Muckenhoupt Class Research Articles

Unified Framework for Continuous and Discrete Relations of Gehring and Muckenhoupt Weights on Time Scales

This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes Mγ;γ>1 and the Gehring classes Gδ;δ>1 of bi-Sobolev weights on a time scale T. In addition, we establish the relations between Muckenhoupt and Gehring classes, where we define a new time scale T˜=v(T), to indicate that if the Δ˜ derivative of the inverse of a bi-Sobolev weight belongs to the Gehring class, then the Δ derivative of a bi-Sobolev weight on a time scale T belongs to the Muckenhoupt class. Furthermore, our results, which will be established by a newly developed technique, show that the study of the properties in the continuous and discrete classes of weights can be unified. As special cases of our results, when T=R, one can obtain classical continuous results, and when T=N, one can obtain discrete results which are new and interesting for the reader.

Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class Ap with p∈(1,∞). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

Weakly porous sets and Muckenhoupt Ap distance functions

We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight w(x)=dist(x,E)−α belongs to the Muckenhoupt class A1, for some α>0, if and only if E⊂Rn is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of w∈Ap, for 1<p<∞, as well. At the end of the paper, we give an example of a set E⊂R which is not weakly porous but for which w∈Ap∖A1 for every 0<α<1 and 1<p<∞.

A T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, and optimal cancellation conditions, II

We extend the T1 theorem of David and Journé, and the corresponding optimal cancellation conditions of Stein, to pairs of doubling measures, completing the weighted theory begun in [20]. For example, when σ and ω are doubling measures satisfying the classical Muckenhoupt condition, and Kλ is a smooth λ-fractional CZ kernel, we show there exists a bounded operator Tλ:L2(σ)→L2(ω) associated with Kλif and only if there is a positive ‘cancellation’ constant AKλ(σ,ω) so that∫|x−x0|<N|∫ε<|x−y|<NKλ(x,y)dσ(y)|2dω(x)≤AKλ(σ,ω)∫|x0−y|<Ndσ(y),for all 0<ε<N and x0∈Rn, along with the dual inequality. These ‘cancellation’ conditions measure the classical L2 norm of integrals of the kernel over shells, and were shown in [20] to be equivalent to the associated T1 conditions,∫Q|Tλ(1Qσ)|2dω≤TTλ(σ,ω)∫Qdσ,for all cubes Q, and its dual, for doubling measures in the presence of the classical Muckenhoupt condition on a pair of measures.The cancellation conditions can be taken with respect to either cubes or balls, the latter resulting in a direct extension of Stein's characterization. More generally, this is extended to a weak form of Tb theorem.

Structure of a generalized class of weights satisfy weighted reverse Hölder’s inequality

In this paper, we will prove some fundamental properties of the power mean operator Mpg(t)=(1ϒ(t)∫0tλ(s)gp(s)ds)1/p,for t∈I⊆R+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M}_{p}g(t)= \\biggl( \\frac{1}{\\Upsilon(t)} \\int _{0}^{t} \\lambda (s)g^{p} ( s ) \\,ds \\biggr) ^{1/p},\\quad\ ext{for }t\\in \\mathbb{I}\\subseteq \\mathbb{R}_{+}, $$\\end{document}of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on mathbb{I} and Upsilon(t)=int _{0}^{t}lambda ( s ) ,ds. Next, we will study the structure of the generalized class mathcal{U}_{p}^{q}(B) of weights that satisfy the reverse Hölder inequality Mqu≤BMpu,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M}_{q}u\\leq B\\mathcal{M}_{p}u, $$\\end{document}for some p< q, p.qneq 0, and B>1 is a constant. For applications, we will prove some self-improving properties of weights in the class mathcal{U}_{p}^{q}(B) and derive the self improving properties of the weighted Muckenhoupt and Gehring classes.

A posteriori error estimates for mixed approximations of degenerate elliptic problems

We consider the Raviart-Thomas mixed finite element approximation of non-uniform elliptic problems of the form −div(a∇u)=f where a=a(x) is in the Muckenhoupt class A2. We introduce an error estimator based on a local post-processing and prove its efficiency and reliability, generalizing to the degenerate case the theory developed in [24]. Finally, we present some numerical computations for a model problem showing the good behaviour of an adaptive procedure based on our estimator.

On weak-type (1, 1) for averaging type operators

It is known that, due to the fact that L1,∞ is not a Banach space, if (Tj)j is a sequence of bounded operators so thatTj:L1⟶L1,∞, with norm less than or equal to ||Tj|| and ∑j||Tj||<∞, nothing can be said about the operator T=∑jTj. This is the origin of many difficult and open problems. However, if we assume thatTj:L1(u)⟶L1,∞(u),∀u∈A1, with norm less than or equal to φ(||u||A1)||Tj||, where φ is a nondecreasing function and A1 the Muckenhoupt class of weights, then we prove that, essentially,T:L1(u)⟶L1,∞(u),∀u∈A1. We shall see that this is the case of many interesting problems in Harmonic Analysis.

Exact exponents for inclusion of discrete Muckenhoupt classes into Gehring classes and reverse

Exact exponents for inclusion of discrete Muckenhoupt classes into Gehring classes and reverse

Properties of Muckenhoupt and Gehring classes via conformable calculus

In this paper, we study the relationship between the Muckenhoupt class \(\mathcal{A}_{1}^{\alpha }(\mathcal{C})\) and the Gehring class \(\mathcal{G}% _{q}^{\alpha }(\mathcal{K})\) via conformable calculus. We also establish the constants of the classes for the power law functions.

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

Morrey Spaces were first introduced by C.B. Morrey in 1938. Morrey space can be considered as a generalization of the Lebesgue spaces. Morrey spaces were then generalized become the generalized Morrey spaces, the weighted Morrey spaces, and the generalized weighted Morrey spaces. One of the studies on Morrey spaces is the boundedness of certain operators on the spaces. One of the operators is the fractional integral. The boundedness of fractional integrals on the classical Morrey spaces, the weighted Morrey spaces, the generalized Morrey spaces, and the generalized weighted Morrey spaces had been known. One of the extensions of fractional integrals is generalized fractional integral. The operator was bounded on the generalized Morrey spaces. The purpose of this study is to investigate the boundedness of generalized fractional integrals on the generalized weighted Morrey spaces. The weight used is Muckenhoupt class. The results obtained show that the generalized fractional integral is bounded from generalized weighted Morrey space to another generalized weighted Morrey space under some assumptions. The main result obtained then implies the boundedness of the generalized fractional maximal operator on generalized weighted Morrey spaces under the same assumptions.

Laplace transform-type multipliers for Kontorovich–Lebedev transform

In this paper, we investigate Poison kernel associated with Kontorovich–Lebedev transform. We significantly simplify the integral yielding a tractable form which is more useful for explicit calculation. We establish also that Kontorovich–Lebedev multipliers of Laplace transform type are bounded from [Formula: see text] into it self when [Formula: see text] and from [Formula: see text] into [Formula: see text] provided that [Formula: see text] is in the Muckenhoupt class [Formula: see text] on [Formula: see text]. At the end, an integral representation of the operator [Formula: see text] is obtained and its boundedness has been discussed in Lebesgue space.

The Kato-Ponce inequality with polynomial weights

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights \(\langle x\rangle ^a = (1+|x|^2)^{a/2}\) for \(a\ge 0\). We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential \(J^s = (1- \Delta )^{{s}/{2}}\) holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree s. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponce inequality. We also consider a few other variants such as commutator and mixed norm estimates, and analogous conclusions are derived. Our results contain strong-type inequalities for both \(L^1\) and \(L^\infty \) endpoints, which extend several existing results.

Weighted norm inequalities for the maximal functions associated to a critical radius function on variable Lebesgue spaces

In this work we obtain boundedness on weighted variable Lebesgue spaces of some maximal functions that come from the localized analysis considering a critical radius function. This analysis appears naturally in the context of the Schrödinger operator L=−Δ+V in Rd, where V a non-negative potential which satisfying a some specific reverse Hölder condition. We consider new classes of weights that locally behave as the Muckenhoupt class for Lebesgue spaces with variable exponents considered in [6] and actually including them.

On a degenerate singular elliptic problem

AbstractIn this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by where Ω is a bounded smooth domain in with , w belongs to the Muckenhoupt class for some , f is a nonnegative function belonging to some Lebesgue space and .

On weighted compactness of variation operators of commutators of Riesz transforms in the Schrödinger setting

In this paper we establish weighted compactness of the variation operators of the truncations of the commutators of Riesz transforms associated with Schrödinger operators and their adjoints. We consider the new weight class defined in [5] which includes the Muckenhoupt class.

Schrödinger operators on Lie groups with purely discrete spectrum

On a Lie group G, we investigate the discreteness of the spectrum of Schrödinger operators of the form L+V, where L is a subelliptic sub-Laplacian on G and the potential V is a locally integrable function which is bounded from below. We prove general necessary and sufficient conditions for arbitrary potentials, and we obtain explicit characterizations when V is a polynomial on G or belongs to a local Muckenhoupt class. We finally discuss how to transfer our results to weighted sub-Laplacians on G.

On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra

We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class \(\varvec{A}_{\varvec{q}}\) for \(\varvec{q} \in (1,\varvec{\infty })\). We show that the Stokes problem is well-posed for all \(\varvec{q}\). In addition, we show that the finite element Stokes projection is stable on weighted spaces. With the aid of these tools, we provide well-posedness and approximation results to some classes of non-Newtonian fluids.

Weighted estimates for Bochner–Riesz operators on Lorentz spaces

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0&lt;\lambda &lt;{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 &lt; r &lt; \infty$.

Γ-convergence and stochastic homogenization of degenerate integral functionals in weighted Sobolev spaces

We study the$\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals$\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.

Parabolic weighted Sobolev–Poincaré type inequalities

We derive weighted Sobolev–Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic Ap-class, where only the parabolic cubes are involved in the definition.