This chapter is based on the presentation “Generalized bilinear CalderónZygmund operators and applications” delivered by the author during the 2008 February Fourier Talks at the Norbert Wiener Center for Harmonic Analysis and Applications, Department of Mathematics, University of Maryland, College Park, on February 21st. In turn, that presentation was based on material from the article “Weighte...

A new proof of a weighted norm inequality for multilinear singular integrals of Calderón-Zygmund type is presented through a more general estimate involving a sharp maximal function. An application is given to the study of certain multilinear commutators.

The Beppo-Levi native spaces which arise when using polyharmonic splines to interpolate in many space dimensions are embedded in Hölder-Zygmund spaces. Convergence rates for radial basis function interpolation are inferred in some special cases.

Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón– Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type. 2000 Mathematics Subject Classification: 42B20, 42B25.

The boundedness and the compactness of generalized weighted composition operators from weighted Bergman spaces into Zygmund-type spaces are investigated in this paper. Moreover, we give some estimates for the essential norm of these operators.