1,070 research outputs found
The Marcinkiewicz multiplier condition for bilinear operators
This article is concerned with the question of whether Marcinkiewicz
multipliers on give rise to bilinear multipliers on . We show that this is not always the case. Moreover we
find necessary and sufficient conditions for such bilinear multipliers to be
bounded. These conditions in particular imply that a slight logarithmic
modification of the Marcinkiewicz condition gives multipliers for which the
corresponding bilinear operators are bounded on products of Lebesgue and Hardy
spaces.Comment: 42 page
Multilinear interpolation between adjoint operators
Multilinear interpolation is a powerful tool used in obtaining strong type
boundedness for a variety of operators assuming only a finite set of restricted
weak-type estimates. A typical situation occurs when one knows that a
multilinear operator satisfies a weak estimate for a single index
(which may be less than one) and that all the adjoints of the multilinear
operator are of similar nature, and thus they also satisfy the same weak
estimate. Under this assumption, in this expository note we give a general
multilinear interpolation theorem which allows one to obtain strong type
boundedness for the operator (and all of its adjoints) for a large set of
exponents. The key point in the applications we discuss is that the
interpolation theorem can handle the case . When , weak
has a predual, and such strong type boundedness can be easily obtained by
duality and multilinear interpolation.Comment: 6 pages, no figures, submitted, J. Funct. Ana
A sharp version of the H\"ormander multiplier theorem
We provide an improvement of the H\"ormander multiplier theorem in which the
Sobolev space with integrability index and smoothness
index is replaced by the Sobolev space with smoothness built upon
the Lorentz space
The multilinear Hormander multiplier theorem with a Lorentz-Sobolev condition
In this article, we provide a multilinear version of the H\"ormander
multiplier theorem with a Lorentz-Sobolev space condition. The work is
motivated by the recent result of the first author and Slav\'ikov\'a where an
analogous version of classical H\"ormander multiplier theorem was obtained;
this version is sharp in many ways and reduces the number of indices that
appear in the statement of the theorem. As a natural extension of the linear
case, in this work, we prove that if , then
for certain with . We also show that
the above estimate is sharp, in the sense that the Lorentz-Sobolev space
cannot be replaced by for , , or by for
Multilinear Calder\'on-Zygmund operators on Hardy spaces
It is shown that multilinear Calder\'on-Zygmund operators are bounded on
products of Hardy spaces.Comment: 10 page
Multilinear Fourier Multipliers with Minimal Sobolev Regularity, I
We find optimal conditions on -linear Fourier multipliers to give rise to
bounded operators from a product of Hardy spaces , , to
Lebesgue spaces . The conditions we obtain are necessary and sufficient
for boundedness and are expressed in terms of -based Sobolev spaces. Our
results extend those obtained in the linear case () by Calder\'on and
Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169]
and in the bilinear case () by Miyachi and Tomita
[http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4].
We also prove a coordinate-type H\"ormander integral condition which we use to
obtain certain extreme cases
The H\"ormander Multiplier Theorem III: The complete bilinear case via interpolation
We develop a special multilinear complex interpolation theorem that allows us
to prove an optimal version of the bilinear H\"ormander multiplier theorem
concerning symbols that lie in the Sobolev space , , uniformly over all annuli. More precisely, given a
smoothness index , we find the largest open set of indices
for which we have boundedness for the associated bilinear multiplier operator
from to when ,
L^p bounds for a maximal dyadic sum operator
We prove bounds in the range for a maximal dyadic sum
operator on \rn. This maximal operator provides a discrete multidimensional
model of Carleson's operator. Its boundedness is obtained by a simple twist of
the proof of Carleson's theorem given by Lacey and Thiele, adapted in higher
dimensions by Pramanik and Terwilleger. In dimension one, the \lp boundedness
of this maximal dyadic sum implies in particular an alternative proof of Hunt's
extension of Carleson's theorem on almost everywhere convergence of Fourier
integrals.Comment: 16 pages, no figures, submitted, Math.
boundedness criteria
We obtain a sharp boundedness criterion for a class
of bilinear operators associated with a multiplier given by a signed sum of
dyadic dilations of a given function, in terms of the integrability of
this function; precisely we show that boundedness holds if and only if .
We discuss applications of this result concerning bilinear rough singular
integrals and bilinear dyadic spherical maximal functions.
Our second result is an optimal boundedness criterion
for bilinear operators associated with multipliers with derivatives.
This result provides the main tool in the proof of the first theorem and is
also manifested in terms of the integrability of the multiplier. The
optimal range is which, in the absence of Plancherel's identity on ,
should be compared to in the classical boundedness for
linear multiplier operators
A remark on an endpoint Kato-Ponce inequality
This note introduces bilinear estimates intended as a step towards an
-endpoint Kato-Ponce inequality. In particular, a bilinear version of
the classical Gagliardo-Nirenberg interpolation inequalities for a product of
functions is proved.Comment: To appear in Differential and Integral Equation
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