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Boundedness of the commutator of the intrinsic square function in variable exponent spaces

Liwei Wang
대한수학회 (The Korean Mathematical Society)|2018.07|대한수학회지 (Journal of the KMS)|제55권 제4호|pp. 939-962

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키워드

commutatorMorrey spacesvariable exponentsthe intrinsic square functionHerz type spaces
주제분류수학

초록

In this paper, we show that the commutator of the intrinsic square function with {\rm BMO} symbols is bounded on the variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M}_{p(\cdot), u}$, Morrey-Herz spaces $M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}({\mathbb { R}}^n)$ and Herz type Hardy spaces $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\mathbb { R}}^n)$, where the exponents $\alpha(\cdot)$ and $p(\cdot)$ are variable. Observe that, even when $\alpha(\cdot)\equiv \alpha$ is constant, the corresponding main results are completely new.

Abstract

In this paper, we show that the commutator of the intrinsic square function with {\rm BMO} symbols is bounded on the variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M}_{p(\cdot), u}$, Morrey-Herz spaces $M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}({\mathbb { R}}^n)$ and Herz type Hardy spaces $H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\mathbb { R}}^n)$, where the exponents $\alpha(\cdot)$ and $p(\cdot)$ are variable. Observe that, even when $\alpha(\cdot)\equiv \alpha$ is constant, the corresponding main results are completely new.

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