Inequalities for some classes of Hardy type operators and compactness in weighted Lebesgue spaces

Abstract: This PhD thesis is devoted to investigate weighted differential Hardyinequalities and Hardy-type inequalities with the kernel when the kernel hasan integrable singularity, and also the additivity of the estimate of a Hardytype operator with a kernel.The thesis consists of seven papers (Papers 1, 2, 3, 4, 5, 6, 7) and anintroduction where a review on the subject of the thesis is given. In Paper 1 weighted differential Hardy type inequalities are investigatedon the set of compactly supported smooth functions, where necessary andsufficient conditions on the weight functions are established for which thisinequality and two-sided estimates for the best constant hold.In Papers 2, 3, 4 a more general class of - order fractional integrationoperators are considered including the well-known classical Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard operators. Here 0 < ∂ < 1. In Papers 2 and 3 the boundedness and compactness of two classesof such operators are investigated namely of Weyl and Riemann-Liouvilletype, respectively, in weighted Lebesgue spaces for 1 < p ≤ q < 1 and 0 < q < p < ∞. As applications some new results for the fractional integrationoperators of Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard aregiven and discussed.In Paper 4 the Riemann-Liouville type operator with variable upper limitis considered. The main results are proved by using a localization methodequipped with the upper limit function and the kernel of the operator.In Papers 5 and 6 the Hardy operator with kernel is considered, wherethe kernel has a logarithmic singularity. The criteria of the boundednessand compactness of the operator in weighted Lebesgue spaces are given for1 < p ≤ q < ∞ and 0 < q < p < ∞, respectively.In Paper 7 we investigated the weighted additive estimatesIIuK± ƒIIq ≤ C (IIρƒIIp + IIνH±ƒIIp), ƒ ≥ 0    (')for integral operators K+ and K¯ defined byK+ ƒ(x) := ∫ K(x,s) ƒ(s)ds,  K¯ ƒ(x) := ∫ K(x,s)ƒ(s)ds.It is assumed that the kernel of the operator K belongs to the general Oinarov class. We derived the criteria for the validity of the inequality (')when 1 ≤ p≤ q < ∞