1 Introduction
The one dimensional Hausdorff operator, for a fixed locally integrable function Φ on
\((0,\infty)\), is defined by
$$ h_{\Phi} (f ) (x)= \int_{0}^{\infty}\frac{\Phi(t)}{t}f \biggl( \frac{x}{t} \biggr)\,dt, \quad x\in\mathbb{R}. $$
This operator has a long history in the study of mathematical analysis aiming to solve classical problems (see [
1‐
3]).
The multidimensional Hausdorff operator on the Euclidean space
\(\mathbb {R}^{n}\), which was studied in [
2] and [
1], is defined by
$$ H_{\Phi,A}(f) (x)= \int_{\mathbb{R}^{n}}\frac{\Phi (y)}{|y|^{n}}f\bigl(A(y)x\bigr)\,dy. $$
(1.1)
Here, in (
1.1) and in the sequel, we fix
\(\Phi\in L_{\mathrm{loc}}^{1}(\mathbb {R}^{n})\) and assume that
\(A=A(y)=(a_{ij})_{i,j=1}^{n}=(a_{ij}(y))_{i,j=1}^{n}\) is an
\(n\times n\) nonsingular matrix almost everywhere in the support of Φ. The entries
\(a_{ij}(y)\) of the matrix
A are measurable functions of
y.
We denote
$$ \Vert A\Vert =\max_{1\leq j\leq n}\sum_{i=1}^{n}|a_{ij}|, $$
to be the operator
ℓ-norm of the operator in an
n dimensional linear space defined by the matrix
A in the corresponding canonical basis (see [
2]).
The boundedness of
\(h_{\Phi} (f )\) as well as its high dimensional extensions in various function spaces, mainly on Lebesgue and Hardy spaces, has been studied by many authors, for example, [
1‐
5] and [
6] are among many others. The aim of this paper is to establish some boundedness conditions for the multidimensional Hausdorff operator on the homogeneous Hardy-Morrey space and the Besov-Morrey space, since these spaces are ‘upgrade version’ of the Lebesgue space and the Hardy space and they play significant roles in the theory of harmonic analysis and partial differential equations (see [
7‐
10] and [
11]).
The Hardy-Morrey space
\(H\mathcal{M}_{q}^{p}\)
\((q\leq1)\), as a generalization of the classical Morrey spaces
\(\mathcal{M}_{q}^{p}\)
\((q>1)\) and Hardy spaces
\(H^{p}\)
\((p\leq1)\), was introduced by Jia and Wang [
7]. Among many features of
\(H\mathcal{M}_{q}^{p}\), particularly Jia and Wang establish the decomposition of
\(H\mathcal{M}_{q}^{p}\) in terms of atoms concentrated on dyadic cubes, which have the same cancelation properties as the atoms in the classical Hardy space. Another space, one that we are also interested in, is the Besov-Morrey space
\(\mathcal{N}_{pqr}^{s}\) for
\(1\leq q\leq p<\infty,1\leq r\leq\infty\) and
\(s\in\mathbb{R}\). This space was originally introduced by Kozono and Yamazaki [
8] in order to investigate time-local solutions of the Navier-Stokes equations with the initial data in Besov-Morrey spaces. Later, Mazzucato [
9] studied the atomic and molecular decompositions on Besov-Morrey spaces. Sawano and Tanaka [
10] further developed a theory of decompositions in Besov-Morrey spaces
\(\mathcal{N}_{pqr}^{s}\) and Triebel-Lizorkin-Morrey spaces
\(\mathcal{E}_{pqr}^{s}\) with
\(0< q\leq p<\infty\),
\(0< r\leq\infty,s\in\mathbb{R}\). The homogeneous Besov-Morrey space
\(\mathcal{N}_{pqr}^{s}\) is a function space whose norm is obtained by replacing the
\(L^{p}\)-norm in the definition of the Besov space with the
\(\mathcal{M}_{q}^{p}\)-norm. Any function in the Besov-Morrey space has a decomposition in terms of atoms supported on dyadic cubes. These atoms have the same smoothness and cancelation properties as those of the classical Besov spaces [
11]. Thus, in our study, we will invoke the atomic decomposition characterization for both Hardy-Morrey space and Besov-Morrey space. Besides the atomic decomposition, the Calderón reproducing formula is another useful tool when we treat the boundedness of
\(H_{\Phi,A } (f )\) on the Besov-Morrey space.
The following definitions and previously obtained results (Theorem
A and Theorem
B) about the function spaces we consider are crucial in our study.
We note that the Morrey space describes local regularity more precisely than
\(L^{p}\) and can be seen as a complement to
\(L^{p}\)-spaces. In fact one easily checks
\(L^{p}={\mathcal{M}_{p}^{p}}\) and
\(L^{p}\subset\mathcal {M}_{q}^{p}\), if
\(1\leq q\leq p<\infty\). Jia and Wang in [
7] give the definition of Hardy-Morrey space as follows.
As stated in [
7], the Hardy-Morrey space
\(H\mathcal{M}_{q}^{p}\) (
\(q\leq1\)) generalizes the Hardy space
\(H^{q}\) (
\(q\leq1\)). Fixing
\(q=1\) in (
1.3), we see that the space
\(H\mathcal{M}_{1}^{p}(\mathbb{R}^{n})\) for
\(1\leq p<\infty\) is a generalization of the Hardy space
\(H^{1}\). Therefore, we may adapt the atomic decomposition method used in [
1] and [
6] for treating the Hardy space to study the newly introduced Hardy-Morrey space
\(H\mathcal{M}_{1}^{p}\) (
\(1\leq p<\infty\)) to deal with the boundedness of the Hausdorff operator (
1.1). However, we must modify the main step since the decompositions of
\(H^{1}\) and
\(H\mathcal{M}_{1}^{p}\) are essentially different when
\(p>1\) (see the following definition). We also note that
\(H\mathcal{M}_{1}^{p}(1\leq p<\infty)\) is a Banach space with the norm in (
1.3).
We remark that, for a dyadic cube
J and
\(q=1\), (
1.2) can also be written as
$$ \Vert f\Vert _{\mathcal{M}_{1}^{p}}\approx\sup_{J:\mathrm{dyadic}}|J|^{1/p-1} \Vert f\Vert _{{L^{1}}(J)}. $$
(1.4)
This equivalent norm of
\(\mathcal{M}_{1}^{p}\) will play a pivotal role in our study.
The next theorem is directly adapted from [
7] when
\(q=1\), and is important in the proof for one of the main theorems (Theorem
2.1). In Theorem
A and in the sequel, we make use of (
1.4) to get the norm of the sequence of complex numbers
\(s=\{s_{Q}:Q \mbox{ dyadic}\}\).
Conversely, every function
\(f\in H\mathcal{M}_{1}^{p}\) has the atomic decomposition (
1.7) in
\(S^{\prime}/P\) with
\(\Vert s\Vert _{p,1}\leq C\Vert f\Vert _{H\mathcal{M}_{1}^{p}}\) for some
\(C=C(n,p)\).
The other important function space in harmonic analysis is the Besov-Morrey space, which has many applications in the study of nonlinear partial differential equations, for example see in [
8‐
10], and [
11].
Again, our interest here is to study the boundedness of
n dimensional Hausdorff operators on the homogeneous Besov-Morrey space. To this end, as we mentioned before, we need to use the smooth atomic decomposition of
\(\mathcal{N}_{pqr}^{s}\), and this will be implemented in the Fourier transform side and by using the results in [
12] and [
13] for classical Besov spaces. Let
S be the Schwartz space on
\(\mathbb{R}^{n}\). For
\(\varphi \in S(\mathbb{R}^{n})\), as in [
14], its Fourier transform is denoted by
φ̂ and its inverse Fourier transform is denoted by
φ̌. Take
\(\varphi_{0}\in S(\mathbb{R}^{n})\) with
\(\varphi_{0}\geq0\) and
$$ \varphi_{0}(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1, & |x|\leq1, \\ 0, & |x|\geq2, \\ \mbox{smooth}, & \mbox{elswhere}.\end{array}\displaystyle \right . $$
Now define \(\varphi(x)=\varphi_{0}(x)-\varphi_{0}(2x)\) and set \(\varphi _{j}(x)=\varphi(2^{-j}x)\) for all \(j\in\mathbb{Z}\). Then \(\{\varphi _{j},j\in\mathbb{Z}\}\) is a homogeneous smooth dyadic resolution of unity in \(\mathbb{R}^{n}\), namely \(\sum_{j\in\mathbb{Z}}\varphi_{j}(x)=1\) for all \(x\in\mathbb{R}^{n}\). Moreover, let \(\psi_{j}=\check{\varphi_{j}}\). Thus, with independence of the choice of \(\psi_{j}\), we give the Fourier analytic definition of \(\mathcal{N}_{pqr}^{s}\) as follows.
One can see that in [
8]
\(\mathcal{N}_{pqr}^{s}\) is a Banach space under the norm (
1.8). This space generalizes the classical homogeneous Besov space. In particular,
\(\mathcal{N}_{ppr}^{s}=\dot{B}_{pr}^{s}\).
The atomic decomposition of
K-times continuously differentiable functions in this space can be obtained in [
9].
The proof of this theorem is obtained in [
9].
To obtain the reverse direction of the second part of Theorem
B we need the molecular decomposition characterization of the space
\(\mathcal{N}_{pqr}^{s}(\mathbb{R}^{n})\), which is less important in this paper. The analogous definition of
\((s,p)_{K,L,M}\)-molecule for
\(M>0\), can be found in [
9] and [
14].
In this paper, we use the notation \(A\preceq B\) to mean that there is a positive constant C (may vary at each appearance) independent of all essential variables such that \(A\leq CB\). We also use the notation \(A\simeq B \) to mean that there are positive constants \(C_{1}\) and \(c_{1}\) independent of all essential variables such that \(c_{1}B\leq A\leq C_{1}B\).
Competing interests
The author declare that there is no conflict of interests regarding the publication of this paper.
Belay Mitiku Damtew is a Doctor in the field of harmonic analysis, applied mathematics.