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...

we derive whole series of new integral inequalities of the hardy-type, with non-conjugate exponents. first, we prove and discuss two equivalent general inequa-li-ties of such type, as well as their corresponding reverse inequalities. general results are then applied to special hardy-type kernel and power weights. also, some estimates of weight functions and constant factors are obtained. ...

Let φ : ℝ(n) × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H(A)(φ)(ℝ(n)), via the grand maximal function. The authors then obtain some real-variable characterizations of H(A)(φ)(ℝ(n)) in terms of the radial, the nontangential, and the tangential maximal functions, which general...

There are different methods for the hydraulic analysis of water supply networks. In the solution process of most of these methods, a large system of linear equations is solved in each iteration. This usually requires a high computational effort. Hardy Cross method is one of the approaches that do not need such a process and may converge to the solution through scalar divisions. However, this me...

see for instance [29]. Notice that the exponent p or q could be larger than N+2 N−2 . Hence the usual Sobolev space H 0 (Ω)×H1 0 (Ω) is not suitable to handle the problem. To study the problem (1.2) under the condition (1.3), a key observation was done by Hulshof and Van de Vorst [16], De Figueiredo and Felmer [9]. In order to solve this problem, the main idea is to destroy the symmetry between...

We give a number field analogue of result Ramanujan, Hardy and Littlewood, thereby obtaining modular relation involving the non-trivial zeros Dedekind zeta function. also provide Riesz-type criterion for Generalized Riemann Hypothesis ζK(s). New elegant transformations are obtained when K is quadratic extension, one which involves modified Bessel function second kind.

We obtain Fourier inequalities in the weighted Lp spaces for any 1<p<∞ involving Hardy–Cesàro and Hardy–Bellman operators. extend these results to product Hardy p⩽1. Moreover, boundedness of Hardy-Cesàro Hardy-Bellman operators various (Lebesgue, Hardy, BMO) is discussed. One our main tools an appropriate version Hardy–Littlewood–Paley inequality ‖fˆ‖Lp′,q≲‖f‖Lp,q.