On the generalized Hardy-Rellich inequalities

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: $$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Projection operators on matrix weighted $L^p$ and a simple sufficient Muckenhoupt condition

Boundedness for a class of projection operators, which includes the coordinate projections, on matrix weighted $L^p$-spaces is completely characterised in terms of simple scalar conditions. Using the projection result, sufficient conditions, which are straightforward to verify, are obtained that ensure that a given matrix weight is contained in the Muckenhoupt matrix $A_p$ class. Applications to singular integral operators with product kernels are considered.

Frame properties of a part of an exponential system with degenerate coefficients in Hardy classes

A part of an exponential system with degenerate coefficients is considered. The frame properties (completeness, minimality, basicity, atomic decomposition) of this system in Hardy classes are studied in the case where the coefficients may not satisfy the Muckenhoupt condition.

A Note on Generalized Hardy-Sobolev Inequalities

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2, u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

Summation of Multiple Fourier Series in Matrix Weighted -Spaces

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.

The maximal operator on weighted variable Lebesgue spaces

We study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L ωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.