18 research outputs found
A Weighted Estimate for the Square Function on the Unit Ball in \C^n
We show that the Lusin area integral or the square function on the unit ball
of \C^n, regarded as an operator in weighted space has a linear
bound in terms of the invariant characteristic of the weight. We show a
dimension-free estimate for the ``area-integral'' associated to the weighted
norm of the square function. We prove the equivalence of the classical
and the invariant classes.Comment: 11 pages, to appear in Arkiv for Matemati
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
On the Scientific Work of Konstantin Ilyich Oskolkov
This chapter is a brief account of the life and the scientific work of K.I. Oskolkov