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Maximal, potential and singular type operators on Herz spaces with variable exponents. (English) Zbl 1250.42077

The main purpose of this paper is to consider the variable exponent homogeneous and inhomogeneous Herz spaces and give boundedness results for a wide class of classical operators, including maximal operators, fractional integral operators and Calderón-Zygmund operators, acting on such Herz spaces. We denote by \(\mathcal{P}(\mathbb{R}^n)\) the set of all measurable functions \(p: \mathbb{R}^n \rightarrow [1, \infty]\). For \(p \in \mathcal{P}(\mathbb{R}^n)\), we use the notation \[ p^+={\text{ ess\;sup}}_{\mathbb{R}^n}p(x),\;\;p^-={\text{ ess\;inf}}_{\mathbb{R}^n}p(x). \] The variable exponent Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\) is the class of all measurable functions \(f\) on \(\mathbb{R}^n\) such that the modular \[ \rho_{p(\cdot)}(f)=\int_{\mathbb{R}^n}|f(x)|^{p(x)}\;dx \] is finite. This is a Banach function space equipped with the norm \[ ||f||_{p(\cdot)}=\inf \{ \mu >0: \rho_{p(\cdot)}(\frac{f}{\mu}) \leq 1 \}. \] We set \(B_{k}=B(0, 2^k),\;\;R_{k}=B_{k}\setminus B_{k-1}\) and \(\chi_{k}=\chi_{R_{k}}, \;\;k \in \mathbb{Z}\).
Let \(0 < q \leq \infty,\;\;p \in \mathcal{P}(\mathbb{R}^n)\) and \(\alpha \in L^{\infty}(\mathbb{R}^n)\). The inhomogeneous Herz space \(K^{\alpha(\cdot)}_{p(\cdot),q}(\mathbb{R}^n)\) consists of all \(f \in L^{p(\cdot)}_{loc}(\mathbb{R}^n)\) such that \[ ||f||_{K^{\alpha(\cdot)}_{p(\cdot),q}}=||f\chi_{B_{0}}||_{p(\cdot)}+ (\sum_{k \geq 1}||2^{k\alpha(\cdot)}f\chi_{k}||_{p(\cdot)}^q)^{1/q} < \infty. \] The homogeneous Herz space \(\dot{K}^{\alpha(\cdot)}_{p(\cdot),q}(\mathbb{R}^n)\) is defined as the set of all \(f \in L^{p(\cdot)}_{loc}(\mathbb{R}^n \setminus \{ 0 \})\) such that \[ ||f||_{\dot{K}^{\alpha(\cdot)}_{p(\cdot),q}}= (\sum_{k \in \mathbb{Z}}||2^{k\alpha(\cdot)}f\chi_{k}||_{p(\cdot)}^q)^{1/q} <\infty. \] We say that a function \(g : \mathbb{R}^n \rightarrow \mathbb{R}\) is log-Hölder continuous at the origin, if \[ |g(x)-g(0)| \leq \frac{c_{log}}{\log (e+1/|x|)} \] for all \(x \in \mathbb{R}^n\). If, for some \(g_{\infty} \in \mathbb{R}\) and \(c_{log}>0\), there holds \[ |g(x)-g_{\infty}|\leq \frac{c_{log}}{\log (e+|x|)} \] for all \(x \in \mathbb{R}^n\), then we say that \(g\) is log-Hölder continuous at infinity. By \(\mathcal{P}^{log}_{0}(\mathbb{R}^n)\) and \(\mathcal{P}^{log}_{\infty}(\mathbb{R}^n)\) we denote the class of all exponents \(p \in \mathcal{P}(\mathbb{R}^n)\) which are log-Hölder continuous at the origin and at infinity, respectivly, with \(p_{\infty}=\lim_{|x| \rightarrow \infty}p(x)\). Let \(1 = \frac{1}{p(x)}+\frac{1}{p'(x)}\) and let \(p^{*}\) be the Sobolev exponent defined by \(\frac{1}{p^*(x)}=\frac{1}{p(x)}-\frac{\lambda}{n},\;\;\;0 < \lambda < n\).
We consider sublinear operators satisfying the size conditions \[ |Tf(x)| \leq C\int_{\mathbb{R}^n}\frac{|f(y)|}{|x-y|^n}\;dy,\;\;\;\;x \notin {\text{ supp}} f\;\;\;(*) \] and \[ |T_{\lambda}f(x)| \leq C\int_{\mathbb{R}^n}\frac{|f(y)|}{|x-y|^{n-\lambda}} \;dy,\;\;\;\;x \notin {\text{ supp}} f\;\;\;(**) \] for integrable and compactly supported functions \(f\). The condition (\(*\)) is satisfied by several classical operators such as Calderón-Zygmund operators, the Carleson maximal operators and Hardy-Littlewood maximal operators. The Riesz potential operators and the fractional maximal operators satisfy the condition (\(**\)). The authors prove the following main results.
Theorem A. Let \(0 < q \leq \infty\).
{(i)} Let \(p \in \mathcal{P}_{\infty}^{log}(\mathbb{R}^n)\) with \(1 <p^- \leq p^+ < \infty\) and let \(\alpha \in L^{\infty}(\mathbb{R}^n)\) be log-Hölder continuous at infinity with \[ -\frac{n}{p_{\infty}} < \alpha_{\infty} < \frac{n}{p'_{\infty}}. \] Suppose that \(T\) is a sublinear operator satisfying \((*)\). If \(T\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^n)\), then \(T\) is bounded on \(K^{\alpha(\cdot)}_{p(\cdot),q}(\mathbb{R}^n)\).
{(ii)} Let \(p \in \mathcal{P}_{0}^{log}(\mathbb{R}^n)\cap \mathcal{P}^{log}_{\infty}(\mathbb{R}^n)\) with \(1 < p^- \leq p^+ < \infty\) and let \(\alpha \in L^{\infty}(\mathbb{R}^n)\) be log-Hölder continuous, both at the origin and at infinity, such that \[ -\frac{n}{p^+} < \alpha^- \leq \alpha^+ < n(1-\frac{1}{p^-}). \] Then every sublinear operator \(T\) satisfying \((*)\) ,which is bounded on \(L^{p(\cdot)}(\mathbb{R}^n)\), is also bounded on \(\dot{K}^{\alpha(\cdot)}_{p(\cdot),q}(\mathbb{R}^n)\).
Theorem B. Let \(0 < \lambda < n,\;0 < q_{0} \leq q_{1} \leq \infty\).
{(i)} Let \(p \in \mathcal{P}^{loc}_{\infty}(\mathbb{R}^n)\) with \(1 < p^- \leq p^+ < \frac{n}{\lambda}\), and let \(\alpha \in L^{\infty}(\mathbb{R}^n)\) be log-Hölder continuous at infinity. If \[ \lambda-\frac{n}{p_{\infty}} < \alpha_{\infty} < \frac{n}{p'_{\infty}}, \] then every sublinear operator \(T_{\lambda}\) satisfying \((**)\) , which is bounded from \(L^{p(\cdot)}(\mathbb{R}^n)\) into \(L^{p^*(\cdot)}(\mathbb{R}^n)\), is also bounded from \(K^{\alpha(\cdot)}_{p(\cdot),q_{0}}(\mathbb{R}^n)\) into \(K^{\alpha(\cdot)}_{p^*(\cdot),q_{1}}(\mathbb{R}^n)\).
{(ii)} Let \(p \in \mathcal{P}^{log}_{0}(\mathbb{R}^n) \cap \mathcal{P}^{log}_{\infty}(\mathbb{R}^n)\) and let \(\alpha \in L^{\infty}(\mathbb{R}^n)\) be log-Hölder continous, both at the origin and at infinity, such that \(1< p^- \leq p^+ < \frac{n}{\lambda}\) and \[ \lambda - \frac{n}{p^+} < \alpha^- \leq \alpha^+ < n(1-\frac{1}{p^-}). \] Then every sublinear operator \(T_{\lambda}\) satisfying \((**)\) and bounded from \(L^{p(\cdot)}(\mathbb{R}^n)\) into \(L^{p^*(\cdot)}(\mathbb{R}^n)\), is also bounded from \(\dot{K}^{\alpha(\cdot)}_{p(\cdot),q_{0}}(\mathbb{R}^n)\) into \(\dot{K}^{\alpha(\cdot)}_{p^*(\cdot),q_{1}}(\mathbb{R}^n)\).

MSC:

42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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