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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 Ω n with values in [1,+). Denote by L p(·) (Ω) the Banach space of measurable functions f on Ω such that for some λ>0, Ω |f(x)/λ| p(x) dx<, with norm f L p(·),Ω =inf{λ>0; Ω |f(x)/λ| p(x) dx1}. In the case p(x):Ω(0,), one defines as above. In general, f L p(·),Ω is not a norm, but quasi-norm. The authors’ main tool (theorem) in this paper is the following: Given a family of ordered pairs of non-negative, measurable functions (f,g), suppose that for some p 0 , 0<p 0 <, and for every weight wA 1 (Muckenhoupt’s weight class), it holds Ω f(x) p 0 w(x)dxC 0 Ω g(x) p 0 w(x)dx, (f,g), where C 0 depends only on p 0 and the A 1 constant of w. Furthermore, let p(x):Ω(0,) satisfy p 0 <essinf xΩ p(x)esssup xΩ p(x)< and for the conjugate exponent (p(·)/p 0 ) ' =(p(·)/p 0 )/(p(·)/p 0 -1) the Hardy-Littlewood maximal operator is bounded on L (p(·)/p 0 ) ' (Ω). Then, for all (f,g) with fL p(·) (Ω), f p(·),Ω Cg p(·),Ω , where C is independent of the pair (f,g).

They give a generalization of this to the case of 0<p 0 q 0 <. They also discuss the case of A 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:
42B25Maximal functions, Littlewood-Paley theory
42B15Multipliers, several variables
42B20Singular and oscillatory integrals, several variables
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation