Josefina Alvarez

The S′-convolution with singular kernels in the Euclidean case and the product domain case

Abstract We characterize those tempered distributions which are S ′-convolvable with a given class of singular convolution kernels. We study both, the Euclidean case and the product domain case. In the Euclidean case, we consider a class of kernels that includes Riesz kernels, Calderon–Zygmund singular convolution kernels, finite part distributions defined by hypersingular convolution kernels, and Hormander multipliers. In the product domain case, we consider a class of singular kernels introduced by Fefferman and Stein as a generalization of the n -dimensional Hilbert kernel.

Optimal codomains for the Laplace operator and the product Laplace operator

An optimal codomain for an operator P (∂) with fundamental solution E, is a maximal space of distributions T for which it is possible to define the convolution E*T and thus to solve the equation P (∂)S=T. We identify optimal codomains for the Laplace operator in the Euclidean case and for the product Laplace operator in the product domain case. The convolution is understood in the sense of the S′-convolution.

On theT(1) theorem on product domains

AbstractThe direct proof by R. R. Coifman and Y. Meyer of theT(1) Theorem of G. David and J. L. Journé is based on the following result.LetT be an operator associated to a kernelk(x, y) satisfying

Spaces of bounded [lambda]-central mean oscillation, Morrey spaces, and [lambda]-central Carleson measures

|k(y,x) - k(z,x)| \leqslant C\frac{{|y - z|^\delta }}{{|x - z|^{n + \delta } }}, if 2|y - z|< |x - z|

Harmonic extensions of distributions

for some 0<δ≤1. Suppose thatT has the weak boundedness property and thatT(1)∈BMO (ℝn). Then, the operator ∫0∞qtQtTPt2dt/t, defined in the weak sense, is continuous onL2 (ℝn).Here the operatorsqt andQt are convolution operators with functions of integral 0, andPt is also a convolution operator similar to the Poisson transform.We prove a product domain version of this result.