76 research outputs found
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
Sparse bilinear forms for Bochner Riesz multipliers and applications
We use the very recent approach developed by Lacey in [23] and extended by
Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by
a sparse bilinear form. In this way, new quantitative weighted estimates, as
well as vector-valued inequalities are deduced
A bilinear Rubio de Francia inequality for arbitrary squares
We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane provided . More exactly, we show that the above operator maps whenever are in the ``local " range, i.e. , , and . Note that we allow for negative values of , which correspond to quasi-Banach spaces
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