Boundedness results for bilinear square functions and vector-valued operators on products of Lebesgue, Sobolev, and other spaces of smooth functions are presented. Bilinear vector-valued Calderón-Zygmund operators are introduced and used to obtain bounds for the optimal range of estimates in target Lebesgue spaces including exponents smaller than one.

Function spaces play a significant role in harmonic analysis and partial differential equations. The integral operators that form a bridge between function spaces and partial differential equations are the Calderón-Zygmund operators. It is well known that Calderón-Zygmund operators are bounded on the Lebesgue space L(R) for 1 < p < ∞. It is also known that the Calderón-Zygmund operators are not...

In this paper we extend an inequality of Littlewood concerning the higher variations of functions of bounded Fréchet variations of two variables (bimeasures) to a class of functions that are p-bimeasures, by using the machinery of vector measures. Using random estimates of Kahane-Salem-Zygmund, we show that the inequality is sharp.

It is shown that the Lw,1< p<∞, operator norms of Littlewood–Paley operators are bounded by a multiple of ‖w‖ Ap , where γp = max{1, p/2} 1 p−1 . This improves previously known bounds for all p > 2. As a corollary, a new estimate in terms of ‖w‖Ap is obtained for the class of Calderón–Zygmund singular integrals commuting with dilations.

ABSTRACT: In this paper, the complete convergence and the complete moment convergence of weighted sums for an array of negatively superadditive dependent random variables are established. The results generalize the Baum-Katz theorem on negatively superadditive dependent random variables. In particular, the Marcinkiewicz-Zygmund type strong law of large numbers of weights sums for sequences of n...

Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L → L for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H1(Sn−1).

We obtain Calderón-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.

Given a weight , we consider the space ML which coincides with L p when ∈ Ap . Sharp weighted norm inequalities on ML for the Calderón–Zygmund and Littlewood–Paley operators are obtained in terms of the Ap characteristic of for any 1<p<∞. © 2005 Elsevier Inc. All rights reserved.

In this paper, we obtain some characterizations of the boundedness and compactness of the products of the radial derivative and multiplication operator RMu between mixed norm spaces H(p, q, φ) and Zygmund-type spaces on the unit ball. Mathematics subject classification (2010): 47B38, 47G10, 32A10, 32A18.

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class [Formula: see text] ([Formula: see text]) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results.