Változó indexű Lebesgue-terek és alkalmazásuk a Fourier-analízisben

Abstract

In the first chapter of my PhD thesis the Musielak-Orlicz spaces were constructed, then proceeded from the Musielak-Orlicz spaces we obtained the variable Lebesgue spaces. Many basic and important properties were presented, emphasized the similarities and differences compared with the classical Lebesgue spaces. In the second chapter of my work the classical Hardy Littlewood maximal operator was generalized for the so-called γ-rectangles. With the help of the generalization of numerous previous results we could prove that the generalized maximal operator (under some certain conditions) bounded on the variable Lebesgue spaces and is of weak-type (p(·),p(·)). The third chapter of my PhD thesis was the spine of the theses. In this chapter the convergence of integral operators were considered and under some general conditions, many convergence theorems for the integral operators were verified in the variable Lebesgue spaces. In the further sections of my dissertation the results of the third thesis were applied. In some special cases we could replace the general conditions (taken on the kernel function of the integral operator) with some easily controllable conditions. Many classical results of the θ-summation of Fourier transforms (which proved by Feichtinger and Weisz) were got back as an immediate consequence of the third thesis. Moreover we got almost everywhere and norm convergence of the θ-summation in the variable Lebesgue spaces. In the fifth chapter of my work we chose the kernel function of the integral operator to get back some projection operators of the wavelet series. Sufficient conditions were given to almost everywhere and norm convergence in the variable Lebesgue spaces. In the last chapter of my PhD thesis the continuous wavelet transforms were studied. In this chapter we showed that the Dirichlet integral of the inversion formula can be represented as a special θ-summation and so the results in the fourth chapter were applicable. We obtained almost everywhere and norm convergence in the variable Lebesgue spaces.