Weighted Norm Inequalities Research Articles

Parabolic weighted norm inequalities and partial differential equations

We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.

The weighted norm inequalities of integral operators with special types of kernels

In this paper, the weighted norm inequalities of the following special type of integral operators are investigated:where or . In particular, those established by E. T. Copson, Hardy-Littlewood-Pólya, S. Lai, N. Levinson, E. R. Love, B. G. Pachpatte and Yang-Hwang will be derived.

Real interpolation with weighted rearrangement invariant Banach function spaces

Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular , weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the K-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of H ∞-functional calculus to this general setting.

Weighted norm inequalities for Toeplitz type operators associated to generalized Calderón-Zygmund operators.

Let T_1 be a generalized Calderón–Zygmund operator or pm I ( the identity operator), let T_2 and T_3 be the linear operators, and let T_3=pm I. Denote the Toeplitz type operator by \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T^b=T_1M^bI_\\alpha T_2+T_3I_\\alpha M^b T_4, \\end{aligned}$$\\end{document}Tb=T1MbIαT2+T3IαMbT4,where M^bf=bf, and I_alpha is fractional integral operator. In this paper, we establish the sharp maximal function estimates for T^b when b belongs to weighted Lipschitz function space, and the weighted norm inequalities of T^b on weighted Lebesgue space are obtained.

TWO‐WEIGHT NORM INEQUALITIES FOR VECTOR‐VALUED OPERATORS

We study two weight norm inequalities for a vector-valued operator from a weighted $L^p(\sigma)$-space to mixed norm $L^q_{l^s}(\mu)$ spaces, $1<q<p$. We apply these results to the boundedness of Wolff's potentials.

Vector-Valued Singular Integrals with Non-smooth Kernels on Spaces of Homogeneous Type

In this article, we obtain weighted norm inequalities for vector-valued singular integrals with non-smooth kernels on spaces of homogeneous type. Vector-valued commutators of singular integrals and BMO functions are also studied.

Extrapolation and weighted norm inequalities in the variable Lebesgue spaces

Extrapolation and weighted norm inequalities in the variable Lebesgue spaces

The convolution for the Kontorovich–Lebedev transform revisited

In this paper we prove a Young type theorem and also a class of weighted norm inequalities for the Kontorovich–Lebedev convolution. As applications we solve some parabolic differential equations (linear and nonlinear) and a convolution integral equation, and give estimates of the obtained solutions via the weighted norm inequalities.

The Mackenhoupt Condition and an Estimating Problem

The paper considers a connection between weighted norm inequalities for the Hilbert transform with matrix valued weights and an estimating problem. A connection of the vector Muckenhoupt condition on the spectral density of the stationary noise and a possibility to transform a difficult estimating problem to another well-studied problem is established. Bibliography: 12 titles

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $\mu(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincar\'e inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities we had obtained in an earlier paper, and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.

Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals

In the present paper, we study the weighted norm inequalities for higher-order commutators formed by a class of one-sided oscillatory singular integrals and $BMO$ functions. We obtain that the boundedness of these commutators can be deduced by that of one-sided Calderon-Zygmund singular integral operators.

Weighted norm inequalities for Toeplitz type operator associated to singular integral operator with non-smooth kernel

Weighted norm inequalities for Toeplitz type operator associated to singular integral operator with non-smooth kernel

Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context

We give extrapolation results starting from weighted inequalities between Lebesgue and Lipschitz spaces, given by(0.1)supB⁡‖wχB‖∞|B|1+δn∫B|f(x)−mB(f)|dx≤C‖gw‖s, where 1<β<∞, 0≤δ<1, δn=1β−1s, f and g are two measurable functions and w belongs to a suitable class of weights. From this hypothesis we obtain a large class of inequalities including weighted Lp–Lq estimates and weighted Lp-Lipschitz integral spaces, generalizing well known results for certain sublinear operator. From the same hypothesis (0.1) we obtain the corresponding results in the setting of variable exponent spaces. Particularly, we obtain estimates of the type Lp(⋅)-variable versions of Lipschitz integral spaces. We also prove a surprising weighted inequalities of the type Lp(⋅)–Lq(⋅). An important tool in order to get the main results is an improvement of an estimate due to Calderon and Scott in [1], which allows us to relate different integral Lipschitz spaces. Our results are new even in the classical context of constant exponents.

An end-point global gradient weighted estimate for quasilinear equations in non-smooth domains

A weighted norm inequality involving A 1 weights is obtained at the natural exponent for gradients of solutions to quasilinear elliptic equations in Reifenberg flat domains. Certain gradient estimates in Lorentz–Morrey spaces below the natural exponent are also obtained as a consequence of our analysis.

Two weight norm inequalities for the bilinear fractional integrals

In this paper, we give a characterization of the two weight strong and weak type norm inequalities for the bilinear fractional integrals. Namely, we give the characterization of the following inequalities, \[ \|\mathcal I_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^q(w)} \le \mathscr N \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)} \] and \[ \|\mathcal I_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^{q,\infty}(w)} \le \mathscr N_{\textup{weak}} \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)}, \] when $q\ge p_1, p_2>1$ and $p_1+p_2\ge p_1p_2$.

Weighted norm inequalities with multiple-weight for singular integral operators with non-smooth kernels

By sharp maximal function, we establish a weighted estimate with multiple-weight for the multilinear singular integral operators with non-smooth kernels.

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p n (x, y) associated to P n satisfy Gaussian upper bounds but do not assume they satisfy the Holder continuity property and the temporal regularity. Denote by L = I − P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L p norm inequalities for the spectral multipliers of L. We also obtain the weighted L p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.

TWO WEIGHT INEQUALITIES FOR THE BERGMAN PROJECTION WITH DOUBLING MEASURES

In this note we show that the problem of characterizing two weight norm inequalities for the Bergman projection under the assumption of doubling measures admits a surprisingly simple solution. Our principal discovery is that Sawyer-type testing can be avoided. This stands in sharp contrast with the current folklore in two weight theory and with the corresponding result for the Hilbert transform.

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators and their Iterated Commutators

Applying the new class of multiple weights functions and new sharp maximal functions, we obtain the pointwise estimates, strong type and weak end-point estmates for certain classes of multilinear operators and their iterated commutators with new BMO functions.

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let denote a weight in which belongs to the Muckenhoupt class and let denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure . The sharp Tauberian constant of with respect to , denoted by , is defined by In this paper, we show that the Solyanik estimate $$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator and a weight : We show that we have if and only if . As a corollary of our methods we obtain a quantitative embedding of into .