Self-improving properties of weighted Gehring classes with applications to partial differential equations

In this paper, we prove that the self-improving property of the weighted Gehring class $G_{\lambda }^{p}$ G λ p with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class $A^{q}$ A q . By using the rearrangement (nonincreasing rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property. Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose.

Self-improving properties of discrete Muckenhoupt weights

In this paper, we will provide a complete study of the self-improving properties of the discrete Muckenhoupt class 𝒜 p ⁢ ( 𝒞 ) {\mathcal{A}^{p}(\mathcal{C})} of weights defined on ℤ + {\mathbb{Z}_{+}} . In addition, we will determine the range of the new constants which are related to the original constants via an algebraic equation. For illustration, we will give an example to prove that the results are sharp. The results will be obtained by employing a discrete version of an inequality due to Hardy–Littlewood and a new discrete Hardy-type inequality with negative powers.

Local Muckenhoupt class for variable exponents

This work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ A p loc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ A p ( ⋅ ) loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes

In this paper, we prove some basic properties of the discrete Muckenhoupt class $\mathcal{A}^{p}$ A p and the discrete Gehring class $\mathcal{G}^{q}$ G q . These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes.

Approximation theorems in weighted Lebesgue spaces with variable exponent

In this work, approximation properties of de la Vall?e-Poussin means are investigated in weighted Lebesgue spaces with variable exponent where weight function belongs to Muckenhoupt class. For this purpose direct, inverse and simultaneous theorems of approximation theory are proved and constructive characterizations of functions are obtained in weighted Lebesgue spaces with variable exponent.

Nuclear Embeddings in Weighted Function Spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong about the nuclearity of diagonal operators acting in $$\ell _p$$ ℓ p spaces. In that way we can further contribute to the characterisation of nuclear embeddings of function spaces on domains.

Continuity of Weighted Operators, Muckenhoupt Ap Weights, and Steklov Problem for Orthogonal Polynomials

We consider weighted operators acting on $L^p({\mathbb{R}}^d)$ and show that they depend continuously on the weight $w\in A_p({\mathbb{R}}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w({\mathbb{T}})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2({\mathbb{T}})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.

On structure of discrete Muchenhoupt and discrete Gehring classes

In this paper, we study the structure of the discrete Muckenhoupt class $\mathcal{A}^{p}(\mathcal{C})$ A p ( C ) and the discrete Gehring class $\mathcal{G}^{q}(\mathcal{K})$ G q ( K ) . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if $u\in \mathcal{A}^{p}(\mathcal{C})$ u ∈ A p ( C ) then there exists $q< p$ q < p such that $u\in \mathcal{A}^{q}(\mathcal{C}_{1})$ u ∈ A q ( C 1 ) . Next, we prove that the power rule also holds, i.e., we prove that if $u\in \mathcal{A}^{p}$ u ∈ A p then $u^{q}\in \mathcal{A}^{p}$ u q ∈ A p for some $q>1$ q > 1 . The relation between the Muckenhoupt class $\mathcal{A}^{1}(\mathcal{C})$ A 1 ( C ) and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose.

Mixed methods for degenerate elliptic problems and application to fractional Laplacian

We analyze the approximation by mixed finite element methods of solutions of equations of the form div [[EQUATION]] , where the coefficient a=a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient $a$ belongs to the Muckenhoupt class [[EQUATION]] . The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart-Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumprtion can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.