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...

This paper focuses on the bounds of weighted multilinear Hardy operators on the product Herz spaces and the product Morrey-Herz spaces, respectively. We present a sufficient condition on the weight function that guarantees weighted multilinear Hardy operators to be bounded on the product Herz spaces. And the condition is necessary under certain assumptions. Finally, we extend the obtained resul...

Córdoba–Fefferman collections are defined and used to characterize functions whose corresponding maximal functions are locally integrable. Córdoba–Fefferman collections are also used to show that, if Mx and My respectively denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in R, MHL denotes the standard Hardy–Littlewood maximal operator in R,...

We introduce a new class of Hardy spaces H(R), called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of Garćıa-Cuerva, Strömberg, and Torchinsky. Here, φ : R × [0,∞) → [0,∞) is a function such that φ(x, ·) is an Orlicz function and φ(·, t) is a MuckenhouptA∞ weight. A function f belongs to H(R) if and only if its maximal fu...

In this paper, we study the asymptotic behaviour of sharp constant in discrete Hardy and Rellich inequality on lattice $\mathbb{Z}^d$ as $d \rightarrow \infty$. process, proved some Hardy-type inequalities for operators $\Delta^m$ $\nabla(\Delta^m)$ non-negative integers $m$ a $d$ dimensional torus. It turns out that grows $d^2$ respectively $ d

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 study various inequalities for numerical radius and Berezin number of a bounded linear operator on Hilbert space. It is proved that the pure two-isometry 1 Crawford 0. In particular, we show any scalar-valued non-constant inner function θ, Toeplitz Tθ Hardy space 0, respectively. also shown multiplicative class isometries sub-multiplicative commutants shift. have illustrated these results wi...

Let $$H_{p(\cdot ),q}$$ be the variable Lorentz–Hardy martingale spaces. In this paper, we give a new atomic decomposition for these spaces via simple $$L_r$$ -atoms $$(1<r \le \infty )$$ . Using decomposition, consider dual of Lorentz-Hardy case $$0<p(\cdot )\le 1$$ , $$0<q\le and )<2$$ $$1<q<\infty $$ respectively, prove that they are equivalent to BMO with exponent. Furthermore, also obtain ...

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.