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.


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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