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The boundedness of classical operators on variable ${L}^{p}$ spaces. (English) Zbl 1100.42012

Let $p\left(x\right)$ be a measurable function on an open set ${\Omega }\subset {ℝ}^{n}$ with values in $\left[1,\phantom{\rule{0.166667em}{0ex}}+\infty \right)$. Denote by ${L}^{p\left(·\right)}\left({\Omega }\right)$ the Banach space of measurable functions $f$ on ${\Omega }$ such that for some $\lambda >0$, ${\int }_{{\Omega }}{|f\left(x\right)/\lambda |}^{p\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx<\infty$, with norm ${\parallel f\parallel }_{{L}^{p\left(·\right),{\Omega }}}=inf\left\{\lambda >0;{\int }_{{\Omega }}{|f\left(x\right)/\lambda |}^{p\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx\le 1\right\}$. In the case $p\left(x\right):{\Omega }\to \left(0,\infty \right)$, one defines as above. In general, ${\parallel f\parallel }_{{L}^{p\left(·\right),{\Omega }}}$ is not a norm, but quasi-norm. The authors’ main tool (theorem) in this paper is the following: Given a family $ℱ$ of ordered pairs of non-negative, measurable functions $\left(f,g\right)$, suppose that for some ${p}_{0}$, $0<{p}_{0}<\infty$, and for every weight $w\in {A}_{1}$ (Muckenhoupt’s weight class), it holds ${\int }_{{\Omega }}f{\left(x\right)}^{{p}_{0}}w\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\le {C}_{0}{\int }_{{\Omega }}g{\left(x\right)}^{{p}_{0}}w\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$, $\left(f,g\right)\in ℱ$, where ${C}_{0}$ depends only on ${p}_{0}$ and the ${A}_{1}$ constant of $w$. Furthermore, let $p\left(x\right):{\Omega }\to \left(0,\infty \right)$ satisfy ${p}_{0}<\text{ess}\phantom{\rule{0.166667em}{0ex}}{\text{inf}}_{x\in {\Omega }}p\left(x\right)\le \text{ess}\phantom{\rule{0.166667em}{0ex}}{\text{sup}}_{x\in {\Omega }}p\left(x\right)<\infty$ and for the conjugate exponent ${\left(p\left(·\right)/{p}_{0}\right)}^{\text{'}}=\left(p\left(·\right)/{p}_{0}\right)/\left(p\left(·\right)/{p}_{0}-1\right)$ the Hardy-Littlewood maximal operator is bounded on ${L}^{{\left(p\left(·\right)/{p}_{0}\right)}^{\text{'}}}\left({\Omega }\right)$. Then, for all $\left(f,g\right)\in ℱ$ with $f\in {L}^{p\left(·\right)}\left({\Omega }\right)$, ${\parallel f\parallel }_{p\left(·\right),{\Omega }}\le C{\parallel g\parallel }_{p\left(·\right),{\Omega }}$, where $C$ is independent of the pair $\left(f,g\right)$.

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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B15 Multipliers, several variables 42B20 Singular and oscillatory integrals, several variables 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation