Summability of Fourier transforms of functions from Lorentz spaces

Abstract: This PhD thesis is devoted to the study of relations between integrability properties of functions and summability properties of its Fourier coefficients and transforms. The relations are given in terms of generalized weighted Lorentz norms, where the weights have some additional growth properties. The thesis contains six papers (papers A-F) together with an introduction, which put these papers into a general frame.In paper A some relations between weighted Lorentz norms and some corresponding sums of Fourier coefficients are studied for the case with a general orthonormal bounded system. Under certain circumstances even two-sided estimates are obtained.In paper B we study relations between summability of Fourier coefficients and integrability of the corresponding functions for generalized weighted Lorentz spaces in the case of a regular system. Some new inequalities of Hardy-Littlewood-Paley type with respect to a regular system for these generalized Lorentz spaces are obtained. It is also proved that the obtained results are in a sense sharp.In paper C we investigate integrability properties of the orthogonal series with coefficients from generalized weighted Lorentz spaces in the case of a regular system. The upper and the lower estimates of some corresponding Lorentz type norms of the Fourier coefficients are obtained.In paper D some new Boas type theorems for generalized weighted Lorentz spaces with respect to regular systems for generalized monotone functions are proved.In paper E inequalities for the Fourier transform of functions from the generalized weighted Lorentz spaces are studied. The upper and the lower estimates of the norm of the Fourier transform in generalized weighted Lorentz spaces are derived.Finally, in paper F a new inequality concerning the Fourier transform is derived. Moreover, it is described conditions so that this result is sharp in the sense that both upper and lower bounds are obtained.