141 research outputs found
On weighted norm inequalities for the Carleson and Walsh-Carleson operators
We prove bounds for the Carleson operator , its
lacunary version , and its analogue for the Walsh series \W
in terms of the constants for . In particular,
we show that, exactly as for the Hilbert transform,
is bounded linearly by for .
We also obtain bounds in terms of , whose sharpness is
related to certain conjectures (for instance, of Konyagin \cite{K2}) on
pointwise convergence of Fourier series for functions near .
Our approach works in the general context of maximally modulated
Calder\'on-Zygmund operators.Comment: A major revision of arXiv: 1310.3352. In particular, the main result
is proved under a different assumption, and applications to the lacunary
Carleson operator and to the Walsh-Carleson operator are give
On the Convergence of Lacunary Walsh-Fourier Series
We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence
of integers {n_j}. Under a suitable integrability condition, we show that the
sequence converges to f a.e. Integral condition is only slightly larger than
what the sharp integrability condition would be, by a result of Konyagin. The
condition is: f is in L loglog L (logloglog L). The method of proof uses four
ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency
Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical
inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality
for lacunary subsequences of integers, and (4) the extrapolation method of
Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.Comment: 18 pages. v2: Several typos corrected. Final version of the paper,
accepted to LM
A Fefferman-Stein inequality for the Carleson operator
We provide a Fefferman-Stein type weighted inequality for maximally modulated
Calder\'on-Zygmund operators that satisfy \textit{a priori} weak type
unweighted estimates. This inequality corresponds to a maximally modulated
version of a result of P\'erez. Applying it to the Hilbert transform we obtain
the corresponding Fefferman-Stein inequality for the Carleson operator
, that is
for any and any weight function , with bound independent of
. We also provide a maximal-multiplier weighted theorem, a vector-valued
extension, and more general two-weighted inequalities. Our proof builds on a
recent work of Di Plinio and Lerner combined with some results on Orlicz spaces
developed by P\'erez.Comment: Revised version. To appear in Rev. Mat. Ibe
Weighted multiple interpolation and the control of perturbed semigroup systems
In this paper the controllabillity and admissibility of perturbed semigroup
systems are studied, using tools from the theory of interpolation and Carleson
measures. In addition, there are new results on the perturbation of Carleson
measures and on the weighted interpolation of functions and their derivatives
in Hardy spaces, which are of interest in their own right
Pointwise localization and sharp weighted bounds for Rubio de Francia square functions
Let be the Fourier restriction of to an
interval . If is an arbitrary collection of
pairwise disjoint intervals, the square function of is termed the Rubio de Francia square function . This
article proves a pointwise bound for by a sparse operator
involving local -averages. A pointwise bound for the smooth version of
by a sparse square function is also proved. These pointwise
localization principles lead to quantified , and weak ,
norm inequalities for . In particular, the obtained weak
norm bounds are new for and sharp for . The proofs rely
on sparse bounds for abstract balayages of Carleson sequences, local
orthogonality and very elementary time-frequency analysis techniques. The paper
also contains two results related to the outstanding conjecture that
is bounded on if and only if . The conjecture is verified
for radially decreasing even weights, and in full generality for the
Walsh group analogue of .Comment: 27 pages; submitted. v2: typos fixed and more details provide
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
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