Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

Given function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

Weighted Norm Inequalities for Certain Pseudo-Differential Operators

We prove in this paper one weight norm inequalities for some positive Bergman-type operators.

In this article we define two-wavelet localization operators corresponding to an irreducible and square-integrable representation of a locally compact Hausdorff group on a Hilbert space. The group structure admitting an irreducible and square-integrable representation which is related to β-Stockwell transform, that we shall use in this article β ∈ R have been introduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis, A group representation related to the Stockwell transform, Indiana Univ. Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norm inequalities of these two-wavelet localization operators are established. The traces and the trace class norm inequalities of the trace class two-wavelet localization operators are given.

Let $({\mathcal~X},~d,~\mu)$ be a metric measurespace satisfying the geometrically doubling condition. We establish some weak type weighted norm inequalitiesfor the multilinear fractional integral operatorand the multilinear fractional maximal function.As an application, we also establish a weak type weighted norm inequalityon Morrey spaces for the multilinear fractional integral.

Conditions on nonnegative weight functions $u(x)$ and $\upsilon (x)$ are given which ensure that an inequality of the form ${(\smallint {\left | {Tf(x)} \right |^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left | {f(x)} \right |^p}\upsilon (x)\;dx)^{1/p}}$ holds for $1 \leqslant q < p < \infty$, where $T$ is an integral operator of the form $\int _{ - \infty }^x {K(x,y)f(y)dy}$ or $\int _x^\infty {K(y,x)f(y)\;dy}$ and $C$ a constant independent of $f$. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for $1 \leqslant p \leqslant q < \infty$ were obtained in [1].

In this paper we discuss the possible generalizations of a lifting theorem of a 2 × 2 2 \times 2 matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.

Let T be a positive linear operator defined for nonnegative functions on a rj-finite measure space {X,m,fi).Given 1 < p < oo and a nonnegative weight function w on X , it is shown that there exists a nonnegative weight function v , finite /-almost everywhere on X , such that (1) I \Tf)*wdfi< j fvd/i, for all/>0, J x J x tere exists <j> posi ( h if and only if there exists <j> positive /-almost everywhere on X with(2) / {T(t>)pwd<oo J xIn case (2) holds, we may take v = 4>l~pT'[(T(t>)P~]w] in (1).This partially answers a question of B. Muckenhoupt in [5].Applications to some specific operators are also given.

It is found class of pairs of exponents (p(\cdot), q(\cdot)) such that for pairs of Banach function spaces (L^{p(\cdot)}(ℝ^n), L^{q(\cdot)}(ℝ^n)) weak Minkowski’s inequality holds. Also some conditions which ensure the boundednness of maximal operator M_φ:L^{p(\cdot)}(ℝ^n) → L^{q(\cdot)}_w(ℝ^n) are found, when for the pair of Banach function spaces (L^{p(\cdot)}(ℝ^n), L^{q(\cdot)}(ℝ^n)) holds weak Minkowski’s inequality.

This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of “testing type,” like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type $(L^{p}, L^{q})$ estimates. We show that in such a space it is possible to characterize these estimates by testing them only over “cubes”. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

We extend our recent results concerning norm inequalities on cones to include the case when 0 &gt; p &gt; 1 0 &gt; p &gt; 1 .

Norm and numerical radius inequalities for Hilbert space operators

Preface

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type