45 research outputs found
Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions
This article is the last in a series of three papers, whose scope is to give
new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and
Meyer. Here we extend the results of the previous two papers to the polydisc
setting. In particular, we solve completely an open question of Coifman from
the early eighties.Comment: 25 page
Sparse domination via the helicoidal method
Using exclusively the localized estimates upon which the helicoidal method
was built, we show how sparse estimates can also be obtained. This approach
yields a sparse domination for multiple vector-valued extensions of operators
as well. We illustrate these ideas for an -linear Fourier multiplier whose
symbol is singular along a -dimensional subspace of , where , and for the
variational Carleson operator.Comment: 60 page
Multiple Vector Valued Inequalities via the Helicoidal Method
We develop a new method of proving vector-valued estimates in harmonic
analysis, which we like to call "the helicoidal method". As a consequence of
it, we are able to give affirmative answers to some questions that have been
circulating for some time. In particular, we show that the tensor product between the bilinear Hilbert transform and a paraproduct
satisfies the same estimates as the itself, solving
completely a problem introduced in a paper of Muscalu, Pipher, Tao and Thiele.
Then, we prove that for "locally exponents" the corresponding vector
valued satisfies (again) the same estimates as the
itself. Before the present work there was not even a single example of
such exponents.
Finally, we prove a bi-parameter Leibniz rule in mixed norm spaces,
answering a question of Kenig in nonlinear dispersive PDE.Comment: 56 pages, 7 figure
Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited I. First commutator and generalizations
This article is the first in a series of three papers, whose scope is to give
new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and
Meyer. Here we treat the case of the first commutator and some of its
generalizations.Comment: 23 pages, 1 figur