In this paper we extend a Calderón-Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces. 2000 Mathematics Subject Classification: 42B15, 42B20, 42B25, 42B35.

In this work, we give several characterizations of the bounded and the compact weighted composition operators from the Lipschitz space into the Zygmund space. Mathematics subject classification (2010): Primary 47B33; secondary 30H05.

In this work, we continue the study of precise functional properties of a linear operator linked with Boltzmann quadratic operator, started in Part I. This is done for singular cross-sections. In particular, we show Calderon-Zygmund type estimates.

We investigate the relationships between the Marcinkiewicz-Zygmund-type inequalities and certain shifted average operators. Applications to the mean boundedness of a quasi-interpolatory operator in the case of trigonometric polynomials, Jacobi polynomials, and Freud polynomials are presented.

In this paper we obtain some limit cases of inequalities of Ul’yanov-type for modulus of smoothness between Lorentz-Zygmund spaces on Tn . Corresponding embedding theorems for the Besov spaces are investigated. Mathematics subject classification (2010): 41A17, 46E30, 46E35, 46M35.

We prove that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability and apply to many operators such as m-linear Calderón-Zygmund operators and their maximal counterparts.

The invertible, compact and Fredholm multiplication operators on generalized Lorentz-Zygmund (GLZ) spaces Lp,q;α, 1 < p ≤ ∞, 1 ≤ q ≤ ∞, α in the Euclidean space R, are characterized in this paper.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L spaces if and only if the variable exponent p(x) ∼ const.

A systematic treatment of multilinear Calderón-Zygmund operators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, and a variety of results regarding multilinear multiplier operators.