1.1. Hardy-Sobolev-Maz’ya inequalities. Hardy inequalities and Sobolev inequalities bound the size of a function, measured by a (possibly weighted) L norm, in terms of its smoothness, measured by an integral of its gradient. Maz’ya [22] proved that for functions on the half-space R+ = {x ∈ R : xN > 0}, N ≥ 3, which vanish on the boundary, the sharp version of the Hardy inequality can be combine...

This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman’s inequality, which may be regarded as a discrete version of Pólya-Knopp’s inequality and also as a natural limiting inequality of the discre...

We established some new α-conformable dynamic inequalities of Hardy–Knopp type. Some generalizations Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions calculus. Our results proved by using two-dimensional Jensen’s and Fubini’s theorem scales. When α=1, then obtain well-known time-scale due to Hardy. As special cases...