Rearrangement-invariant norm inequalities for convolution operators

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

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.

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

Norm and numerical radius inequalities for Hilbert space operators

Several inequalities for norms of operators are extended to more operators and/or to more norms. These include results of Halmos and Bouldin on approximating a normal operator by another with restricted spectrum, the Powers-Stormer and the van Hemmen-Ando inequalities for the distance between the square roots of two positive operators and also some recent generalisations of these latter results by Kittaneh.

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calder'on and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting questions remain unanswered.

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.

Weighted Norm Inequalities for Certain Pseudo-Differential Operators

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.

The aim of this paper is to study two-weight norm inequalities for fractional maximal functions defined on the upper-half plane. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak-type inequalities. Our characterizations are in terms of Sawyer and Bekolle–Bonami-type conditions. We also obtain a $$\Phi $$ -bump characterization for these maximal functions, where $$\Phi $$ is a Orlicz function. As a consequence, we obtain two-weight norm inequalities for fractional Bergman operators. Finally, we provide some sharp weighted inequalities for the fractional maximal functions.

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.