Seminário de Análise Funcional, Estruturas Lineares e Aplicações 

13/05/2016, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade Nova de Lisboa

On convolution type operator with piecewise continuous symbols on variable Lebesgue spaces

In this talk we discuss three results on convolution type operators acting on variable Lebesgue spaces  under mild assumptions on variable exponents  : we assume only that the exponent  is bounded away from one and infinity, and is such that the Hardy-Littlewood maximal operator is bounded on . The first result says that the set of the Fourier multipliers  on the space  forms a Banach algebra. The second result is the generalization of the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces saying that the algebra  of functions  of finite total variation  is continuously embedded into , that is, Now let  denote the set of all continuous functions on the one-point compactification of   and let  be the set of all piecewise continuous functions on . The completeness of  and the Stechkin inequality allows us to define the classes  and  of continuous and piecewise continuous Fourier multipliers as the closure of  and  with respect to the norm of . The third result concerns the compactness of the commutator of the Fourier convolution operator  and the operator of multiplication  by a function  on variable Lebesgue spaces . This result is proved under the assumption that  or  and generalizes the corresponding result by Roland Duduchava proved for constant exponents in 1970's.