1 Introduction
In this study, we investigate the boundedness of composition operators on Morrey spaces and weak Morrey spaces. The composition operator
\(C_{\varphi }\) induced by a mapping
φ is a linear operator defined by
\(C_{\varphi }f\equiv f\circ \varphi \), where
\(f\circ \varphi \) represents the function composition. The composition operator is also called the Koopman operator in the fields of dynamical systems, physics, and engineering [
12]. Recently, it has attracted attention in various scientific fields [
10,
11]. It becomes more and more important recently to prove the properties of composition operators mathematically.
Let \((X,\mu )\) be a σ-finite measure space, and \(L^{0}(X,\mu )\) be the set of all μ-measurable functions on X. We provide a precise definition of the composition operators induced by a measurable map \(\varphi :X\rightarrow X\).
Subsequently, we employ the result obtained by Singh [
16] for the boundedness of the composition operator on the Lebesgue space
\(L^{p}(X,\mu )\).
Singh [
16] provided the following necessary and sufficient condition for the map
φ to generate a bounded mapping acting on Lebesgue spaces:
The boundedness of the composition operator on
\(L^{\infty }(X,\mu )\) easily follows from the definition. Theorem
1.2 was extended to several important function spaces, such as Lorentz spaces [
1,
6], Orlicz spaces [
3,
13], mixed Lebesgue spaces [
5,
7], Musielak–Orlicz spaces [
14], and reproducing kernel Hilbert spaces [
9]. However, there are no previous results on the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces.
The first aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator \(C_{\varphi }\) on Morrey spaces. Subsequently, we discuss the boundedness of the operator on weak Morrey spaces.
Hereafter, we consider the Euclidean space \(\mathbb{R}^{n}\); μ is the Lebesgue measure dx. We denote by \(|E|\) the volume of a measurable set \(E \subset \mathbb{R}^{n}\). Let \(\chi _{A}:\mathbb{R}^{n}\rightarrow \mathbb{R}_{\ge 0}\) be an indicator function for a subset \(A\subset \mathbb{R}^{n}\), which is defined as \(\chi _{A}(x)=1\) if \(x\in A\) and \(\chi _{A}(x)=0\), otherwise.
Now, we recall the definition of Morrey spaces on \(\mathbb{R}^{n}\).
A standard argument in functional analysis shows that \({\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\) is a quasi-Banach space.
From the Hölder inequality, we observe that the Lebesgue space \(L^{p}({\mathbb{R}}^{n})\) is embedded into the Morrey space \({\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\), where \(0< q \le p<\infty \).
We now state the main results of the present paper. The following theorem provides a sufficient condition on the boundedness of the composition operator \(C_{\varphi }\) on the Morrey space \(\mathcal{M}^{p}_{q}(\mathbb{R}^{n})\).
Conversely, as stated in the following theorem, if
\(\varphi :\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is a diffeomorphism, then the
\({\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\)-boundedness of the composition operators
\(C_{\varphi }\) and
\(C_{\varphi ^{-1}}\) indicates that
φ is bi-Lipschitz. Note that any bi-Lipschitz mapping satisfies the assumption of Theorem
1.5.
Subsequently, we investigate the characterization of the boundedness of the composition operators acting on weak Morrey spaces, which are defined as follows:
Another standard argument in functional analysis shows that \({\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\) is a quasi-Banach space. The weak Morrey space \({\mathrm{W}}{\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\) has the following basic properties:
The following theorem provides a necessary and sufficient condition on the boundedness of the composition operator on weak Morrey spaces.
In fact, we will establish the boundedness of the composition operator in a more general framework.
For example, the space
\({\mathrm{W}}L^{p}({\mathbb{R}}^{n})\) is the weak Lebesgue space whose norm is given by
\(\|f\|_{{\mathrm{W}}L^{p}}=\sup_{\lambda >0}\lambda \|\chi _{f^{-1}(( \lambda ,\infty ])}\|_{L^{p}}\) (see [
8, Chap. 1] for more).
Now, we can rewrite Theorem
1.10 as follows:
Here is a list of standard notation used in this paper in addition to that which has already appeared above:
-
The space \(L^{\infty }_{\mathrm{c}}({\mathbb{R}}^{n})\) stands for the set of all \(L^{\infty }({\mathbb{R}}^{n})\) functions with compact support.
-
The linear space \({\mathrm{M}}_{n}({\mathbb{R}}^{n})\) is the set of all \(n\times n\)-matrices.
-
For \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\), the matrix \(\operatorname{diag}(\alpha _{1},\ldots ,\alpha _{n})\) is the diagonal matrix with entries \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\).
-
The matrix \(E\in {\mathrm{M}}_{n}(\mathbb{R}^{n})\) denotes the identity matrix.
-
For \(A \in {\mathrm{M}}_{n}({\mathbb{R}}^{n})\), the quantity \(\|A\|_{\mathrm{F}}\) denotes the Frobenius norm defined by \(\sqrt{\operatorname{tr}(A^{\mathrm{T}}A)}\).
-
The space \(C^{\infty }_{\mathrm{c}}({\mathbb{R}}^{n})\) is the set of all smooth functions with compact support.
-
For a cube Q, \(\ell (Q)\) stands for its side-length.
-
Let \(A,B \ge 0\). Then \(A \lesssim B\) and \(B \gtrsim A\) mean that there exists a constant \(C>0\) such that \(A \le C B\), where C depends only on the parameters of importance. The symbol \(A \sim B\) means that \(A \lesssim B\) and \(B \lesssim A\) happen simultaneously. For example, for functions \(A(x)\) and \(B(x)\) of x, we use shorthand \(A(x)\lesssim B(x)\) to denote an estimate \(A(x)\leq CB(x)\) with some constant \(C>0\) independent of x. As a result, notation \(A(x)\sim B(x)\) represents \(A(x)\lesssim B(x)\) and \(B(x)\lesssim A(x)\).
-
For a differentiable vector-valued function
\(\varphi =(\varphi _{1},\ldots ,\varphi _{n})^{\mathrm{T}}\) on
\(\mathbb{R}^{n}\), we denote by
Dφ the Jacobi matrix of
φ, that is,
$$ D\varphi \equiv \biggl(\frac{\partial \varphi _{i}}{\partial x_{j}} \biggr)_{1\le i,j\le n} \equiv ( \varphi _{i,j})_{1\le i,j\le n}. $$
The remainder of this paper is organized as follows: In Sect.
2, we prove Theorems
1.5 and
1.6. In Sect.
3, we present some examples and counterexamples of the mapping that induces the composition operator to be bounded on Morrey spaces. In Sect.
4, we prove Theorem
1.13.
4 Boundedness of composition operators on weak type spaces
To prove Theorem
1.13, we use the following identity.
The weak type spaces generated by the Banach lattice are essential.
Now, as the special case of the Morrey space
\(B({\mathbb{R}}^{n})={\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\), in Theorem
1.13, we have Theorem
1.10.
In Theorem
1.10, through real interpolation, it is known that
$$ {\mathrm{W}} {\mathcal{M}}^{p}_{q}\bigl({ \mathbb{R}}^{n}\bigr)=\bigl[{\mathcal{M}}^{pr}_{qr} \bigl({ \mathbb{R}}^{n}\bigr),L^{\infty }\bigl({ \mathbb{R}}^{n}\bigr)\bigr]_{1-r,\infty } $$
as long as
\(1< q \le p<\infty \) and
\(0< r<1\) satisfy
\(q r>1\) (see [
4]). Here, as the
\(L^{\infty }({\mathbb{R}}^{n})\)-boundedness of the composition operators is trivial and the
\({\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\)-boundedness and
\({\mathcal{M}}^{pr}_{qr}({\mathbb{R}}^{n})\)-boundedness of composition operators, for
\(r>0\), are equivalent owing to the fact that
\(|C_{\varphi }f|^{r}=C_{\varphi }[|f|^{r}]\) for mapping
φ, we obtain that the boundedness “
\(C_{\varphi }:{\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\to { \mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\) implies the boundedness
\(C_{\varphi }:{\mathrm{W}}{\mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\to {\mathrm{W}}{ \mathcal{M}}^{p}_{q}({\mathbb{R}}^{n})\)”.