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The boundedness of classical operators on variable $L^p$ spaces. (English) Zbl 1100.42012
Let $p(x)$ be a measurable function on an open set $\Omega\subset \Bbb R^n$ with values in $[1,\,+\infty)$. Denote by $L^{p(\cdot)}(\Omega)$ the Banach space of measurable functions $f$ on $\Omega$ such that for some $\lambda>0$, $\int_{\Omega}\vert f(x)/\lambda\vert ^{p(x)}\,dx<\infty$, with norm $\Vert f\Vert _{L^{p(\cdot),\Omega}}=\inf\{\lambda>0; \int_{\Omega}\vert f(x)/\lambda\vert ^{p(x)}\,dx\le1\}$. In the case $p(x):\Omega\to (0,\infty)$, one defines as above. In general, $\Vert f\Vert _{L^{p(\cdot),\Omega}}$ is not a norm, but quasi-norm. The authors’ main tool (theorem) in this paper is the following: Given a family $\Cal F$ of ordered pairs of non-negative, measurable functions $(f, g)$, suppose that for some $p_0$, $0<p_0<\infty$, and for every weight $w\in A_1$ (Muckenhoupt’s weight class), it holds $\int_{\Omega}f(x)^{p_0}w(x)\,dx\le C_0\int_{\Omega}g(x)^{p_0}w(x)\,dx$, $(f,g)\in \Cal F$, where $C_0$ depends only on $p_0$ and the $A_1$ constant of $w$. Furthermore, let $p(x): \Omega\to (0,\infty)$ satisfy $p_0<\text{ess inf}_{x\in\Omega}p(x)\le \text{ess sup}_{x\in\Omega}p(x)<\infty$ and for the conjugate exponent $(p(\cdot)/p_0)'=(p(\cdot)/p_0)/(p(\cdot)/p_0-1)$ the Hardy-Littlewood maximal operator is bounded on $L^{(p(\cdot)/p_0)'}(\Omega)$. Then, for all $(f,g)\in\Cal F$ with $f\in L^{p(\cdot)}(\Omega)$, $\Vert f\Vert _{p(\cdot),\Omega}\le C\Vert g\Vert _{p(\cdot),\Omega}$, where $C$ is independent of the pair $(f,g)$. They give a generalization of this to the case of $0<p_0\le q_0<\infty$. They also discuss the case of $A_\infty$ or $A_{p_0}$ in place of $A_1$ and give vector-valued inequalities. As applications of these main results, they discuss a vector-valued inequality for the Hardy-Littlewood maximal operator, the sharp maximal functions, singular integral operators, commutators of a singular integral operator and a BMO function, Fourier multipliers, Littlewood-Paley’s functions, fractional integrals, the Calderón-Zygmund inequality for the solutions of Poisson’s equation, and the Calderón extension theorem for variable Sobolev spaces.

42B25Maximal functions, Littlewood-Paley theory
42B15Multipliers, several variables
42B20Singular and oscillatory integrals, several variables
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
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