Seminário Análise Funcional e Aplicações 

22/03/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro

The Singular Type Operators in the Lebesgue Spaces with Variable Exponent

Last decade there was intensively developed the theory of Lebesgue spaces with variable exponent when the order $p$ of integrability depends on $x$. (These spaces have interesting applications in fluid mechanics). The corresponding theory proved to be difficult to develop because these spaces are not invariant neither with respect to translation nor dilation. It suffices to mention that for example Young type theorems for convolutions are not already valid in these spaces. In general, convolution operators have a "bad" behaviour in such spaces. A progress was recently made by proving the uniform boundedness of dilation convolution operators under some natural assumptions on the kernel of the convolution. This result, presented in particular, in the talk allowed us to prove that "nice" functions (infinitely differentiable with compact support) are dense not only in the Lebesgue spaces with variable exponent, but also in Sobolev spaces generated by them. However, boundedness of the singular integral operators remained an open question for a long time. We show that some modification of the method developed in the above investigation allows us to prove also that the singular operator along a bounded Lyapunov curve is bounded in the space with a variable exponent $p(x)$ under some natural assumptions on $p(x)$. The last topic of the talk is based on the joint research with Prof. Vakhtang Kokilashvili.