## 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 $$\Omega\subset \mathbb R^n$$ with values in $$[1,\,+\infty)$$. Denote by $$L^{p(\cdot)}(\Omega)$$ the Banach space of measurable functions $$f$$ on $$\Omega$$ such that for some $$\lambda>0$$, $$\int_{\Omega}| f(x)/\lambda| ^{p(x)}\,dx<\infty$$, with norm $$\| f\| _{L^{p(\cdot),\Omega}}=\inf\{\lambda>0; \int_{\Omega}| f(x)/\lambda| ^{p(x)}\,dx\leq1\}$$. In the case $$p(x):\Omega\to (0,\infty)$$, one defines as above. In general, $$\| f\| _{L^{p(\cdot),\Omega}}$$ is not a norm, but quasi-norm. The authors’ main tool (theorem) in this paper is the following: Given a family $$\mathcal F$$ 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 $$\int_{\Omega}f(x)^{p_0}w(x)\,dx\leq C_0\int_{\Omega}g(x)^{p_0}w(x)\,dx$$, $$(f,g)\in \mathcal F$$, where $$C_0$$ depends only on $$p_0$$ and the $$A_1$$ constant of $$w$$. Furthermore, let $$p(x): \Omega\to (0,\infty)$$ satisfy $$p_0<\text{ess\,inf}_{x\in\Omega}p(x)\leq \text{ess\,sup}_{x\in\Omega}p(x)<\infty$$ 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 $$(f,g)\in\mathcal F$$ with $$f\in L^{p(\cdot)}(\Omega)$$, $$\| f\| _{p(\cdot),\Omega}\leq C\| g\| _{p(\cdot),\Omega}$$, where $$C$$ is independent of the pair $$(f,g)$$.
They give a generalization of this to the case of $$0<p_0\leq q_0<\infty$$. 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 for harmonic analysis in several variables 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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