau _1=\frac{1}{p_1}$, then $\tau _2\le \frac{1}{p_2}$. If $\tau _2<\frac{1}{p_2}$, then it follows from Substep 1.2 that

$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \le C\, \Big \Vert t \mid {\widetilde{ b^{s_2,\frac{1}{p_2}}_{p_2,q_2}}}\Big \Vert , \end{aligned}$$

so the final statement follows from the last two inequalities.

Substep 1.4.  Now let $\gamma = \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _1\frac{p_1}{p_2} > 0$ and $0\le \tau _1\le \frac{1}{p_1}$. Please note that this assumption implies $\tau _1\frac{p_1}{p_2} \ge \tau _2$, $p_1\tau _1<1$ and $p_1<p_2$.

The case $\tau _1\frac{p_1}{p_2}= \tau _2$ is covered by (4.22) since $\tau _2=\tau _1\frac{p_1}{p_2} < \tau _1$. Let $\gamma >0$ and $\tau _2 < \frac{p_1}{p_2}\tau _1$. We take $\tau _0$ such that $\tau _0=\frac{p_1}{p_2}\tau _1$. The above considerations show that

$$\begin{aligned} \Vert t \mid {\widetilde{b^{s_2,\tau _0}_{p_2,q_2}}}\Vert \le C \, \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert \end{aligned}$$

if $q_1\le \frac{p_1}{p_2}q_2$. Since $\tau _2< \tau _0$ it follows from Substep 1.2 that

$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \le C \,\Vert t \mid {\widetilde{b^{s_2,\tau _0}_{p_2,q_2}}}\Vert . \end{aligned}$$

Step 2.  Now we come to the necessity. We do some preparation first.

By the diffeomorphic properties of Besov-type spaces, using translations and dilations if necessary we can assume that the domain $\Omega$ satisfies the following conditions: there exists some number $\nu _0\in {{\mathbb {Z}}}$ such that

• $Q_{\nu _0,0} \subset \Omega$,

• if $Q_{j,m} \subset Q_{\nu _0,0}, \quad j\ge 0, \quad$ then $\quad {\mathop {\textrm{supp}}\nolimits }\psi _{i,j,m} \subset \Omega$,

• if $Q_{0,m} \subset Q_{\nu _0,0}, \quad$ then $\quad {\mathop {\textrm{supp}}\nolimits }\phi _{0,m} \subset \Omega$.

Due to the isomorphism resulting from the wavelet decomposition between function and sequence spaces, and similar to the explanation given in Substep 2.1 of the proof of [11, Theorem 3.1], one can equivalently prove the necessary conditions for the embedding

$$\begin{aligned} \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}(Q_{\nu _0,0}) \hookrightarrow \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}(Q_{\nu _0,0}), \end{aligned}$$

with $\nu _0<0$. For convenience, let us denote $\widetilde{Q}=Q_{\nu _0,0}$.

Substep 2.1.  We show that $q_1\le q_2$ is necessary when $s_1=s_2$. We assume $q_1 >q_2$. Then we can choose a sequence of positive numbers $\{\gamma _j\}_{j\in {{\mathbb {N}}}_0} \in \ell _{q_1}({{\mathbb {N}}}_0){\setminus }\ell _{q_2}({{\mathbb {N}}}_0)$. Let us define the sequence $t=\{t_{i,j,m}\}_{i,j,m}$, $i=1,...,2^d-1, j\in {{\mathbb {N}}}_0, m \in {{{\mathbb {Z}}}^d},$ by

$$\begin{aligned} t_{i,j,m} {:=} {\left\{ \begin{array}{ll} 2^{-j(s_1 + \frac{d}{2})} \gamma _j \quad &{}\text{ if }\quad i=1 \quad \text{ and } \quad Q_{j,m}\subset [0,1)^d,\\ 0 &{}\text{ otherwise }. \end{array}\right. } \end{aligned}$$

(4.31)

Then,

$$\begin{aligned}&\Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2})q_1} \sum _{i=1}^{2^d-1} \left[ \int _P \left( \sum _{m \in {{{\mathbb {Z}}}^d}} |t_{i,j,m}|\chi _{j,m}(x)\right) ^{p_1}\;\textrm{d}x \right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2})q_1} 2^{-j(s_1 + \frac{d}{2})q_1} |\gamma _j|^{q_1} |P\cap [0,1)^{d}|^{\frac{q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \Vert \{\gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_1}\Vert < \infty , \end{aligned}$$

where the last equality holds because $\tau _1 \le \frac{1}{p_1}$. On the other hand, we obtain similarly that

$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _2}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\right) ^\frac{1}{q_2}\\&=\Vert \{\gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_2}\Vert = \infty , \end{aligned}$$

which contradicts the embedding.

Substep 2.2. Now we show that the condition $q_1\le q_2$ is also necessary when $\frac{s_1-s_2}{d}= \frac{1}{p_1}-\tau _1 - \frac{1}{p_2}+ \frac{p_1}{p_2}\tau _1>0$. Let us assume $q_1>q_2$. We adapt the counter-example used in Substep 2.4 of the proof [10, Theorem 3.2]. For any $0>\nu \ge \nu _0$, we put

$$\begin{aligned} k_\nu {:=} \lfloor 2^{d|\nu |p_1 \tau _1}\rfloor , \end{aligned}$$

where $\lfloor x \rfloor =\max \{l \in {{\mathbb {Z}}}: l\le x\}$. Then $1\le k_\nu <2^{d|\nu |}$ and

$$\begin{aligned} k_\nu \le c_{p_1, \tau _1} \, 2^{d(\mu - \nu )}\,k_{\mu }, \quad \text{ if }\quad \nu \le \mu <0. \end{aligned}$$

(4.32)

For convenience, we assume $c_{p_1, \tau _1}=1$ (otherwise the proper modifications have to be done). As there, we define a sequence $t^{(\nu )}= \{t_{i,j,m}^{(\nu )}\}_{i,j,m}$, $i=1,...,2^d-1, j\in {{\mathbb {N}}}_0, m \in {{{\mathbb {Z}}}^d},$ in the following way: we assume that $k_\nu$ elements of the sequence equal 1 and the rest equals 0. If $j\ne 0$, $i\ne 1$ or $Q_{0,m} \nsubseteq Q_{\nu ,0}$, then $t_{i,j,m}^{(\nu )}=0$. Because of (4.32), we can choose the elements that equal 1 in such a way that the following property holds:

$$\begin{aligned} \text{ if } \quad Q_{\mu , l}\subseteq Q_{\nu ,0} \quad \text{ and } \quad Q_{\mu ,l}= \bigcup _{i=1}^{2^{-d \mu }} Q_{0,m_i}, \quad \text{ then } \text{ at } \text{ most } k_\mu \text{ elements } t_{1,0,m_i}^{(\nu )} \text{ equal } 1. \end{aligned}$$

Now we define a new sequence $t=\{t_{i,j,m}\}_{i,j,m} \in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$ by

$$\begin{aligned} t_{i,j,m}= \gamma _j \, t_{i,0,m}^{(\nu )}, \quad \text{ if }\quad j=\nu -\nu _0 \quad \text{ and } \quad Q_{0,m}\subset Q_{\nu ,0}, \end{aligned}$$

where $\{\gamma _j\}_{j \in {{\mathbb {N}}}_0}$ is a sequence of positive numbers with $\{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0} \in \ell _{q_1}({{\mathbb {N}}}_0){\setminus }\ell _{q_2}({{\mathbb {N}}}_0)$. If $Q_{\mu ,l}\subset Q_{\nu _0, 0}$, then for fixed $j \ge \mu$, there are at most $k_{\mu -l}$ non-zero elements $t_{i,j,m}$ such that $Q_{j,m} \subset Q_{\mu ,l}$. Thus

$$\begin{aligned} \sum _{m: Q_{j,m} \subset Q_{\mu ,l}} |t_{i,j,m}|^{p_1} \le \gamma _j^{p_1}\, 2^{d(j-\mu )\tau _1 p_1} \end{aligned}$$

and the last sum is $k_{\mu -j}\gamma _j^{p_1}$ if $\mu =\nu _0$. Therefore,

$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\nonumber \\&\le \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1}\, \gamma _j^{q_1}\, 2^{d(j-j_P)\tau _1q_1} \right) ^\frac{1}{q_1}\nonumber \\&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)q_1} \,\gamma _j^{q_1} \right) ^\frac{1}{q_1}\nonumber \\&= \Vert \{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)}\, \gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_1}\Vert <\infty . \end{aligned}$$

(4.33)

Similarly we obtain

$$\begin{aligned} \left( \sum _{m:\ Q_{j,m}\subset Q_{\mu ,l}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \le \left( \gamma _j^{p_2}\, k_{\mu -j}\right) ^{\frac{q_2}{p_2}}\le \gamma _j^{q_2}\, 2^{d(j-\mu )\tau _1 p_1\frac{q_2}{p_2}}, \end{aligned}$$

and when $\mu =\nu _0$

$$\begin{aligned} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} =\left( \gamma _j^{p_2}\, k_{\nu _0-j}\right) ^{\frac{q_2}{p_2}}\ge C\, \gamma _j^{q_2} \,2^{d(j-\nu _0)\tau _1 p_1\frac{q_2}{p_2}}, \end{aligned}$$

for some constant C independent of $\gamma$. Then,

$$\begin{aligned} C\, \gamma _j^{q_2}\, 2^{\nu _0 d \frac{p_1 \tau _1}{p_2}q_2} \le \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} 2^{-j d\frac{p_1 \tau _1}{p_2}q_2}, \end{aligned}$$

which yields

$$\begin{aligned}&\Vert \{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _{q_2}\Vert \\&\quad \le C \left\{ \sum _{j=0}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \, 2^{\nu _0 d\frac{p_1 \tau _1}{p_2}q_2} \sum _{i=1}^{2^d-1} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \right\} ^\frac{1}{q_2}\\&\quad \sim 2^{\nu _0 d\frac{p_1 \tau _1}{p_2}} \left\{ \sum _{j=\max \{0,j_P\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \right\} ^\frac{1}{q_2}\\&\quad \le \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \lesssim \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert <\infty , \end{aligned}$$

using (4.33) in the last step and [47, Lemma 3.3] in the second, since $\tau _2<\frac{1}{p_2}$ here. This contradicts our assumption on the sequence $\{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0}$, and completes the proof in this case. $\square$

Before we turn our interest to Triebel-Lizorkin-type spaces, we shall discuss some special case and compare it with the classical result as recalled in Remark 2.15. We concentrate on the limiting case (4.11) under the assumptions (4.10) again. Let us assume now $\tau _1=\tau _2=:\tau$, i.e., $0<\tau \le \min \{\frac{1}{p_1},\frac{1}{p_2}\}$ with $q_i<\infty$ if $\tau =\frac{1}{p_i}$, $i=1,2$. In that case we find that (4.11) reads as $s_1-s_2 = d \max \{0, \frac{1}{p_1}-\frac{1}{p_2}\}$. Then Theorem 4.9 implies the following.

Corollary 4.11

Let $0< p_i\le \infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0< \tau \le \min \{\frac{1}{p_1}, \frac{1}{p_2}\}$, with $q_i<\infty$ if $\tau =\frac{1}{p_i}=\min \{\frac{1}{p_1},\frac{1}{p_2}\}$, $i=1,2$. Assume that

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau ,\tau ,p_1,p_2) =\max \left\{ 0, \frac{1}{p_1}-\frac{1}{p_2}\right\} . \end{aligned}$$

Then the embedding

$$\begin{aligned} \text {id}_{\tau }: B^{s_1,\tau }_{p_1,q_1}(\Omega )\hookrightarrow B^{s_2,\tau }_{p_2,q_2}(\Omega ) \end{aligned}$$

(4.34)

is continuous if, and only if, either $p_1<p_2$, or $p_1\ge p_2$ with $q_1\le q_2$.

Proof

The sufficiency follows from (4.13) in case of $p_1<p_2$, and from (4.15) for $p_1\ge p_2$. Note that the case (4.14) is not applicable in this situation. The necessity is implied by (4.16) in case of $p_1\ge p_2$, and the last statement in (ii) if $p_1<p_2$. Again, (4.17) is not possible in this context. $\square$

Remark 4.12

Let us explicitly comment on the difference between the above result for $\tau >0$ and the classical one for $\tau =0$ as recalled in Remark 2.15. Only in case of embeddings of spaces with the same smoothness $s_1=s_2$ (and thus $p_1\ge p_2$) we have an influence of the fine parameters $q_i$, that is, $q_1\le q_2$. This is parallel to the classical case $\tau =0$ and could thus be expected. However, what is far more surprising, is the outcome for $s_1>s_2$ and $p_1<p_2$: in contrast to the classical setting for $\tau =0$ we do not have any q-dependence here as long as $\tau >0$ (and small enough, such that we are still in the new Morrey-type situation, unlike in Propositions 4.3 and 4.6). Again this explains the special rôle of the hybrid parameter $\tau$ which influences both smoothness and integrability.

Remark 4.13

Note that Proposition 3.9 can be obtained also as an immediate consequence of Theorem 4.9 and the Propositions 4.6 and 4.3.

  We collect now the counterpart of Theorem 4.9 for the Triebel-Lizorkin-type spaces. When $\tau _i<\frac{1}{p_i}$, $i=1,2$, the result follows immediately from [12, Theorem 5.2] and the coincidence of ${F}_{p,q}^{s,\tau }$ and ${{\mathcal {E}}}^{s}_{u,p,q}$ spaces if $\tau =\frac{1}{p}-\frac{1}{u}$, and it reads as follows.

Corollary 4.14

Let $0< p_1,p_2<\infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0\le \tau _i< \frac{1}{p_i}$, $i=1,2$. Assume that

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$

(i):

The embedding

$$\begin{aligned} \text {id}_{\tau }: {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$

(4.35)

is continuous if one of the following conditions holds:

$$\begin{aligned}&\frac{1}{p_1}-\frac{1}{p_2}> \tau _1-\tau _2 \end{aligned}$$

(4.36)

$$\begin{aligned} \text {or}\qquad&\frac{1}{p_1}-\frac{1}{p_2}\le \tau _1-\tau _2\quad \text {and}\quad q_1\le \min \bigg \{1,\frac{p_1}{p_2}\bigg \}q_2. \end{aligned}$$

(4.37)

(ii):

If there is a continuous embedding $\text {id}_{\tau }$ in (4.35), then the parameters satisfy the condition (4.36), or (4.37) holds with $q_1\le q_2$.

We return to the situation $\tau _1=\tau _2>0$ studied in Corollary 4.11, but now in case of F-spaces.

Corollary 4.15

Let $0< p_i<\infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $i=1,2$, and $0\le \tau < \min \{\frac{1}{p_1}, \frac{1}{p_2}\}$. Assume that

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau ,\tau ,p_1,p_2) =\max \left\{ 0, \frac{1}{p_1}-\frac{1}{p_2}\right\} . \end{aligned}$$

Then the embedding

$$\begin{aligned} \text {id}_{\tau }: F^{s_1,\tau }_{p_1,q_1}(\Omega )\hookrightarrow F^{s_2,\tau }_{p_2,q_2}(\Omega ) \end{aligned}$$

(4.38)

is continuous if, and only if, either $p_1<p_2$, or $p_1\ge p_2$ with $q_1\le q_2$.

Proof

This is an immediate consequence of Corollary 4.14 when $\tau >0$ and of the classical situation when $\tau =0$, cf. Remark 4.12. $\square$

Remark 4.16

In contrast to Remark 4.12 concerning Besov-type spaces, we thus obtain the natural counterpart of the well-known classical situation ($\tau =0$) to the situation $\tau >0$, recall Remark 2.15.

We study now some more possible situations regarding Triebel-Lizorkin-type spaces.

Corollary 4.17

Let $0< p_1,p_2<\infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0\le \tau _i\le \frac{1}{p_i}$, with $q_i<\infty$ if $\tau _i=\frac{1}{p_i}$ $i=1,2$. Assume that

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$

(i):

The embedding (4.35) is continuous if one of the following conditions holds:

$$\begin{aligned}&\tau _1=\frac{1}{p_1},\quad {\tau _2 \le \frac{1}{p_2}}\quad \text {and}\quad q_1\le q_2, \end{aligned}$$

(4.39)

$$\begin{aligned} \text {or}\qquad&\tau _1<\frac{1}{p_1}\quad \text {and}\quad \tau _2=\frac{1}{p_2}. \end{aligned}$$

(4.40)

(ii):

If the embedding (4.35) is continuous and $\tau _1=\frac{1}{p_1}$ and $\tau _2=\frac{1}{p_2}$, then $q_1\le q_2$. Moreover, if $\tau _1=\frac{1}{p_1}$ and $\tau _2<\frac{1}{p_2}$, then the continuity of the embedding (4.35) implies $q_1 \le \max \{p_2,q_2\}$.

Proof

The case $\tau _1 = \frac{1}{p_1}$ and $\tau _2 = \frac{1}{p_2}$ can be reduced to Theorem 4.9 due to (2.17). So we are left with the cases $\tau _1=\frac{1}{p_1}$ and $\tau _2<\frac{1}{p_2}$ in (4.39), and $\tau _1<\frac{1}{p_1}$ and $\tau _2=\frac{1}{p_2}$ in (4.40). In both cases we can use the coincidence (2.17). To prove that the condition (4.40) is sufficient we first take a sufficiently small number $q_3$ such that $\tau _1<\frac{1}{q_3}$. We consider the following factorisation

$$\begin{aligned} F^{s_1,\tau _1}_{p_1,q_1}(\Omega )\hookrightarrow B^{s_1,\tau _1}_{p_1,\infty }(\Omega ) \hookrightarrow B^{s_2,1/q_3}_{q_3,q_3}(\Omega ) = F^{s_2,1/p_2}_{p_2,q_3}(\Omega ) \end{aligned}$$

(4.41)

since

$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\frac{1}{p_2}, p_1,p_2)= \frac{1}{p_1}-\tau _1= \gamma (\tau _1,\frac{1}{q_3}, p_1,q_3) > 0. \end{aligned}$$

Please note that the continuity of the second embedding follows from Theorem 4.9 since the condition (4.13) is satisfied. By elementary embeddings the statement holds for any $q_2\ge q_3$.

Regarding the case $\tau _1 = \frac{1}{p_1}$ and $\tau _2< \frac{1}{p_2}$, we have now $\gamma (\tau _1,\tau _2,p_1,p_2)=0$, so $s_1=s_2$. Then

$$\begin{aligned} F^{s_1,1/p_1}_{p_1,q_1}(\Omega ) = B^{s_1,1/q_1}_{q_1,q_1}(\Omega )\hookrightarrow B^{s_2,\frac{1}{q_2}}_{q_2,q_2}(\Omega )= F^{s_2,\frac{1}{p_2}}_{p_2,q_2}(\Omega ) \hookrightarrow F^{s_2,\tau _2}_{p_2,q_2}(\Omega ), \end{aligned}$$

(4.42)

where we made use of Proposition 3.9 in the last embedding. Moreover,

$$\begin{aligned} \gamma \left( \frac{1}{q_1}, \frac{1}{q_2}, q_1,q_2\right) = 0, \end{aligned}$$

so the statement follows once more from (2.17) and Theorem 4.9.

Now we come to the necessity part of our result. The necessity of the condition $q_1\le q_2$ in the case of $\tau _1=\frac{1}{p_1}$ and $\tau _2=\frac{1}{p_2}$ follows from the second part of Theorem 4.9. So we are left with the case $\tau _1 = \frac{1}{p_1}$ and $\tau _2< \frac{1}{p_2}$. Here we can use the following factorisation

$$\begin{aligned} B^{s_1,1/q_1}_{q_1,q_1}(\Omega )= F^{s_1,1/p_1}_{p_1,q_1}(\Omega ) \hookrightarrow F^{s_2,\tau _2}_{p_2,q_2}(\Omega ) \hookrightarrow B^{s_2,\tau _2}_{p_2,\max \{p_2,q_2\}}(\Omega ). \end{aligned}$$

(4.43)

Now Theorem 4.9 implies $q_1\le \max \{p_2,q_2\}$ since

$$\begin{aligned} \gamma \left( \frac{1}{p_1}, \tau _2, p_1,p_2\right) = \gamma \left( \frac{1}{q_1},\tau _2, q_1,p_2\right) =0. \end{aligned}$$

$\square$

Remark 4.18

We return to the special situation of (4.7), i.e., to $\textrm{bmo}(\Omega )$ as a source or target space of $\text {id}_\tau$. In continuation of Remarks 4.5 and 4.8 we now concentrate on the situation covered by Theorem 4.9 and Corollary 4.17, that is, we assume in this remark that $\tau \le \frac{1}{p}$ with $q<\infty$ if $\tau =\frac{1}{p}$ for the target space ${A}_{p,q}^{s,\tau }(\Omega )$ in (4.44) and the source space ${A}_{p,q}^{s,\tau }(\Omega )$ in (4.45). First we deal with

$$\begin{aligned} \text {id}_\tau : \textrm{bmo}(\Omega )\hookrightarrow {A}_{p,q}^{s,\tau }(\Omega ) \end{aligned}$$

(4.44)

such that the limiting situation reads as $s = 0$ in that case. Then Theorem 4.9 and Corollary 4.17 imply that $\text {id}_\tau$ is continuous if

$$\begin{aligned} s=0, \quad \tau \le \frac{1}{p}, \quad \text {and}\quad {\left\{ \begin{array}{ll} q\ge \max \{p,2\}, &{} A=B, \\ q\ge 2, &{} A=F. \end{array}\right. } \end{aligned}$$

Regarding the necessity, when $A=B$, the condition $q\ge 2$ is necessary for the continuity of $\text {id}_\tau$. In addition, when $A=F$, then $q\ge 2$ is also necessary when $\tau =\frac{1}{p}$, while for $\tau <\frac{1}{p}$ the condition $\max \{p,q\}\ge 2$ is necessary. Note that one can also use (4.7) and Proposition 3.9 for the sufficiency argument. In the second setting,

$$\begin{aligned} \text {id}_\tau : {A}_{p,q}^{s,\tau }(\Omega )\hookrightarrow \textrm{bmo}(\Omega ), \end{aligned}$$

(4.45)

the limiting case means $s=d(\frac{1}{p}-\tau )$. Then for the continuity of $\text {id}_\tau$ it is sufficient that

$$\begin{aligned} \text {either}\quad \tau <\frac{1}{p}, \qquad \text {or}\qquad \tau =\frac{1}{p}\quad \text {and}\quad {\left\{ \begin{array}{ll} q\le \min \{p,2\}, &{} A=B, \\ q\le 2, &{} A=F, \end{array}\right. } \end{aligned}$$

where for $\tau =\frac{1}{p}$ the condition $q\le 2$ is also necessary for the continuity of $\text {id}_\tau$.

Remark 4.19

Note that one can formulate counterparts of our above embedding results for spaces of type ${A}_{p,q}^{s,\tau }(\Omega )$ in terms of the hybrid spaces $L^rA^s_{p,q}(\Omega )$ as introduced in Remark 2.9 using the coincidence (2.19). We define the spaces $L^rA^s_{p,q}(\Omega )$ by restriction, parallel to the approach in Definition 2.12. In [8, Remark 3.4] we have explicated the compactness condition for the embedding

$$\begin{aligned} \text {id}_L: L^{r_1}{A}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow L^{r_2}{A}_{p_2,q_2}^{s_2}(\Omega ), \end{aligned}$$

which in the special case $r_1=r_2=r$ reads as

$$\begin{aligned} s_1-s_2 > {\left\{ \begin{array}{ll} 0, &{} r\ge 0, \\ r\, \min \{\frac{p_1}{p_2}-1,0\}, &{} r<0,\end{array}\right. } \end{aligned}$$

where $0<p_i<\infty$, $0<q_i\le \infty$, $s_i\in {{\mathbb {R}}}$, $i=1,2$, and $-d \min \{\frac{1}{p_1}, \frac{1}{p_2}\} \le r<\infty$ is always assumed. For convenience we only discuss this special setting $r_1=r_2=r$ and $A=B$ here, but the other cases can be done in a parallel way.

So the limiting case for the continuity of the embedding is just

$$\begin{aligned} s_1-s_2 = {\left\{ \begin{array}{ll} 0, &{} r\ge 0, \\ r\, \min \{\frac{p_1}{p_2}-1,0\}, &{} r<0.\end{array}\right. } \end{aligned}$$

(4.46)

Corollary 4.20

Let $0< p_i<\infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $i=1,2$, and $-d \min \left\{ \frac{1}{p_1}, \frac{1}{p_2}\right\}< r< \infty$. Let

$$\begin{aligned} \text {id}_L: L^r{B}_{p_1,q_1}^{s_1}(\Omega )\hookrightarrow L^r{B}_{p_2,q_2}^{s_2}(\Omega ). \end{aligned}$$

(4.47)

(i):

If $r>0$, then $\text {id}_L$ is continuous if, and only if, $s_1\ge s_2$.

(ii):

Let $r=0$.

(ii $_a$):

If $q_2=\infty$, then $\text {id}_L$ is continuous if, and only if, $s_1\ge s_2$.

(ii $_b$):

If $q_2<\infty$ and $q_1=\infty$, then $\text {id}_L$ is continuous if, and only if, $s_1>s_2$.

(ii $_c$):

If $q_i<\infty$, $i=1,2$, then $\text {id}_L$ is continuous if $s_1>s_2$ or $s_1=s_2$ and $q_1\le \min \{1, \frac{p_1}{p_2}\}q_2$. Conversely, if $\text {id}_L$ is continuous, then either $s_1>s_2$ or $s_1=s_2$ and $q_1\le q_2$.

(iii):

Let $r<0$.

(iii $_a$):

Assume that $p_1\ge p_2$. Then $\text {id}_L$ is continuous if, and only if, $s_1>s_2$ or $s_1=s_2$ and $q_1\le q_2$.

(iii $_b$):

Assume $p_1< p_2$. Then $\text {id}_L$ is continuous if $s_1-s_2>r(\frac{p_1}{p_2}-1)$, or $s_1-s_2=r(\frac{p_1}{p_2}-1)$ and $q_1\le \frac{p_1}{p_2} q_2$. Conversely, the continuity of $\text {id}_L$ implies $s_1-s_2>r(\frac{p_1}{p_2}-1)$, or $s_1-s_2=r(\frac{p_1}{p_2}-1)$ and $q_1\le q_2$.

Proof

In view of Remark 4.19 we only need to consider the limiting case (4.46), the rest is covered by Theorem 4.1 and the coincidence (2.19), extended to spaces on domains. Then (i) and (ii$_a$) are covered by Proposition 4.3 together with $r=d(\tau _i-\frac{1}{p_i})$, $i=1,2$. Likewise (ii$_b$) is a consequence of Proposition 4.6 since there is no continuous embedding in the limiting case $s_1=s_2$. Part (ii$_c$) follows from Theorems 4.9 and 4.1, as well as part (iii). $\square$

We finish our paper by collecting some immediate extensions of Theorems 2.10 and 2.11 regarding the embeddings on ${{{\mathbb {R}}}^d}$. The first result improves part (b) of Theorem 2.10 (iii), and it follows from Theorem 3.6 and Theorem 4.9.

Corollary 4.21

Let $0< p_1,p_2\le \infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0\le \tau _i\le \frac{1}{p_i}$, with $q_i<\infty$ if $\tau _i=\frac{1}{p_i}$, $i=1,2$. If

$$\begin{aligned} \text {id}_{\tau }^{{{{\mathbb {R}}}^d}}: {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$

(4.48)

is continuous and $\frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+ \frac{p_1}{p_2}\tau _1>0$, then

$$\begin{aligned} q_1\le q_2. \end{aligned}$$

Proof

By Theorem 3.6, we have

$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \xrightarrow {\textrm{ext}} {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \xrightarrow {\text {id}_\tau ^{{{{\mathbb {R}}}^d}}} {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \xrightarrow {\textrm{re}} {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega ), \end{aligned}$$

i.e., $\text {id}_\tau ^\Omega = \textrm{re}\circ \text {id}_\tau ^{{{{\mathbb {R}}}^d}} \circ \textrm{ext}$. Hence, the continuity of $\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$ implies the continuity of $\text {id}_\tau ^\Omega$, which, in turn, by Theorem 4.9 implies $q_1\le q_2$. $\square$

Next we show that, in case of $\tau _1=\tau _2$, the embedding $\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$ in (4.48) holds under weaker assumptions on the parameters than the ones stated in Theorem 2.10 (iii)-(a). Namely, in this case, we do not need any condition on the parameters $q_1, q_2$.

Corollary 4.22

Let $0< p_1,p_2\le \infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $0\le \tau _i\le \frac{1}{p_i}$, with $q_i<\infty$ if $\tau _i=\frac{1}{p_i}$, $i=1,2$. If

$$\begin{aligned} \frac{s_1-s_2}{d}= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+ \tau _2>0 \quad \text{ and } \quad \tau _1=\tau _2, \end{aligned}$$

then the embedding $\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$ in (4.48) is continuous.

Proof

This result can be proved in the same way as its counterpart for embeddings on domains, cf. Substep 1.2 of the proof of Theorem 4.9. Therefore, we omit the argument here. $\square$

Lastly, we turn to the Triebel-Lizorkin-type spaces and state a result which gives us sufficient and necessary conditions for the continuity of the embedding on ${{{\mathbb {R}}}^d}$, when $\tau _2$ is large and $\tau _1$ is small. Specifically, we assume that

$$\begin{aligned} \tau _2 \ge \frac{1}{p_2}\,\,\,\, \text{ with } \,\,\,\, q_2=\infty \,\,\,\, \text{ if } \,\,\,\, \tau _2=\frac{1}{p_2} \qquad \text{ and } \qquad \tau _1\le \frac{1}{p_1} \,\,\,\, \text{ with } \,\,\,\, q_1<\infty \,\,\,\, \text{ if } \,\,\,\, \tau _1=\frac{1}{p_1},\nonumber \\ \end{aligned}$$

(4.49)

as this case that was not considered in Theorem 2.11 (i).

Corollary 4.23

Let $0< p_1,p_2<\infty$, $s_i\in {{\mathbb {R}}}$, $0<q_i\le \infty$, $\tau _i\ge 0$, $i=1,2$. Assume that condition (4.49) holds. Then the embedding

$$\begin{aligned} \text {id}_{\tau }^{{{{\mathbb {R}}}^d}}: {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$

(4.50)

holds if, and only if, $\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$.

Proof

The sufficiency part follows from the fact that

$$\begin{aligned} {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d})\hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty , \infty }({{{\mathbb {R}}}^d}) \hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty , \infty }({{{\mathbb {R}}}^d}) = {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}), \end{aligned}$$

due to (2.6) and Proposition 2.3, and the corresponding result for the classical Besov spaces.

For the necessity, we use a similar argument as in the proof of Corollary 4.21, via the extension operator from Theorem 3.6. In this case, Theorem 4.1 (i) and Proposition 4.3 will give us the complete result. $\square$