1 Introduction

Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain, that is, \(\Omega \) is of the form \(\{(x',x_n):x_n>\rho (x')\}\) where \(\rho :\mathbb {R}^{n-1}\rightarrow \mathbb {R}\) is a Lipschitz function such that \(\Vert \nabla \rho \Vert _{L^\infty }<\infty \). (See also [14, Definition 1.103].)

In [9], based on the construction of his extension operator, Rychkov gave a Littlewood-Paley type intrinsic characterization of the Triebel-Lizorkin spaces on \(\Omega \): for \(0<p<\infty \), \(0<q\le \infty \) and \(s\in \mathbb {R}\), \(\mathscr {F}_{pq}^s(\Omega )\) has the following equivalent (quasi-)norm (see [9, Theorem 3.2]):

$$\begin{aligned} f\mapsto \Vert (2^{js}\phi _j*f)_{j=0}^\infty \Vert _{\ell ^q(\mathbb {Z}_{\ge 0};L^p(\Omega ))}=\bigg (\int _\Omega \Big (\sum _{j=0}^\infty 2^{jsq}|\phi _j*f(x)|^q\Big )^{p/q}dx\bigg )^{1/p}. \end{aligned}$$
(1)

We take obvious modification for \(q=\infty \). Here \((\phi _j)_{j=0}^\infty \) is a carefully chosen family of Schwartz functions such that the convolution \(\phi _j*f\) is defined on \(\Omega \), see Definition 4.

In [12, version 3, Proposition 6.6], we used Rychkov’s construction to prove that \(\Vert f\Vert _{\mathscr {F}_{pq}^s(\Omega )}\) have equivalent (quasi-)norms via their derivatives. More precisely, let \(m\ge 1\), for every \(0<p<\infty \), \(0<q\le \infty \) and \(s\in \mathbb {R}\) there is a \(C=C(\Omega ,p,q,s,m)>0\) such that

$$\begin{aligned} C^{-1}\Vert f\Vert _{\mathscr {F}_{pq}^s(\Omega )}\le \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {F}_{pq}^{s-m}(\Omega )}\le C\Vert f\Vert _{\mathscr {F}_{pq}^s(\Omega )},\quad \forall f\in \mathscr {F}_{pq}^s(\Omega ). \end{aligned}$$
(2)

Both (1) and (2) miss the endpoint: do we have the analogy of (1) and (2) for \(p=\infty \)? In this paper, we give the positive answers to both cases, by using the recently developed Triebel-Lizorkin-type spaces \(\mathscr {F}_{pq}^{s\tau }\): we have the coincidences \(\mathscr {F}_{\infty q}^s=\mathscr {F}_{pq}^{s,\frac{1}{p}}=\mathscr {B}_{qq}^{s,\frac{1}{q}}\) for \(0<p<\infty \) (see (9)).

To make the results more general, we include the discussions of Besov-type spaces \(\mathscr {B}_{pq}^{s\tau }\) and the Besov-Morrey spaces \(\mathscr {N}_{pq}^{s\tau }\), see Definition 6.

We denote by \(\mathcal {Q}\) the set of dyadic cubes in \(\mathbb {R}^n\), that is

$$\begin{aligned} \mathcal {Q}:=\{Q_{J,v}:J\in \mathbb {Z},v\in \mathbb {Z}^n\},\quad \text {where}\quad Q_{J,v}:=2^{-J}v+(0,2^{-J})^n. \end{aligned}$$
(3)

Our result for (1) is the following:

Theorem 1

(Littlewood-Paley type characterizations) Let \(\Omega =\{(x',x_n):x_n>\rho (x')\}\subset \mathbb {R}^n\) be a special Lipschitz domain and let \((\phi _j)_{j=0}^\infty \) be a Littlewood-Paley family associated with \(\Omega \) (see Definition 4). Then for \(0<p,q\le \infty \), \(s\in \mathbb {R}\) and \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases), we have the following equivalent (quasi-)norms:

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau }\Vert (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f))_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }\\&= \sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Big (\sum _{j=\max (0,J)}^\infty 2^{jsq}\Vert \phi _j*f\Vert _{L^p(Q_{J,v}\cap \Omega )}^q\Big )^\frac{1}{q}; \\ \Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau }\Vert (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f))_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}\\&= \sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Big (\int _{Q_{J,v}\cap \Omega }\Big (\sum _{j=\max (0,J)}^\infty 2^{jsq}|\phi _j*f(x)|^q\Big )^\frac{p}{q}dx\Big )^\frac{1}{p}; \\ \Vert f\Vert _{\mathscr {N}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau }\Vert (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f))_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }\\&=\Big (\sum _{j=0}^\infty \sup _{Q_{J,v}\in \mathcal {Q}} 2^{(js+nJ\tau )q}\Vert \phi _j*f\Vert _{L^p(Q_{J,v}\cap \Omega )}^q\Big )^\frac{1}{q}. \end{aligned}$$

(See Definition 5 for \(\ell ^qL^p_\tau \), \(L^p_\tau \ell ^q\) and \(\ell ^qM^p_\tau \).) In particular for \(0<q\le \infty \) and \(s\in \mathbb {R}\),

$$\begin{aligned} \Vert f\Vert _{\mathscr {F}_{\infty q}^s(\Omega )}\approx _{\phi ,q,s}\sup _{J\in \mathbb {Z},v\in \mathbb {Z}^n}2^{J\frac{n}{q}}\int _{Q_{J,v}\cap \Omega }\Big (\sum _{j=\max (0,J)}^\infty 2^{jqs}|\phi _j*f(x)|^qdx\Big )^\frac{1}{q}. \end{aligned}$$

One can also get some characterizations on bounded Lipschitz domain, whose expressions are less elegant however. See Remark 24.

Similar to [9, Theorem 2.3], we also have the corresponding characterizations using Peetre maximal functions, see Proposition 21 and Corollary 23.

Our result for (2) is the following:

Theorem 2

(Equivalent norm characterizations via derivatives) Let \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\), \(0<p,q\le \infty \), \(s\in \mathbb {R}\) and \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases). Let \(\Omega \subset \mathbb {R}^n\) be either a special Lipschitz domain or a bounded Lipschitz domain. Then for any positive integer m, the space \(\mathscr {A}_{pq}^{s\tau }(\Omega )\) has the following equivalent (quasi-)norm:

$$\begin{aligned} \Vert f\Vert _{\mathscr {A}_{p,q}^{s,\tau }(\Omega )}\approx _{p,q,s,m,\tau ,\Omega }\sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )}. \end{aligned}$$
(4)

In particular \(\Vert f\Vert _{\mathscr {F}_{\infty ,q}^{s}(\Omega )}\approx _{q,s,m,\Omega }\sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {F}_{\infty ,q}^{s-m}(\Omega )}\) for all \(0<q\le \infty \) and \(s\in \mathbb {R}\).

The Besov-Morrey case \(\mathscr {A}=\mathscr {N}\) of Theorem 2 was stated in [25, Proposition 4.15]. However, the key step in their proof requires [15, (4.70)] (see [25, Remark 4.14]), which cannot be achieved.

Remark 3

In the proof of [15, Proposition 4.21], Triebel claimed the following statement:

$$\begin{aligned}{} & {} \Vert f\Vert _{\mathscr {A}_{pq}^s(\Omega )}\approx \Vert Ef\Vert _{\mathscr {A}_{pq}^s(\mathbb {R}^n)}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha E f\Vert _{\mathscr {A}_{pq}^s(\mathbb {R}^n)}\nonumber \\{} & {} \quad =\sum _{|\alpha |\le m}\Vert E\partial ^\alpha f\Vert _{\mathscr {A}_{pq}^s(\mathbb {R}^n)}\lesssim \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {A}_{pq}^s(\Omega )}. \end{aligned}$$
(5)

Here \(E=E_\Omega \) is an extension operator which is bounded on \(\mathscr {A}_{pq}^s(\Omega )\rightarrow \mathscr {A}_{pq}^s(\mathbb {R}^n)\) and \(\mathscr {A}_{pq}^{s-m}(\Omega )\rightarrow \mathscr {A}_{pq}^{s-m}(\mathbb {R}^n)\).

However, the commutativity \(\partial ^\alpha \circ E=E\circ \partial ^\alpha \) in (5) (see [15, (4.70)]) cannot be achieved. In [12, Remark 1.6] we borrowed some facts from several complex variables to show that \(\partial ^\alpha \circ E=E\circ \partial ^\alpha \) can never be true: if it is true (even locally) then \(\overline{\partial }\)-equation for \(\Omega \) can gain 1 derivative. To prove Theorem 2 (also to fix the proof of [25, Proposition 4.15]), simply using the boundedness of \(E_\Omega \) is not enough.

By observing (5) more carefully, the argument still works if \(\partial ^\alpha \circ E=E^\alpha \circ \partial ^\alpha \) hold for some extension operators \(E^\alpha {:}\,\mathscr {A}^{s-m}_{pq}(\Omega )\rightarrow \mathscr {A}^{s-m}_{pq}(\Omega )\). This can be done if E is the standard half space extension.Footnote 1 Using the operators \(E^\alpha \) Triebel proved the equivalent norms via derivatives for \(\mathbb {R}^n_+\) and for smooth domains, see [16, Section 3.3.5].

In our case E is Rychkov’s extension operator (see (31)). Even on special Lipschitz domain, it is not known to the author whether \(\partial ^\alpha \circ E=E^\alpha \circ \partial ^\alpha \) can be achieved (which in general should have the form (27)). Nevertheless, a weaker form \(\partial ^\alpha \circ E=\sum _\beta E^{\alpha ,\beta }\circ \partial ^\beta \) is enough to fix (5). In the proof we introduce \(E^{\alpha ,\beta }\) in (41) and get the proof using (42).

See also [12, Section 2.2 and Remark 6.5].

2 Function Spaces and Notations

Let \(U\subseteq \mathbb {R}^n\) be an open set, we define \(\mathscr {S}'(U)\) to be the space of restricted tempered distributions: \(\mathscr {S}'(U):=\{\tilde{f}|_U:\tilde{f}\in \mathscr {S}'(\mathbb {R}^n)\}\). See also [9, Proposition 3.1].

We use the notation \(A \lesssim B\) to mean that \(A \le CB\) where C is a constant independent of A, B. We use \(A \approx B\) for “\(A \lesssim B\) and \(B \lesssim A\)”. And we use \(A\lesssim _xB\) to emphasize that the constant depends on the quantity x.

When p or \(q<1\), we use “norms” (for \(\mathscr {A}_{pq}^{s\tau }\) etc.) as the abbreviation to the usual “quasi-norms”.

In the paper we use the following Littlewood–Paley family, whose elements do not have compact supports in the Fourier side. It is crucially useful in the construction of Rychkov’s extension operator.

Definition 4

Let \(\Omega =\{x_n>\rho (x')\}\) be a special Lipschitz domain, a Littlewood-Paley family associated with \(\Omega \) is a sequence \(\phi =(\phi _j)_{j=0}^\infty \subset \mathscr {S}(\mathbb {R}^n)\) of Schwartz functions that satisfies the following:

  1. (P.a)

    Moment condition: \(\int x^\alpha \phi _1(x)dx=0\) for all multi-indices \(\alpha \in \mathbb {Z}_{\ge 0}^n\).

  2. (P.b)

    Scaling condition: \(\phi _j(x)=2^{(j-1)n}\phi _1(2^{j-1}x)\) for all \(j\ge 2\).

  3. (P.c)

    Approximate identity: \(\sum _{j=0}^\infty \phi _j=\delta _0\) is the Direc delta measure.

  4. (P.d)

    Support condition: \({\text {supp}}\phi _j\subset \{(x',x_n): x_n<-\Vert \nabla \rho \Vert _{L^\infty }\cdot |x'|\}\) for all \(j\ge 0\).

In the paper we use the sequence spaces \(\ell ^qL^p_\tau \), \(L^p_\tau \ell ^q\), \(\ell ^qM^p_\tau \) given by the following:

Definition 5

Let \(0<p,q\le \infty \) and \(\tau \ge 0\). We denote by \(\ell ^qL^p_\tau (\mathbb {R}^n)\) and \(L^p_\tau \ell ^q(\mathbb {R}^n)\) the spaces of vector valued measurable functions \((f_j)_{j=0}^\infty \subset L^p_\textrm{loc}(\mathbb {R}^n)\) such that the following (quasi-)norms are finite respectively:

$$\begin{aligned} \Vert (f_j)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }&:=\sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Vert (f_j)_{j=\max (0,J)}^\infty \Vert _{\ell ^q(L^p(Q_{J,v}))}\\&=\sup _{J\in \mathbb {Z},v\in \mathbb {Z}^n}2^{nJ\tau }\Big (\sum _{j=\max (0,J)}^\infty \Vert f_j\Vert _{L^p(Q_{J,v})}^q\Big )^\frac{1}{q}; \\ \Vert (f_j)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}&:=\sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Vert (f_j)_{j=\max (0,J)}^\infty \Vert _{L^p(Q_{J,v};\ell ^q)}\\&=\sup _{J\in \mathbb {Z},v\in \mathbb {Z}^n}2^{nJ\tau }\bigg (\int _{Q_{J,v}}\Big (\sum _{j=\max (0,J)}^\infty |f_j(x)|^q\Big )^\frac{p}{q}dx\bigg )^\frac{1}{p}. \end{aligned}$$

We define the Morrey space.Footnote 2\(M^p_\tau (\mathbb {R}^n)\) to be the set of all \(f\in L^p_\textrm{loc}(\mathbb {R}^n)\) whose (quasi-)norm below is finite:

$$\begin{aligned} \Vert f\Vert _{M^p_\tau }:=\textstyle \sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Vert f\Vert _{L^p(Q_{J,v})}. \end{aligned}$$

We define \(\ell ^qM^p_\tau (\mathbb {R}^n):=\ell ^q(\mathbb {Z}_{\ge 0};M^p_\tau (\mathbb {R}^n))\) with \(\Vert (f_j)_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }:=\big (\sum _{j=0}^\infty \Vert f_j\Vert _{M^p_\tau (\mathbb {R}^n)}^q\big )^\frac{1}{q}\).

Our Besov-type spaces \(\mathscr {B}_{pq}^{s\tau }\), Triebel-Lizorkin-type spaces \(\mathscr {F}_{pq}^{s\tau }\) and Besov-Morrey spaces \(\mathscr {N}_{pq}^{s\tau }\) are given by the following:

Definition 6

Let \(\lambda =(\lambda _j)_{j=0}^\infty \) be a sequence of Schwartz functions satisfying:

  1. (P.a’)

    The Fourier transform \(\hat{\lambda }_0(\xi )=\int _{\mathbb {R}^n}\lambda _0(x)2^{-2\pi ix\xi }dx\) satisfies \({\text {supp}}\hat{\lambda }_0\subset \{|\xi |<2\}\) and \(\hat{\lambda }_0|_{\{|\xi |<1\}}\equiv 1\).

  2. (P.b’)

    \(\lambda _j(x)=2^{jn}\lambda _0(2^jx)-2^{(j-1)n}\lambda _0(2^{j-1}x)\) for \(j\ge 1\).

Let \(0<p,q\le \infty \), \(s\in \mathbb {R}\) and \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases). We define the Besov-type Morrey space \(\mathscr {B}_{pq}^{s\tau }(\mathbb {R}^n)\), the Triebel-Lizorkin-type Morrey space \(\mathscr {F}_{pq}^{s\tau }(\mathbb {R}^n)\) and the Besov-Morrey space \(\mathscr {N}_{pq}^{s\tau }(\mathbb {R}^n)\), to be the sets of all tempered distributions \(f\in \mathscr {S}'(\mathbb {R}^n)\) such that the following norms are finite, respectively:

$$\begin{aligned} {} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\mathbb {R}^n)}:=&\, \Vert (2^{js}\lambda _j*f)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau };\nonumber \\ {}\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\mathbb {R}^n)}:=\,&\Vert (2^{js}\lambda _j*f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q};\nonumber \\ {}\Vert f\Vert _{\mathscr {N}_{pq}^{s\tau }(\mathbb {R}^n)}:=\,&\Vert (2^{js}\lambda _j*f)_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }. \end{aligned}$$
(6)

Let \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\). For an (arbitrary) open subset \(U\subseteq \mathbb {R}^n\), we define \(\mathscr {A}_{pq}^{s\tau }(U):=\{\tilde{f}|_U:\tilde{f}\in \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\}\) (\(p<\infty \) for \(\mathscr {F}\)-cases) with the norm

$$\begin{aligned} \textstyle \Vert f\Vert _{\mathscr {A}_{pq}^{s\tau }(U)}:=\inf \{\Vert \tilde{f}\Vert _{\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)}:\tilde{f}\in \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n), \tilde{f}|_U=f\}. \end{aligned}$$
(7)

The definitions of the spaces \(\mathscr {A}_{pq}^{s\tau }(U)\) do not depend on the choice of \((\lambda _j)_{j=0}^\infty \) which satisfies (P.a’) and (P.b’). See [24, Page 39, Corollary 2.1] and [20, Theorem 2.8].

Remark 7

We remark some known results and different notations for these spaces in \(\mathbb {R}^n\) from the literature:

  1. (i)

    Clearly \(\mathscr {B}_{pq}^s(\mathbb {R}^n)=\mathscr {B}_{pq}^{s0}(\mathbb {R}^n)=\mathscr {N}_{pq}^{s0}(\mathbb {R}^n)\) and \(\mathscr {F}_{pq}^{s0}(\mathbb {R}^n)=\mathscr {F}_{pq}^s(\mathbb {R}^n)\) (provided \(p<\infty \)).

  2. (ii)

    In applications only \(0\le \tau \le \frac{1}{p}\) is interesting: by [27, Theorem 2] and [10, Lemma 3.4],

    $$\begin{aligned}{} & {} \mathscr {B}_{p,q}^{s,\tau }(\mathbb {R}^n)=\mathscr {F}_{p,q}^{s,\tau }(\mathbb {R}^n)=\mathscr {B}_{\infty ,\infty }^{s+n(\tau -\frac{1}{p})}(\mathbb {R}^n),\nonumber \\{} & {} \quad \mathscr {N}_{p,q}^{s,\tau }(\mathbb {R}^n)=\{0\},\quad \forall \,0<p,q\le \infty ,\ s\in \mathbb {R},\ \tau >\tfrac{1}{p}. \end{aligned}$$
    (8)
  3. (iii)

    For the case \(\tau =1/p\), by [27, Theorem 2] and [10, Remark 11(ii)],

    $$\begin{aligned}{} & {} \mathscr {B}_{p,\infty }^{s,\frac{1}{p}}(\mathbb {R}^n)=\mathscr {F}_{p,\infty }^{s,\frac{1}{p}}(\mathbb {R}^n)=\mathscr {B}_{\infty ,\infty }^{s}(\mathbb {R}^n),\\{} & {} \quad \mathscr {N}_{p,q}^{s,\frac{1}{p}}(\mathbb {R}^n)=\mathscr {B}_{\infty ,q}^s(\mathbb {R}^n),\quad \forall \,0<p,q\le \infty ,\ s\in \mathbb {R}. \end{aligned}$$
  4. (iv)

    Although \(\mathscr {F}_{pq}^{s\tau }\)-spaces are only defined for \(p<\infty \), we have a description for \(\mathscr {F}_{\infty q}^s\)-spaces as the following (see [24, Page 41, Proposition 2.4(iii)] and [2, Section 5]):

    $$\begin{aligned} \mathscr {F}_{\infty q}^s(\mathbb {R}^n)=\mathscr {F}_{p,q}^{s,\frac{1}{p}}(\mathbb {R}^n)=\mathscr {B}_{q,q}^{s,\frac{1}{q}}(\mathbb {R}^n),\quad \forall \,0<p<\infty ,\ 0<q\le \infty ,\ s\in \mathbb {R}. \end{aligned}$$
    (9)
  5. (v)

    Our notation \(\mathscr {N}_{pq}^{s\tau }\) corresponds to the \(\mathcal B_{pq}^{s\tau }\) in [10, Definition 5]. For the classical notationsFootnote 3\(\mathcal {N}_{uqp}^s\) we have correspondence (see [10, Remark 13(iii)] for example):

    $$\begin{aligned} \mathcal {N}_{u,q,p}^s(\mathbb {R}^n)=\mathscr {N}_{p,q}^{s,\frac{1}{p}-\frac{1}{u}}(\mathbb {R}^n),\quad \forall \,0<p\le u\le \infty ,\ 0<q\le \infty ,\ s\in \mathbb {R}. \end{aligned}$$
  6. (vi)

    We do not talk about the Triebel-Lizorkin-Morrey spaces \(\mathcal {E}_{uqp}^s\) in the paper, because they are special cases of the Triebel-Lizorkin-type spaces: we have \(\mathcal {E}_{u,q,p}^s(\mathbb {R}^n)=\mathscr {F}_{p,q}^{s,1/p-1/u}(\mathbb {R}^n)\) for all \(p\in (0,\infty )\), \(q\in (0,\infty ]\), \(u\in [p,\infty ]\) and \(s\in \mathbb {R}\). See [24, Corollary 3.3].

  7. (vii)

    There are also papers that use the notations \(\Lambda ^\varrho \mathscr {A}_{pq}^s\) and \(\Lambda _{\varrho }\mathscr {A}_{pq}^s\) for \(\mathscr {A}\in \{\mathscr {B},\mathscr {F}\}\) and \(-n\le \varrho \le 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases), for example [6, 19]. These spaces describe the same collection to \(\mathscr {A}_{pq}^{s\tau }\) for \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\), see [6, Remarks 2.7 and 2.9] for example.

For more discussions, we refer the reader to [6, 18, 24].

3 Proof of the Theorems

Our proof follows from some results in [9] and [26].

The key ingredient is the Peetre maximal operators introduced in [8].

Definition 8

Let \(N>0\), \(U\subseteq \mathbb {R}^n\) be an open set and let \(\eta =(\eta _j)_{j=0}^\infty \) be a sequence of Schwartz functions. The associated Peetre maximal operators \((\mathcal {P}^{\eta ,N}_{U,j})_{j=0}^\infty \) are given by

$$\begin{aligned} \mathcal {P}^{\eta ,N}_{U,j}f(x):=\sup \limits _{y\in U}\frac{|\eta _j*f(y)|}{(1+2^j|x-y|)^{N}},\quad f\in \mathscr {S}'(\mathbb {R}^n),\quad x\in \mathbb {R}^n,\quad j\ge 0. \end{aligned}$$

Lemma 9

Let \(\phi =(\phi _j)_{j=0}^\infty \) be a Littlewood–Paley family associated with a special Lipschitz domain \(\Omega \) (see Definition 4). Then there is a \(\psi =(\psi _j)_{j=0}^\infty \subset \mathscr {S}'(\mathbb {R}^n)\) satisfying (P.a) and (P.b) such that \((\psi _j*\phi _j)_{j=0}^\infty \) is also associated with \(\Omega \).

Proof

The assumptions \(\phi _j(x)=2^{(j-1)n}\phi _1(2^{j-1}x)\) for \(j\ge 1\) and \(\sum _{j=0}^\infty \phi _j=\delta _0\) imply \(\phi _1(x)=2^n\phi _0(2x)-\phi _0(x)\), i.e. \(\hat{\phi }_1(\xi )=\hat{\phi }_0(\xi /2)-\hat{\phi }_0(\xi )\). We can take \(\psi =(\psi _j)_{j=0}^\infty \) via the Fourier transforms:

$$\begin{aligned}{} & {} \hat{\psi }_0(\xi ):=2\hat{\phi }_0(\xi )-\hat{\phi }_0(\xi )^3;\\{} & {} \hat{\psi }_j(\xi ):=(\hat{\phi }_0(2^{-j}\xi )+\hat{\phi }_0(2^{1-j}\xi ))(2-\hat{\phi }_0(2^{-j}\xi )^2-\hat{\phi }_0(2^{1-j}\xi )^2),\text { for }j\ge 1. \end{aligned}$$

See [9, Proposition 2.1] for details.

Lemma 10

([1, Lemma 2.1]) Let \(\eta =(\eta _j)_{j=0}^\infty \) and \(\theta =(\theta _j)_{j=0}^\infty \subset \mathscr {S}(\mathbb {R}^n)\) both satisfy conditions (P.a) and (P.b). Then for any \(N>0\) there exists a \(C=C(\eta ,\theta ,N)>0\) such that

$$\begin{aligned} \int _{\mathbb {R}^n}|\eta _j*\theta _k(x)|(1+2^k|x|)^Ndx\lesssim _{\eta ,\theta ,N}2^{-N|j-k|},\qquad \forall j,k\ge 0. \end{aligned}$$

Lemma 11

Let \(0<p,q\le \infty \), \(\tau \ge 0\) and \(\delta >n\tau \). There is a \(C=C(n,p,q,\tau ,\delta )>0\) such that for every \((g_j)_{j=0}^\infty \subset L^p_\textrm{loc}(\mathbb {R}^n)\),

$$\begin{aligned} \Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}g_k\Big )_{j=0}^\infty \Big \Vert _{\ell ^qL^p_\tau }&\le C\Vert (g_j)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(10)
$$\begin{aligned} \Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}g_k\Big )_{j=0}^\infty \Big \Vert _{L^p_\tau \ell ^q}&\le C\Vert (g_j)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q},\qquad \text {provided }p<\infty ; \end{aligned}$$
(11)
$$\begin{aligned} \Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}g_k\Big )_{j=0}^\infty \Big \Vert _{\ell ^qM^p_\tau }&\le C\Vert (g_j)_{j=0}^\infty \Vert _{\ell ^q M^p_\tau }. \end{aligned}$$
(12)

Proof

(10) and (11) have been done in [26, Lemma 2.3]. We only prove (12).

Using the case \(\tau =0\) in (10) we have

$$\begin{aligned}{} & {} \textstyle \big \Vert \big (\sum _{k\ge 0}2^{-\delta |j-k|}f_k\big )_{j=0}^\infty \big \Vert _{\ell ^q(L^p)}\\ {}{} & {} \quad \lesssim _{p,q,\delta }\Vert (f_j)_{j=0}^\infty \Vert _{\ell ^q(L^p)},\quad \forall (f_j)_{j=0}^\infty \in \ell ^q(\mathbb {Z}_{\ge 0};L^p(\mathbb {R}^n)). \end{aligned}$$

Note that \(\Vert g_k\Vert _{M^p_\tau }=\Vert \sup _{Q_{J,v}}|2^{nJ\tau }\textbf{1}_{Q_{J,v}}\cdot g_k|\Vert _{L^p(\mathbb {R}^n)}\). By taking \(f_k:=\sup _{Q_{J,v}}|2^{nJ\tau }\textbf{1}_{Q_{J,v}}\cdot g_k|\) above we have

$$\begin{aligned}&\Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}|g_k|\Big )_{j=0}^\infty \Big \Vert _{\ell ^qM^p_\tau }\\&\quad =\Big \Vert \Big (\sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\textbf{1}_{Q_{J,v}}\cdot \sum _{k\ge 0}2^{-\delta |j-k|}|g_k|\Big )_j\Big \Vert _{\ell ^q(L^p)}\\&\quad \le \Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}\sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\textbf{1}_{Q_{J,v}}\cdot |g_k|\Big )_j\Big \Vert _{\ell ^q(L^p)}\\&\quad =\Big \Vert \Big (\sum _{k\ge 0}2^{-\delta |j-k|}f_k\Big )_j\Big \Vert _{\ell ^q(L^p)} \\&\quad \lesssim _{p,q,\delta }\Vert (f_j)_{j=0}^\infty \Vert _{\ell ^q(L^p)}=\Vert (g_j)_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }. \\ \end{aligned}$$

\(\square \)

Lemma 12

Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain, let \(\phi =(\phi _j)_{j=0}^\infty \) be a Littlewood-Paley family associated with \(\Omega \), and let \(\theta =(\theta _j)_{j=0}^\infty \) satisfies conditions (P.a), (P.b) and (P.d). Then for any \(N>0\) and \(\gamma \in (0,\infty ]\) there is a \(C=C(\theta ,\phi ,N)>0\), such that, for every \(f\in \mathscr {S}'(\mathbb R^n)\), \(j\ge 0\) and \(x\in \Omega \),

$$\begin{aligned}&\mathcal {P}^{\theta ,N}_{\Omega ,j}f(x)\le C\bigg (\sum _{k=0}^\infty 2^{-N\gamma |j-k|}\int _{\Omega }\frac{2^{kn}|\phi _k*f(y)|^\gamma dy}{(1+2^k|x-y|)^{N\gamma }}\bigg )^{1/\gamma }. \end{aligned}$$
(13)

Proof

The special case \(\theta =\phi \) of (13) is proved in [9, Proof of Theorem 3.2, Step 1]. Namely, we have

$$\begin{aligned}&\mathcal {P}^{\phi ,N}_{\Omega ,j}f(x)\lesssim _{\phi ,N}\bigg (\sum _{k=0}^\infty 2^{-N\gamma |j-k|}\int _{\Omega }\frac{2^{kn}|\phi _k*f(y)|^\gamma dy}{(1+2^k|x-y|)^{N\gamma }}\bigg )^{1/\gamma }. \end{aligned}$$
(14)

Also see [21, Proof of Theorem 2.6, Step 1] for the argument. Thus it suffices to prove the case \(\gamma =\infty \):

$$\begin{aligned} \mathcal {P}^{\theta ,N}_{\Omega ,j}f(x)\lesssim _{\theta ,\phi ,N}\sup _{k\ge 0}2^{-N|j-k|}\mathcal {P}^{\phi ,N}_{\Omega ,k}f(x),\quad \forall f\in \mathscr {S}'(\mathbb {R}^n),\quad j\ge 0,\quad x\in \Omega . \end{aligned}$$
(15)

Let \(\psi =(\psi _j)_{j=0}^\infty \) satisfies the consequence of Lemma 9, so \(\theta _j*f=\sum _{k=0}^\infty (\theta _j*\psi _k)*(\phi _k*f)\) for \(j\ge 0\). By assumption \(\phi _j,\psi _j,\theta _j\) are supported in \(K=\{x_n<-\Vert \nabla \rho \Vert _{L^\infty }\cdot |x'|\}\) where \(\rho \) is the defining function for \(\Omega =\{x_n>\rho (x')\}\). Using the property \(\Omega -K\subseteq \Omega \), we have

$$\begin{aligned}\textbf{1}_\Omega \cdot (\theta _j*f)&\textstyle =\textbf{1}_\Omega \cdot \sum _{k=0}^\infty (\theta _j*\psi _k)*(\textbf{1}_\Omega \cdot (\phi _k*f)); \\ \text {and thus}\quad \mathcal {P}^{\theta ,N}_{\Omega ,j}f(x)&=\sup _{z\in \Omega }\frac{|\theta _j*f(z)|}{(1+2^j|x-z|)^N}\\ {}&\le \sup _{z\in \Omega }\sum _{k=0}^\infty \int _\Omega \frac{|\theta _j*\psi _k(z-y)||\phi _k*f(y)|dy}{(1+2^j|x-z|)^N}. \end{aligned}$$

The elementary inequality yields

$$\begin{aligned} {} \frac{1}{(1+2^j|x-z|)^N}&\le \frac{2^{N|j-k|}}{(1+2^k|x-z|)^N}\frac{(1+2^k|z-y|)^N}{(1+2^k|z-y|)^N}\\ {}&\le 2^{N|j-k|}\frac{(1+2^k|z-y|)^N}{(1+2^k|x-y|)^N}. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&\mathcal {P}^{\theta ,N}_{\Omega ,j}f(x)\\ {}&=\sup _{z\in \Omega }\frac{|\phi _k*f(z)|}{(1+2^k|x-z|)^N}\sum _{k=0}^\infty \int _\Omega 2^{N|j-k|}|\theta _j*\psi _k(z-y)|(1+2^k|z-y|)^Ndy \\&\le \sup _{k\ge 0}2^{-N|j-k|}\mathcal {P}_{\Omega ,k}^{\phi ,N}f(x)\sum _{l=0}^\infty \int _\Omega 2^{2N|j-l|}|\theta _j*\psi _l(y)|(1+2^l|y|)^Ndy \\&\lesssim _{\theta ,\phi ,N}\sup _{k\ge 0}2^{-N|j-k|}\mathcal {P}_{\Omega ,k}^{\phi ,N}f(x)\sum _{l=0}^\infty 2^{(2N-(2N+1))|j-l|}\lesssim \sup _{k\ge 0}2^{-N|j-k|}\mathcal {P}_{\Omega ,k}^{\phi ,N}f(x). \end{aligned} \end{aligned}$$
(16)

Here the last inequality is obtained by applying Lemma 10.

Therefore we get (15). Combining it with (14) we complete the proof.

Recall the Hardy–Littlewood maximal function \(\mathcal {M}f(x):=\sup _{R>0}|B(0,R)|^{-1}\int _{B(x,R)}|f(y)|dy\) for \(f\in L^1_\textrm{loc}\).

Lemma 13

Let \(N>n\). There is a \(C=C(N)>0\) such that for any \(g\in L^1_\textrm{loc}(\mathbb {R}^n)\), \(J\in \mathbb Z\), \(v\in \mathbb Z^n\), \(k\ge J\) and \(x\in Q_{J,v}\),

$$\begin{aligned} {}&{} \int _{\mathbb {R}^n}\frac{2^{kn}|g(y)|dy}{(1+2^k|x-y|)^N} \le C\sum _{w\in \mathbb {Z}^n}\frac{\mathcal {M}({\textbf {1}}_{Q_{J,w}}\cdot g)(x)}{(1+|v-w|)^{N-n}}. \end{aligned}$$
(17)

Our lemma here is weaker than the corresponding estimate in [26, Proof of Theorem 1.2, Step 3].

Proof

By taking a translation, it suffices to prove the estimate on \(x\in Q_{J,0}\), i.e for \(v=0\). Note that if \(y\in Q_{J,w}\), then \(|x-y|\ge {\text {dist}}(Q_{J,w},Q_{J,0})\ge \frac{1}{\sqrt{n}}2^{-J}\max (0,|w|-\sqrt{n})\) and \(|x-y|\le |w|+\sqrt{n}\). Therefore

$$\begin{aligned}&\int _{\mathbb {R}^n}\frac{2^{kn}|g(y)| dy}{(1+2^k|x-y|)^N}\\ {}&\quad \le \int _{B(x,3\sqrt{n} 2^{-J})}\frac{2^{kn}|g(y)| dy}{(1+2^k|x-y|)^N}+\sum _{|w|>2\sqrt{n}}\int _{Q_{J,w}}\frac{2^{kn}|g(y)| dy}{(1+2^k|x-y|)^N} \\ {}&\quad \lesssim \Big \Vert \frac{2^{n(k-J)}}{(1+2^k|y|)^N}\Big \Vert _{L^1(\mathbb {R}^n_y)}\mathcal {M}({\textbf {1}}_{B(0,4\sqrt{n} 2^{-J})}\cdot g)(x)\\ {}&\qquad +\sum _{|w|>2\sqrt{n}}\frac{2^{kn}}{\big (1+2^k 2^{-J}(\frac{|w|}{\sqrt{n}}-1)\big )^N}\int _{Q_{J,w}}|g(y)| dy \\ {}&\quad \lesssim \sum _{|w|<4\sqrt{n}}\mathcal {M}({\textbf {1}}_{Q_{J,w}}\cdot g)(x)\\ {}&\qquad +\sum _{|w|>2\sqrt{n}}\frac{2^{-(k-J)(N-n)}}{|w|^{N-n}}\cdot \frac{2^{nJ}}{|w|^n}\int _{B(x,2^{-J}(|w|+\sqrt{n}))}|{\textbf {1}}_{Q_{J,w}}\cdot g(y)| dy \\ {}&\quad \lesssim \sum _{w\in \mathbb {Z}^n}\frac{1}{(1+|w|)^{N-n}}\cdot \mathcal {M}({\textbf {1}}_{Q_{J,w}}\cdot g)(x). \end{aligned}$$

\(\square \)

Combining Lemmas 11 - 13 we have the following Morrey–type estimates for Peetre maximal functions.

Proposition 14

Keeping the assumptions of Lemma 12, for every \(0<p,q\le \infty \), \(s\in \mathbb {R}\), \(\tau \ge 0\) and \(N>~\!\max (2n/\min (p,q),|s|+n\tau )\), there is a \(C=C(\theta ,\phi ,p,q,s,\tau ,N)>0\) such that for every \(f\in \mathscr {S}'(\Omega )\),

$$\begin{aligned} \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\theta ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{\ell ^qL^p_\tau }&\le C\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(18)
$$\begin{aligned} \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\theta ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}&\le C\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q},\qquad \text {provided }p<\infty ; \end{aligned}$$
(19)
$$\begin{aligned} \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\theta ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{\ell ^qM^p_\tau }&\le C\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{\ell ^qM^p_\tau }. \end{aligned}$$
(20)

Remark 15

It is possible that the assumption \(N>\max (\frac{2n}{\min (p,q)},|s|+n\tau )\) can be relaxed to \(N>\frac{n}{\min (p,q)}\). In applications, we only need a large enough N that does not depend on f.

A similar result for (20) can be found in [20, Proposition 2.12]. Note that we require \(\theta _j\) to have Fourier compact supports in that proposition.

Proof

We use a convention \(\phi _j:\equiv 0\) for \(j\le -1\). Thus in the computations below every sequence \((a_j)_{j=J}^\infty \) is identical to \((a_j)_{j=\max (0,J)}^\infty \).

By the assumption on N we can take \(\gamma \in (0,\min (p,q))\) such that \(N\gamma >2n\). We first prove (19).

Since \(N>|s|+n\tau \). By Lemma 12 and using \(2^{j\gamma s}2^{-N\gamma |j-k|}\le 2^{-(N-|s|)\gamma |j-k|}2^{k\gamma s} \),

$$\begin{aligned}{} & {} \Vert (2^{js}\mathcal {P}^{\theta ,N}_{\Omega ,j}f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q} =\big \Vert \big (2^{j\gamma s}(\mathcal {P}^{\theta ,N}_{\Omega ,j}f)^\gamma \big )_{j=0}^\infty \big \Vert _{L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }}^\frac{1}{\gamma }\\{} & {} \quad \lesssim \bigg \Vert \Big (\sum _{k=0}^\infty 2^{(|s|-N)\gamma |j-k|}\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{j=0}^\infty \bigg \Vert _{L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }}^\frac{1}{\gamma }. \end{aligned}$$

By Lemma 11 and since \((N-|s|)\gamma >n\tau \gamma \),

$$\begin{aligned}{} & {} \bigg \Vert \Big (\sum _{k=0}^\infty 2^{(|s|-N)\gamma |j-k|}\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{j=0}^\infty \bigg \Vert _{L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }}\\{} & {} \quad \lesssim \bigg \Vert \Big (\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{k=0}^\infty \bigg \Vert _{L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }}. \end{aligned}$$

Applying Lemma 13 with \(g(x)=\textbf{1}_\Omega (x)\cdot |2^{ks}\phi _k*f(x)|^\gamma \) for each \(k\ge 0\) and expanding the \(L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }\)-norm,

$$\begin{aligned}&\bigg \Vert \Big (\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{k=0}^\infty \bigg \Vert _{L^{\frac{p}{\gamma }}_{\tau \gamma }\ell ^\frac{q}{\gamma }}\\ {}&\quad =\sup _{J\in \mathbb {Z},v\in \mathbb {Z}^n}2^{nJ\tau \gamma \cdot \frac{1}{\gamma }}\bigg \Vert \Big (\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{k=J}^\infty \bigg \Vert _{L^\frac{p}{\gamma }(Q_{J,v};\ell ^\frac{q}{\gamma })}^\frac{1}{\gamma }\\ {}&\quad \lesssim _{N,\gamma }\sup _{J\in \mathbb {Z},v\in \mathbb {Z}^n}2^{nJ\tau }\bigg \Vert \Big (\sum _{w\in \mathbb {Z}^n}\frac{\mathcal {M}({\textbf {1}}_{Q_{J,w}}\cdot {\textbf {1}}_\Omega \cdot |2^{ks}\phi _k*f|^\gamma )}{(1+|w-v|)^{N\gamma -n}}\Big )_{k=J}^\infty \bigg \Vert _{L^\frac{p}{\gamma }(Q_{J,v};\ell ^\frac{q}{\gamma })}^\frac{1}{\gamma }\\ {}&\quad \le \Big (\sum _{v\in \mathbb {Z}^n}\frac{1}{(1+|v|)^{N\gamma -n}}\Big )^{1/\gamma }\\ {}&\qquad \times \sup _{J\in \mathbb {Z},w\in \mathbb {Z}^n}2^{nJ\tau }\big \Vert \big (\mathcal {M}({\textbf {1}}_{Q_{J,w}\cap \Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=J}^\infty \big \Vert _{L^\frac{p}{\gamma }(\mathbb {R}^n;\ell ^\frac{q}{\gamma })}^\frac{1}{\gamma }. \end{aligned}$$

Since \(N\gamma -n>n\) the sum \(\sum _{v\in \mathbb {Z}^n}(1+|v|)^{n-N\gamma }\) is finite.

Finally, applying Fefferman-Stein’s inequality to \(\big (\mathcal {M}(\textbf{1}_{Q_{J,w}\cap \Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=J}^\infty \) in \(L^\frac{p}{\gamma }(\mathbb {R}^n;\ell ^\frac{q}{\gamma })\) for each \(J\in \mathbb {Z}\) (see [3, Theorem 1(1)] and also [5, Remark 5.6.7]), since \(1<p/\gamma <\infty \) and \(1< q/\gamma \le \infty \),

$$\begin{aligned}&\sup _{Q_{J,w}\in \mathcal {Q}}2^{nJ\tau }\big \Vert \big (\mathcal {M}(\textbf{1}_{Q_{J,w}\cap \Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=J}^\infty \big \Vert _{L^\frac{p}{\gamma }(\mathbb {R}^n;\ell ^\frac{q}{\gamma })}^\frac{1}{\gamma }\\&\quad \lesssim \sup _{Q_{J,w}}2^{nJ\tau }\big \Vert \big (\textbf{1}_{Q_{J,w}\cap \Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=J}^\infty \big \Vert _{L^\frac{p}{\gamma }(\mathbb {R}^n;\ell ^\frac{q}{\gamma })}^\frac{1}{\gamma }\\&\quad =\sup _{Q_{J,w}}2^{nJ\tau }\big \Vert \big (\textbf{1}_{\Omega }\cdot (2^{ks}\phi _k*f)\big )_{k=J}^\infty \big \Vert _{L^p(Q_{J,w};\ell ^q)}\\&\quad =\big \Vert \big (2^{ks}\textbf{1}_\Omega \cdot (\phi _k*f)\big )_{k=0}^\infty \big \Vert _{L^p_\tau \ell ^q}. \end{aligned}$$

This completes the proof of (19).

The proof of (18) and (20) are similar but simpler: by assumption \(1<p/\gamma \le \infty \) we have

$$\begin{aligned} \mathcal {M}:L^\frac{p}{\gamma }(\mathbb {R}^n)\rightarrow L^\frac{p}{\gamma }(\mathbb {R}^n). \end{aligned}$$
(21)

Therefore, we prove (18) by the following:

$$\begin{aligned}&\Vert (2^{js}\mathcal {P}^{\theta ,N}_{\Omega ,j}f)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau } \\ {}&\quad \lesssim _{\theta ,\phi ,s,\tau ,N,\gamma } \bigg \Vert \Big (\sum _{k=0}^\infty 2^{-(n\tau +1)\gamma |j-k|}\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{j=0}^\infty \bigg \Vert _{\ell ^\frac{q}{\gamma }L^{\frac{p}{\gamma }}_{\tau \gamma }}^\frac{1}{\gamma }&\text{ by } \text{(13) } \\ {}&\quad \lesssim _{p,q,s,\tau }\bigg \Vert \Big (\int _\Omega \frac{2^{kn}|2^{ks}\phi _k*f(y)|^\gamma dy}{(1+2^k|\cdot -y|)^{N\gamma }}\Big )_{k=0}^\infty \bigg \Vert _{\ell ^\frac{q}{\gamma }L^\frac{p}{\gamma }_{\tau \gamma }}^\frac{1}{\gamma }&\text{ by } \text{(10) } \\ {}&\quad \lesssim _{N,\gamma }\Big (\sum _{v\in \mathbb {Z}^n}\frac{1}{(1+|v|)^{N\gamma -n}}\Big )^{1/\gamma }\big \Vert \big (\mathcal {M}({\textbf {1}}_{\Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=0}^\infty \big \Vert _{\ell ^\frac{q}{\gamma }L^\frac{p}{\gamma }_{\tau \gamma }}^\frac{1}{\gamma }&\text{ by } \text{(17) } \\ {}&\quad \lesssim _{p,\gamma }\big \Vert \big ({\textbf {1}}_{\Omega }\cdot |2^{ks}\phi _k*f|^\gamma )\big )_{k=0}^\infty \big \Vert _{\ell ^{q/\gamma } L^{p/\gamma }_{\tau \gamma }}^{1/\gamma }=\big \Vert \big (2^{ks}{} {\textbf {1}}_\Omega \cdot (\phi _k*f)\big )_{k=0}^\infty \big \Vert _{\ell ^qL^p_\tau }&\text{ by } \text{(21) }. \end{aligned}$$

Finally we prove (20). Using (15) and (12) (since \(N>|s|+n\tau \)) we have

$$\begin{aligned}{} & {} \Vert (2^{js}\mathcal {P}^{\theta ,N}_{\Omega ,j}f)_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }\lesssim _{\theta ,\phi ,s,N}\nonumber \\{} & {} \quad \Big \Vert \Big (\sum _{k=0}^\infty 2^{(N-|s|)|j-k|}2^{ks}\mathcal {P}^{\phi ,N}_{\Omega ,k}f\Big )_{j=0}^\infty \Big \Vert _{\ell ^qM^p_\tau }\lesssim _{p,q,\tau ,N}\Vert (2^{js}\mathcal {P}^{\phi ,N}_{\Omega ,j}f)_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }. \nonumber \\ \end{aligned}$$
(22)

Taking \(\gamma \in (n/N,\min (p,q))\), we have \(2^{js}(\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\lesssim _{N,\gamma }\mathcal {M}(|2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)|^\gamma )^{1/\gamma }\) pointwise in \(\mathbb {R}^n\).

When \(p<\infty \) and \(\tau <1/p\), by [20, Lemma 2.5] we have

$$\begin{aligned}&\Vert 2^{js}\mathcal {P}^{\phi ,N}_{\Omega ,j}f\Vert _{M^p_\tau }\lesssim _{N,\gamma }\big \Vert \mathcal {M}\big (|2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)|^\gamma \big )^{1/\gamma }\big \Vert _{M^p_\tau }\nonumber \\ {}&\quad \lesssim _{p,\gamma ,\tau }\Vert 2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\Vert _{M^p_\tau },\quad j\ge 0. \end{aligned}$$
(23)

We see that (23) is valid for all \(1<p/\gamma \le \infty ,\tau \ge 0\).

When \(\tau =1/p\), we have \(M^p_\tau =L^\infty \) by [10, Remark 11(ii)], so (23) follows from (21). When \(\tau >1/p\) we have \(M^p_\tau =\{0\}\), so (23) holds trivially.

Thus by taking \(\ell ^q\)-sum of (23), we get (20), completing the proof.

Proposition 16

Let \(\theta =(\theta _j)_{j=0}^\infty \) satisfies (P.a) and (P.b), and let \(\lambda =(\lambda _j)_{j=0}^\infty \) satisfies (P.a’) and (P.b’). For any \(0<p,q\le \infty \), \(s\in \mathbb {R}\), \(\tau \ge 0\) and \(N>\max (2n/\min (p,q),|s|+n\tau )\), there is a \(C=C(\theta ,\lambda ,p,q,s,\tau ,N)>0\) such that for every \(\tilde{f}\in \mathscr {S}'(\mathbb {R}^n)\),

$$\begin{aligned} \Vert (2^{js}\mathcal {P}^{\theta ,N}_{\mathbb {R}^n,j}\tilde{f})_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }&\le C\Vert (2^{js}\lambda _j*\tilde{f})_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(24)
$$\begin{aligned} \Vert (2^{js}\mathcal {P}^{\theta ,N}_{\mathbb {R}^n,j}\tilde{f})_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}&\le C\Vert (2^{js}\lambda _j*\tilde{f})_{j=0}^\infty \Vert _{L^p_\tau \ell ^q},\qquad \text {provided }p<\infty ; \end{aligned}$$
(25)
$$\begin{aligned} \Vert (2^{js}\mathcal {P}^{\theta ,N}_{\mathbb {R}^n,j}\tilde{f})_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }&\le C\Vert (2^{js}\lambda _j*\tilde{f})_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }. \end{aligned}$$
(26)

Proof

The proof is the same as that for Proposition 14, except that we replace every \(\Omega \) by \(\mathbb {R}^n\) in the arguments. We leave the details to readers.

Based on Proposition 14, we can prove a boundedness result of Rychkov-type operators on \(\mathscr {A}_{pq}^{s\tau }\)-spaces.

Proposition 17

Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain and let \(\gamma \in \mathbb {R}\). Let \(\eta =(\eta _j)_{j=0}^\infty \) and \(\theta =(\theta _j)_{j=0}^\infty \) satisfy conditions (P.a), (P.b) and (P.d) with respect to \(\Omega \). We define an operator.Footnote 4\(T^{\eta ,\theta ,\gamma }_\Omega \) as

$$\begin{aligned} T^{\eta ,\theta ,\gamma }_\Omega f:=\sum _{j=0}^\infty 2^{j\gamma }\eta _j*(\textbf{1}_\Omega \cdot (\theta _j*f)),\quad f\in \mathscr {S}'(\Omega ). \end{aligned}$$
(27)

Then for \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\), \(0<p,q\le \infty \), \(s\in \mathbb {R}\) and \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases), we have the boundedness

$$\begin{aligned} T^{\eta ,\theta ,\gamma }_\Omega :\mathscr {A}_{p,q}^{s,\tau }(\Omega )\rightarrow \mathscr {A}_{p,q}^{s-\gamma ,\tau }(\mathbb {R}^n). \end{aligned}$$

Proof

Recall \(\mathscr {S}'(\Omega )=\{\tilde{f}|_\Omega :\tilde{f}\in \mathscr {S}'(\mathbb {R}^n)\}\) is defined via restrictions. We see that \(T^{\eta ,\theta ,\gamma }_\Omega :\mathscr {S}'(\Omega )\rightarrow \mathscr {S}'(\mathbb {R}^n)\) is well-defined in the sense that, for every extension \(\tilde{f}\in \mathscr {S}'(\mathbb {R}^n)\) of f, the summation \(\sum _{j=0}^\infty 2^{j\gamma }\eta _j*(\textbf{1}_\Omega \cdot (\theta _j*\tilde{f}))\) converges \(\mathscr {S}'(\mathbb {R}^n)\) and does not depend on the choice of \(\tilde{f}\). See [12, Propositions 3.11 and 3.16] for example.

Let \(\lambda =(\lambda _j)_{j=0}^\infty \) be as in Definition 6 that defines the \(\mathscr {A}_{pq}^{s\tau }\)-norms. By Lemma 10, for every \(j,k\ge 0\), \(\int _{\mathbb {R}^n}|\lambda _j*\eta _k(y)|(1+2^k|y|)^Ndy\lesssim _{\lambda ,\eta ,N}2^{-N|j-k|}\). Thus by the similar argument to (16), for every \(N>|s-\gamma |\),

$$\begin{aligned}&2^{j(s-\gamma )} 2^{k\gamma }|\lambda _j*\eta _k*(\textbf{1}_\Omega \cdot (\theta _k*f))(x)|\\&\quad \le 2^{j(s-\gamma )} 2^{k\gamma }\int _\Omega |\lambda _j*\eta _k(y)|(1+2^k|y|)^Ndy\cdot \sup \limits _{t\in \Omega }\frac{|\theta _k*f(t)|}{(1+2^k|x-t|)^N} \\&\quad \lesssim _{\lambda ,\eta ,N}2^{-(N-|s-\gamma |)|j-k|}2^{ks}(\mathcal {P}_{\Omega ,k}^{\theta ,N}f)(x). \end{aligned}$$

Therefore, by Lemma 11, for any \(N>|s-\gamma |+n\tau \),

$$\begin{aligned}&\Vert (2^{j(s-\gamma )}\lambda _j*T^{\eta ,\theta ,\gamma }_\Omega f)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }\lesssim _{\lambda ,\eta ,p,q,s,\gamma ,\tau ,N}\Vert (2^{ks}\mathcal {P}_{\Omega ,k}^{\theta ,N}f)_{k=0}^\infty \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(28)
$$\begin{aligned}&\Vert (2^{j(s-\gamma )}\lambda _j*T^{\eta ,\theta ,\gamma }_\Omega f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}\lesssim _{\lambda ,\eta ,p,q,s,\gamma ,\tau ,N}\Vert (2^{ks}\mathcal {P}_{\Omega ,k}^{\theta ,N} f)_{k=0}^\infty \Vert _{L^p_\tau \ell ^q},\ (p<\infty ); \end{aligned}$$
(29)
$$\begin{aligned}&\Vert (2^{j(s-\gamma )}\lambda _j*T^{\eta ,\theta ,\gamma }_\Omega f)_{j=0}^\infty \Vert _{\ell ^q M^p_\tau }\lesssim _{\lambda ,\eta ,p,q,s,\gamma ,\tau ,N}\Vert (2^{ks}\mathcal {P}_{\Omega ,k}^{\theta ,N} f)_{k=0}^\infty \Vert _{\ell ^q M^p_\tau }. \end{aligned}$$
(30)

Let \(\tilde{f}\in \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) be an extension of f. Clearly \(\mathcal {P}^{\theta ,N}_{\Omega ,k} f(x)=\mathcal {P}^{\theta ,N}_{\Omega ,k}\tilde{f}(x)\le \mathcal {P}^{\theta ,N}_{\mathbb {R}^n,k}\tilde{f}(x)\) holds pointwise for \(x\in \mathbb {R}^n\). Therefore, by choosing \(N>2n/\min (p,q)\) and combining (28) and (24), we have

$$\begin{aligned}{} & {} \Vert T^{\eta ,\theta ,\gamma }_\Omega f\Vert _{\mathscr {B}_{pq}^{s\tau }(\mathbb {R}^n)}=\Vert (2^{j(s-\gamma )}\lambda _j*T^{\eta ,\theta ,\gamma }_\Omega f)_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }\\{} & {} \quad \lesssim _{\eta ,\theta ,\lambda ,p,q,s,\gamma ,\tau }\Vert (2^{js}\lambda _j*\tilde{f})_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }=\Vert \tilde{f}\Vert _{\mathscr {B}_{pq}^{s\tau }(\mathbb {R}^n)}. \end{aligned}$$

Taking the infimum over all extensions \(\tilde{f}\) of f we get the boundedness \( T^{\eta ,\theta ,\gamma }_\Omega :\mathscr {B}_{p,q}^{s,\tau }(\Omega )\rightarrow \mathscr {B}_{p,q}^{s-\gamma ,\gamma }(\mathbb {R}^n)\). Similarly using (29), (25) and (30), (26) we get \(T^{\eta ,\theta ,\gamma }_\Omega :\mathscr {A}_{p,q}^{s,\tau }(\Omega )\rightarrow \mathscr {A}_{p,q}^{s-\gamma ,\gamma }(\mathbb {R}^n)\) for \(\mathscr {A}\in \{\mathscr {F},\mathscr {N}\}\).

Remark 18

Under the definition (7), the operator norms of \(T^{\eta ,\theta ,\gamma }_\Omega \) do not dependFootnote 5 on \(\Omega \). This is due to the same reason as mentioned in [12, Remark 3.14]:

One can see that the constants in Proposition 14 depend on everything except on \(\Omega \). The same hold for the implied constants in (28), (29) and (30). After the pointwise inequality \(\mathcal {P}^{\theta ,N}_{\Omega ,k} f\le \mathcal {P}^{\theta ,N}_{\mathbb {R}^n,k}\tilde{f}\), it remains to estimate \((2^{js}\mathcal {P}^{\theta ,N}_{\mathbb {R}^n,j}\tilde{f})_{j=0}^\infty \) (which is Proposition 16), where \(\Omega \) is not involved.

Corollary 19

([25, 28, 29]) Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain. Let \(\phi =(\phi _j)_{j=0}^\infty \) and \(\psi =(\psi _j)_{j=0}^\infty \) be as in the assumption and conclusion of Lemma 9 with respect to \(\Omega \). Then the Rychkov’s extension operator

$$\begin{aligned} E_\Omega f=E^{\psi ,\phi }_\Omega f: =\sum _{j=0}^\infty \psi _j*(\textbf{1}_{\Omega }\cdot (\phi _j*f)),\quad f\in \mathscr {S}'(\Omega ), \end{aligned}$$
(31)

is well-defined and has boundedness \(E_\Omega :\mathscr {A}_{pq}^{s\tau }(\Omega )\rightarrow \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) for \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\) and all \(0<p,q\le \infty \), \(s\in \mathbb {R}\), \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases).

Proof

\(E_\Omega \) is an extension operator because by assumption \(E_\Omega f|_\Omega =\sum _{j=0}^\infty \psi _j*\phi _j *f=f\). The boundedness is immediate since \(E_\Omega =T^{\psi ,\phi ,0}_\Omega \) from (27).

Remark 20

Corollary 19 is not new. See [25, Proposition 4.13] for \(\mathscr {A}=\mathscr {N}\), [28, Section 4] for \(\mathscr {A}=\mathscr {F}\) and [29, Section 4] for \(\mathscr {A}=\mathscr {B}\). For the proof we also refer [4, Theorem 3.6] to readers.

The key to prove Theorem 1 is to use the following analog of [9, Theorem 2.3].

Proposition 21

(Characterizations via Peetre’s maximal functions) Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain and let \(\phi =(\phi _j)_{j=0}^\infty \) be a Littlewood-Paley family associated with \(\Omega \). Then for \(0<p,q\le \infty \), \(s\in \mathbb {R}\) and \(\tau \ge 0\) (\(p<\infty \) for \(\mathscr {F}\)-cases), we have the following intrinsic characterizations: for every \(N>\max (\frac{2n}{\min (p,q)},|s|+n\tau )\),

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau ,N}\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(32)
$$\begin{aligned} \Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau ,N}\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q},\qquad \text {provided }p<\infty ; \end{aligned}$$
(33)
$$\begin{aligned} \Vert f\Vert _{\mathscr {N}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau ,N}\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{\ell ^q M^p_\tau }. \end{aligned}$$
(34)

Remark 22

(32) and (33) are not new as well. The case \(\mathscr {A}=\mathscr {F}\) is done in [13, Theorem 1.7], where a more general setting is considered. See also [4, Proof of Theorem 3.6, Step 2] for a proof of \(\mathscr {A}\in \{\mathscr {B},\mathscr {F}\}\).

As already mentioned in Remark 15, it is possible that the assumption of N can be weakened.

Proof of Proposition 21

Let \(\lambda =(\lambda _j)_{j=0}^\infty \) be as in Definition 6 that defines the \(\mathscr {A}_{pq}^{s\tau }\)-norms. We only prove (33) since the proof of (32) and (34) are the same by replacing \(L^p_\tau \ell ^q\) with \(\ell ^qL^p_\tau \) and \(\ell ^q M^p_\tau \), and including the discussion of \(p=\infty \).

(\(\gtrsim \)) For \(f\in \mathscr {F}_{pq}^{s\tau }(\Omega )\), let \(\tilde{f}\in \mathscr {F}_{pq}^{s\tau }(\mathbb {R}^n)\) be an extension of f. We see that pointwisely

$$\begin{aligned} (\textbf{1}_\Omega \cdot \mathcal {P}^{\phi ,N}_{\Omega ,j}f)(x)\le \mathcal {P}^{\phi ,N}_{\Omega ,j}f(x)= \mathcal {P}^{\phi ,N}_{\Omega ,j}\tilde{f}(x)\le \mathcal {P}^{\phi ,N}_{\mathbb {R}^n,j}\tilde{f}(x),\quad j\ge 0,\quad x\in \mathbb {R}^n. \end{aligned}$$

Thus by Proposition 14,

$$\begin{aligned} {}&{} \big \Vert \big (2^{js}{} {\textbf {1}}_\Omega \cdot (\mathcal {P}^{\phi ,N}_jf)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\le \big \Vert \big (2^{js}\mathcal {P}^{\phi ,N}_{\Omega ,j}\tilde{f}\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\\{}&{} \quad \lesssim _{\lambda ,\phi ,p,q,s,\gamma ,\tau ,N}\big \Vert \big (2^{js}\lambda _j*\tilde{f}\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}=\Vert \tilde{f}\Vert _{\mathscr {F}_{pq}^{s\tau }(\mathbb {R}^n)}. \end{aligned}$$

Taking infimum over all extensions \(\tilde{f}\) of f, we get \(\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}\gtrsim \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\).

(\(\lesssim \)) By Corollary 19 we have \(\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}\approx \Vert E_\Omega f\Vert _{\mathscr {F}_{pq}^{s\tau }(\mathbb {R}^n)}=\Vert (2^{js}\lambda _j*E_\Omega f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}\). Therefore using (28) with the fact that \(E_\Omega =T^{\psi ,\phi ,0}_\Omega \),

$$\begin{aligned}&\Vert (2^{js}\lambda _j*E_\Omega f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}=\Vert (2^{js}\lambda _j*T^{\psi ,\phi ,0}_\Omega f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}\nonumber \\ {}&\quad \lesssim _{\psi ,\phi ,\lambda ,p,q,s,\tau }\Vert (2^{js}\mathcal {P}^{\phi ,N}_{\Omega ,j} f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}. \end{aligned}$$
(35)

Write \(\Omega =\{(x',x_n):x_n>\rho (x')\}\). We define a “fold map” \(L=L_\Omega :\mathbb {R}^n\twoheadrightarrow \overline{\Omega }\) as

$$\begin{aligned} L(x):=x\quad \text {if }x\in \Omega ;\qquad L(x):=(x',2\rho (x')-x_n),\quad \text {if }x\notin \Omega . \end{aligned}$$

Recall \(\Omega =\{x_n>\rho (x')\}\). By direct computation, we have

$$\begin{aligned} \textstyle |L(x)-y|\le \big (\Vert \nabla \rho \Vert _{L^\infty }+\sqrt{1+\Vert \nabla \rho \Vert _{L^\infty }^2}\big )^2|x-y|\lesssim _\Omega |x-y|\qquad x\in \mathbb {R}^n,\quad y\in \Omega . \end{aligned}$$
(36)

Therefore

$$\begin{aligned} \mathcal {P}_{\Omega ,j}^{\phi ,N}f(x){} & {} =\sup _{y\in \Omega }\frac{|\phi _j*f(y)|}{(1+2^j|x-y|)^N}\lesssim _{\Omega ,N}\sup _{y\in \Omega }\frac{|\phi _j*f(y)|}{(1+2^j|L(x)-y|)^N}\\{} & {} =\big (\mathcal {P}_{\Omega ,j}^{\phi ,N}f\big )\big (L(x)\big ),\quad x\in \mathbb {R}^n. \end{aligned}$$

Clearly for \(0<p\le \infty \) we have the following estimate for cube \(Q\in \mathcal {Q}\) and function \(g\in L^p_\textrm{loc}(\Omega )\):

$$\begin{aligned}{} & {} \Vert g\circ L\Vert _{L^p(Q)}\lesssim _p\Vert g\Vert _{L^p(\Omega \cap L^{-1}(Q))} \lesssim _p\sum _{P\in \mathcal I_Q}\Vert \textbf{1}_\Omega \cdot g\Vert _{L^p(P)},\\{} & {} \quad \text {where }\mathcal I_Q:=\{P\in \mathcal {Q}:|P|=|Q|,\ P\cap L^{-1}(Q)\ne \varnothing \}. \end{aligned}$$

By (36) we have control of the cardinality \(\# \mathcal I_Q\lesssim _n(1+\Vert \nabla \rho \Vert _{L^\infty })^{2n}\lesssim _\Omega 1\), which is uniform in \(Q\in \mathcal {Q}\). Therefore,

$$\begin{aligned}&\Vert (2^{js}\mathcal {P}^{\phi ,N}_{\Omega ,j} f)_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}\lesssim _N\big \Vert \big (2^{js}(\mathcal {P}^{\phi ,N}_{\Omega ,j} f)\circ L\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\nonumber \\ {}&\quad \lesssim _{p,q,\Omega }\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j} f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}. \end{aligned}$$
(37)

Combining (35) and (37) we get \(\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}\lesssim \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\), finishing the proof.

We can now prove Theorem 1:

Proof of Theorem 1

The \(\mathscr {F}_{\infty q}^s\)-cases follow immediately from the \(\mathscr {F}_{pq}^{s\tau }\)-cases using (9).

Fix a \(N>\max (2n/\min (p,q),|s|+n\tau )\). We only prove the \(\mathscr {F}_{pq}^{s\tau }\)-cases. The proofs of the \(\mathscr {B}_{pq}^{s\tau }\)-cases and the \(\mathscr {N}_{pq}^{s\tau }\)-cases are the same, except that we replace every \(L^p_\tau \ell ^q\) with \(\ell ^qL^p_\tau \) and \(\ell ^q M^p_\tau \).

By Proposition 21 we have \(\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}\approx \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\). Therefore, it suffices to show that \(\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\approx \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\).

Clearly \(\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\ge \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\) since \(\phi _j*f(x)\le \mathcal {P}^{\phi ,N}_{\Omega ,j}f(x)\) holds for all \(f\in \mathscr {S}'(\Omega )\), \(x\in \Omega \) and \(j\ge 0\). The converse \(\big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\mathcal {P}^{\phi ,N}_{\Omega ,j}f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\lesssim _{\phi ,p,q,s,\tau ,N} \big \Vert \big (2^{js}\textbf{1}_\Omega \cdot (\phi _j*f)\big )_{j=0}^\infty \big \Vert _{L^p_\tau \ell ^q}\) follows from (18). Thus, we prove the \(\mathscr {F}_{pq}^{s\tau }\)-cases.

We have the immediate analogy of [26, Theorem 1.1] on Lipschitz domains:

Corollary 23

Keeping the assumptions in Proposition 21, we have the following intrinsic characterizations: for every \(N>\max (2n/\min (p,q),|s|+n\tau )\),

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau ,N}\sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau }\Big (\sum _{j=\max (0,J)}^\infty 2^{jsq}\Vert \mathcal {P}^{\phi ,N}_{(Q_{J,v}\cap \Omega ),j}f\Vert _{L^p(Q_{J,v}\cap \Omega )}^q\Big )^\frac{1}{q}; \\ \Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}&\approx \sup _{Q_{J,v}}2^{nJ\tau }\Big (\!\int _{ Q_{J,v}\cap \Omega }\Big (\!\!\!\!\sum _{j=\max (0,J)}^\infty 2^{jsq}|\mathcal {P}^{\phi ,N}_{(Q_{J,v}\cap \Omega ),j}f(x)|^q\Big )^\frac{p}{q}dx\Big )^\frac{1}{p}, (p<\infty ) ; \\ \Vert f\Vert _{\mathscr {N}_{pq}^{s\tau }(\Omega )}&\approx _{\phi ,p,q,s,\tau ,N}\Big (\sum _{j=0}^\infty \sup _{Q_{J,v}\in \mathcal {Q}}2^{nJ\tau q+jsq}\Vert \mathcal {P}^{\phi ,N}_{(Q_{J,v}\cap \Omega ),j}f\Vert _{L^p(Q_{J,v}\cap \Omega )}^q\Big )^\frac{1}{q}. \end{aligned}$$

Proof

Since \(|\phi _j*f(x)|\le \mathcal {P}^{\phi ,N}_{(Q_{J,v}\cap \Omega ),j}f(x)\le \mathcal {P}^{\phi ,N}_{\Omega ,j}f(x)\) pointwisely for every \(Q_{J,v}\in \mathcal Q\) and \(x\in Q_{J,v}\cap \Omega \), the results follow immediately by combining Theorem 1 and Proposition 21.

Remark 24

By the standard partition of unity argument, we can give the analogy of Theorem 1 on a bounded Lipschitz domain. An example is the following:

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}&\approx \sum _{\nu =1}^N\Vert (2^{js}\textbf{1}_{\Omega \cap U_\nu }\cdot (\phi _j^\nu *(\chi _\nu f)))_{j=0}^\infty \Vert _{\ell ^qL^p_\tau }; \end{aligned}$$
(38)
$$\begin{aligned} \Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}&\approx \sum _{\nu =1}^N\Vert (2^{js}\textbf{1}_{\Omega \cap U_\nu }\cdot (\phi _j^\nu *(\chi _\nu f)))_{j=0}^\infty \Vert _{L^p_\tau \ell ^q}; \end{aligned}$$
(39)
$$\begin{aligned} \Vert f\Vert _{\mathscr {N}_{pq}^{s\tau }(\Omega )}&\approx \sum _{\nu =1}^N\Vert (2^{js}\textbf{1}_{\Omega \cap U_\nu }\cdot (\phi _j^\nu *(\chi _\nu f)))_{j=0}^\infty \Vert _{\ell ^qM^p_\tau }; \end{aligned}$$
(40)

Here \(\{U_\nu ,(\phi _j^\nu )_{j=0}^\infty ,\chi _\nu \}_{\nu =1}^N\) satisfy the following:

  • \(\{U_\nu \}_{\nu =1}^N\) is an open cover of \(\overline{\Omega }\), and there are cones \(K_\nu \subset \mathbb {R}^n\) such that \(U_\nu \cap (\Omega -K_\nu )\subseteq U_\nu \cap \Omega \) for each \(\nu =1,\dots ,N\).

  • For \(\nu =1,\dots ,N\), \((\phi _j^\nu )_{j=0}^\infty \) satisfies (P.a) - (P.c) in Definition 4, with support condition \({\text {supp}}\phi _j^\nu \subset K_\nu \) for \(j\ge 0\).

  • \(\chi _\nu \in C_c^\infty (U_\nu )\) for \(\nu =1,\dots ,N\), and satisfyFootnote 6\(\sum _{\nu =1}^N\chi _\nu |_{\overline{\Omega }}\equiv 1\).

To prove (38), (39) and (40) the only thing we need are the following standard results (\(p<\infty \) for \(\mathscr {F}\)-cases):

(\(\Psi \).a):

Let \(\chi \in C_c^\infty (\mathbb {R}^n)\). Then \([\tilde{f}\mapsto \chi \tilde{f}]:\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\rightarrow \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) is bounded.

(\(\Psi \).b):

Let \(\Phi \) be an invertible affine linear transform. Then \([\tilde{f}\mapsto \tilde{f}\circ \Phi ]:\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\rightarrow \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) is bounded.

(\(\Psi \).c):

For every \(m\ge 1\), we have equivalent norms \(\Vert f\Vert _{\mathscr {A}_{p,q}^{s,\tau }(\mathbb {R}^n)}\approx _{p,q,s,\tau ,m}\sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\mathbb {R}^n)}\).

One can see [24, Sections 6.1.1 and 6.2], [22, Theorem 1.6] and [11, Theorem 3.3] for their proof. See also [6, Sections 3.4, 4.2 and 4.3]. We remark that because of (8) it is enough to consider the case \(0\le \tau \le \frac{1}{p}\). We leave the details to the readers.

One can also write down the analogy of Proposition 21 and Corollary 23 similar to (38), (39) and (40), we leave the details to the readers as well.

Finally, we prove Theorem 2 using the following fact:

Proposition 25

([12, Theorem 1.5 (ii)]) Let \((\phi _j)_{j=1}^\infty \) be a familyFootnote 7 of Schwartz functions satisfying (P.a), (P.b) and (P.d). Recall that for every \(j\ge 1\), \(\phi _j(x)=2^{(j-1)n}\phi _1(2^{j-1}x)\), \(\int x^\alpha \phi _j(x)dx=0\) for all \(\alpha \), and \({\text {supp}}\phi _j\subset \{x_n<-A|x'|\}\) for some \(A>0\).

Then for any \(m\ge 1\), there are families of Schwartz functions \(\tilde{\phi }^\beta =(\tilde{\phi }^\beta _j)_{j=1}^\infty \) for \(|\beta |=m\) that also satisfy (P.a), (P.b) and (P.d), such that

$$\begin{aligned} \phi _j=2^{-jm}\sum _{|\beta |=m}\partial ^\beta \tilde{\phi }_j^\beta ,\quad \text {for every }j\ge 1. \end{aligned}$$

Proof of Theorem 2

Once the case of special Lipschitz domains is done, the proof of the case of bounded Lipschitz domains follows from the standard partition of unity argument (one can read [12, Section 6] for details) along with the facts (\(\Psi \).a), (\(\Psi \).b) and (\(\Psi \).c) mentioned in Remark 24.

Let \(\Omega \subset \mathbb {R}^n\) be a special Lipschitz domain. Let \(f\in \mathscr {A}_{pq}^{s\tau }(\Omega )\) and let \(\tilde{f}\in \mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) be an extension of f. By (\(\Psi \).c) we have \(\Vert \partial ^\alpha \tilde{f}\Vert _{\mathscr {A}_{p,q}^{s-|\alpha |,\tau }(\mathbb {R}^n)}\lesssim _{p,q,s,\tau ,\alpha }\Vert \tilde{f}\Vert _{\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)}\). Since \(\partial ^\alpha \tilde{f}\) is also an extension of \(\partial ^\alpha f\), by (7) in Definition 5, taking the infimum over all extensions \(\tilde{f}\) of f we get \(\sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )}\lesssim \Vert f\Vert _{\mathscr {A}_{pq}^{s\tau }(\Omega )}\).

To prove the converse inequality \(\Vert f\Vert _{\mathscr {A}_{pq}^{s\tau }(\Omega )}\lesssim \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )}\), let \((\phi _j,\psi _j)_{j=0}^\infty \) be as in (31).

We let \((\tilde{\phi }^\beta _j)_{j=1}^\infty \subset \mathscr {S}(\mathbb {R}^n)\) (\(|\beta |>0\)) be given in Proposition 25. Thus \(\phi _j=2^{-jq}\sum _{\beta :|\beta |=q}\partial ^\beta \tilde{\phi }^\beta _j\) for all \(j,q\ge 1\).

For \(\alpha \ne 0\), we define \(\psi ^\alpha =(\psi ^\alpha _j)_{j=1}^\infty \) by \(\psi ^\alpha _j(x):=2^{-j|\alpha |}\partial ^\alpha \psi _j(x)\) (for \(j\ge 1\)). Thus the sequences \(\psi ^\alpha \) (for \(\alpha \ne 0\)) all satisfy (P.a), (P.b) and (P.d).

We define a family of linear operators,

$$\begin{aligned}{} & {} E^{\alpha ,0} f=E^{\alpha ,0}_\Omega f:=\partial ^\alpha \psi _0*(\textbf{1}_\Omega \cdot (\phi _0*f)),\nonumber \\{} & {} E^{\alpha ,\beta }f=E^{\alpha ,\beta }_\Omega f:=\sum _{j=1}^\infty \psi _j^\alpha *(\textbf{1}_\Omega \cdot (\tilde{\phi }_j^\beta *f)),\text { for }|\alpha |=|\beta |>0. \end{aligned}$$
(41)

For every \(f\in \mathscr {S}'(\Omega )\) and for every multi-index \(\alpha \ne 0\), we see that

$$\begin{aligned} \begin{aligned} \partial ^\alpha E f&=\sum _{j=0}^\infty \partial ^\alpha \psi _j*(\textbf{1}_\Omega \cdot (\phi _j*f))=\partial ^\alpha \psi _0*(\textbf{1}_\Omega \cdot (\phi _0*f))\\&\quad +\sum _{j=1}^\infty \sum _{\beta :|\beta |=|\alpha |}2^{j|\alpha |}\psi _j^\alpha *(\textbf{1}_\Omega \cdot 2^{-j|\alpha |}(\partial ^\beta \tilde{\phi }_j^\beta *f)) \\&=\partial ^\alpha \psi _0*(\textbf{1}_\Omega \cdot (\phi _0*f))+\sum _{\beta :|\beta |=|\alpha |}\sum _{j=1}^\infty \psi _j^\alpha *(\textbf{1}_\Omega \cdot (\tilde{\phi }_j^\beta *\partial ^\beta f))\\&=E^{\alpha ,0}f+\sum _{\beta :|\beta |=|\alpha |}E^{\alpha ,\beta }[\partial ^\beta f]. \end{aligned} \end{aligned}$$
(42)

By Proposition 17, \(E^{\alpha ,0},E^{\alpha ,\beta }:\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )\rightarrow \mathscr {A}_{p,q}^{s-m,\tau }(\mathbb {R}^n)\) are all bounded. Therefore

$$\begin{aligned} \begin{aligned}&\Vert f\Vert _{\mathscr {A}_{pq}^{s\tau }(\Omega )}\approx \Vert E f\Vert _{\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)}\overset{(\Psi .\text{ c})}{\approx }\sum _{|\alpha |\le m} \Vert \partial ^\alpha Ef\Vert _{\mathscr {A}_{pq}^{s-m,\tau }(\mathbb {R}^n)}\\ \overset{\text{(42) }}{\lesssim }&\Vert Ef\Vert _{\mathscr {A}_{pq}^{s-m,\tau }}+\sum _{0<|\alpha |\le m}\Big (\Vert E^{\alpha ,0}f\Vert _{\mathscr {A}_{pq}^{s-m,\tau }}+\sum _{\beta :|\beta |=|\alpha |}\Vert E^{\alpha ,\beta }[\partial ^\beta f]\Vert _{\mathscr {A}_{pq}^{s-m,\tau }}\Big ) \\ \lesssim&\Vert f\Vert _{\mathscr {A}_{pq}^{s-m,\tau }(\Omega )}+\sum _{0<|\alpha |\le m}\Big (\Vert f\Vert _{\mathscr {A}_{pq}^{s-m,\tau }(\Omega )}+\sum _{\beta :|\beta |=|\alpha |}\Vert \partial ^\beta f\Vert _{\mathscr {A}_{pq}^{s-m,\tau }(\Omega )}\Big )\\ \lesssim&\sum _{|\beta |\le m}\Vert \partial ^\beta f\Vert _{\mathscr {A}_{pq}^{s-m,\tau }(\Omega )}. \end{aligned} \end{aligned}$$

This completes the proof of (4) for the case of special Lipschitz domains.

The \(\mathscr {F}_{\infty q}^s\)-cases follow immediately from (9) since we have \(\mathscr {F}_{\infty q}^s(\mathbb {R}^n)=\mathscr {F}_{qq}^{s,\frac{1}{q}}(\mathbb {R}^n)\).

4 Further Open Questions

By the same method, using Lemma 10 - Proposition 14, it is possible for us to get the analogs of Theorems 1 and 2 on the so-called local spaces.

The local version of \(\mathscr {A}_{pq}^{s\tau }(\mathbb {R}^n)\) for \(\mathscr {A}\in \{\mathscr {B},\mathscr {F},\mathscr {N}\}\), denoted by \(\mathscr {A}_{p,q,\text {unif}}^{s,\tau }(\mathbb {R}^n)\), is defined by replacing the supremum among the set of dyadic cubes \(\mathcal {Q}\) with \(\{Q_{J,v}\in \mathcal {Q}:J\ge 0\}\). See [10, Section 3.4] for example. For an open subset \(\Omega \subseteq \mathbb {R}^n\) we use \(\mathscr {A}_{p,q,\text {unif}}^{s,\tau }(\Omega ):=\{\tilde{f}|_\Omega :\tilde{f}\in \mathscr {A}_{p,q,\text {unif}}^{s,\tau }(\mathbb {R}^n)\}\) similarly. For more details we refer [17] to readers.

One can also consider the analog of Theorems 1 and 2 on \(\mathscr {A}_{p(\cdot ),q(\cdot )}^{s(\cdot ),\phi }\), the spaces with variable exponents. For example [13], which may require certain assumptions on the exponents.

In Definition 6, it is known that the norms are equivalent if \((\lambda _j)_{j=0}^\infty \) only satisfies the scaling condition (P.b) and the Tauberian condition:

$$\begin{aligned}{} & {} \text {There exist }\varepsilon _0,c>0\quad \text {such that}\quad |\hat{\lambda }_0(\xi )|>c\text { for }|\xi |<\varepsilon _0,\nonumber \\{} & {} \quad \text {and }|\hat{\lambda }_1(\xi )|>c\text { for }\varepsilon _0/2<|\xi |<2\varepsilon _0. \end{aligned}$$
(43)

See [22, Theorems 2.5 and 2.6] and [23, Theorem 1] for example.

It is not known to the author whether we can replace the assumption (P.c) for \((\phi _j)_{j=0}^\infty \) in Theorem 1 with the Tauberian condition (43).

For Theorem 2, we do not know whether (4) has the following improvement:

Question 26

Keeping the assumptions of Theorem 2, can we find a \(C=C(\Omega ,p,q,s,\tau ,m)>0\) such that the following holds?

$$\begin{aligned} \Vert f\Vert _{\mathscr {A}_{pq}^{s\tau }(\Omega )}\le C\Big (\Vert f\Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )}+\sum _{k=0}^n\Big \Vert \frac{\partial ^m f}{\partial x_k^m}\Big \Vert _{\mathscr {A}_{p,q}^{s-m,\tau }(\Omega )}\Big ),\quad \forall \,f\in \mathscr {A}_{pq}^{s\tau }(\Omega ). \end{aligned}$$

Cf. [22, Theorem 1.6]. The question is open even for the classical Besov and Triebel-Lizorkin spaces \(\mathscr {A}_{pq}^s(\Omega )\) when \(\Omega \) is a (special or bounded) Lipschitz domain.