2,903 research outputs found
Extrapolation and Factorization
A modestly revised version of lecture notes that were distributed to
accompany my four lectures at the 2017 Spring School of Analysis at Paseky,
sponsored by Charles University, Prague. They are an introductory survey of
Rubio de Francia extrapolation, Jones factorization, and applications
Two weight norm inequalities for fractional integral operators and commutators
In these lecture notes we describe some recent work on two weight norm
inequalities for fractional integral operators, also known as Riesz potentials,
and for commutators of fractional integrals. These notes are based on three
lectures delivered at the 6th International Course of Mathematical Analysis in
Andalucia, held in Antequera, Spain, September 8-12, 2014. They are, however,
greatly expanded to include both new results and many details that I did not
present in my lectures due to time constraints
The Boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type
Given a space of homogeneous type we give sufficient conditions on a variable
exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps
{L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the
endpoint case we also prove the corresponding weak type inequality. As an
application we prove norm inequalities for the fractional integral operator
{I_{\eta}}. Our proof for the fractional maximal operator uses the theory of
dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting
it is simpler than existing proofs. For the fractional integral operator we
extend a pointwise inequality of Welland to spaces of homogeneous type. Our
work generalizes results from the Euclidean case and extends recent work by
Adamowicz, et al. on the Hardy-Littlewood maximal operator on spaces of
homogeneous type.Comment: 31 page
Kato-Ponce inequalities on weighted and variable Lebesgue spaces
We prove fractional Leibniz rules and related commutator estimates in the
settings of weighted and variable Lebesgue spaces. Our main tools are uniform
weighted estimates for sequences of square-function-type operators and a
bilinear extrapolation theorem. We also give applications of the extrapolation
theorem to the boundedness on variable Lebesgue spaces of certain bilinear
multiplier operators and singular integrals.Comment: Revised version contains corrections from referees report and new
section Kato-Ponce inequalities in Lorentz and Morrey space
Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces
We extend the theory of weighted norm inequalities on variable Lebesgue
spaces to the case of bilinear operators. We introduce a bilinear version of
the variable \A_\pp condition, and show that it is necessary and sufficient
for the bilinear maximal operator to satisfy a weighted norm inequality. Our
work generalizes the linear results of the first author, Fiorenza and
Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the
bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical
Lebesgue spaces. As an application we prove weighted norm inequalities for
bilinear singular integral operators in the variable Lebesgue spaces.Comment: Revised based on anonymous referee's reports. A number of typos and
small errors corrected. One conjecture added to introductio
Regularity results for weak solutions of elliptic PDEs below the natural exponent
We prove regularity estimates for weak solutions to the Dirichlet problem for
a divergence form elliptic operator. We give estimates for the second
derivative for . Our work generalizes results due to Miranda [28].Comment: Several results corrected and revised. We would like to thank Doyoon
Kim for pointing out an error in the previous versio
Regularity of solutions to degenerate -Laplacian equations
We prove regularity results for solutions of the equation , where is a family of
vector fields satisfying H\"ormander's ellipticity condition, is an
symmetric matrix that satisfies degenerate ellipticity conditions.
If the degeneracy is of the form , then we show that
solutions are locally H\"older continuous. If the degeneracy is of the form ,where depends on the homogeneous dimension, then the
solutions are continuous almost everywhere, and we give examples to show that
this is the best result possible. We give an application to maps of finite
distortion.Comment: v3 several revisions. Final version. To appear in JMA
Poincare Inequalities and Neumann Problems for the p-Laplacian
We prove an equivalence between weighted Poincare inequalities and the
existence of weak solutions to a Neumann problem related to a degenerate p-
Laplacian. The Poincare inequalities are formulated in the context of
degenerate Sobolev spaces defined in terms of a quadratic form, and the
associated matrix is the source of the degeneracy in the p-Laplacian
On the Kato problem and extensions for degenerate elliptic operators
We study the Kato problem for degenerate divergence form operators. This was
begun by Cruz-Uribe and Rios who proved that given an operator , where and is a -degenerate elliptic measure
(i.e, with an bounded, complex-valued, uniformly
elliptic matrix), then satisfies the weighted estimate
. Here we solve the
-Kato problem: under some additional conditions on the weight , the
following unweighted -Kato estimates hold
This extends the celebrated solution to the Kato conjecture by Auscher,
Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator
to have some degeneracy in its ellipticity. For example, we consider the family
of operators , where
is any bounded, complex-valued, uniformly elliptic matrix. We prove that
there exists , depending only on dimension and the ellipticity
constants, such that This
gives a range of 's for which the classical Kato square root
is an interior point.
Our main results are obtained as a consequence of a rich Calder\'on-Zygmund
theory developed for some operators associated with . These results, which
are of independent interest, establish estimates on , and also on
with , for the associated semigroup, its
gradient, the functional calculus, the Riesz transform, and square functions.
As an application, we solve some unweighted -Dirichlet, Regularity and
Neumann boundary value problems for degenerate elliptic operators
Logarithmic bump conditions for Calder\'on-Zygmund Operators on spaces of homogeneous type
We establish two-weight norm inequalities for singular integral operators
defined on spaces of homogeneous type. We do so first when the weights satisfy
a double bump condition and then when the weights satisfy separated logarithmic
bump conditions. Our results generalize recent work on the Euclidean case, but
our proofs are simpler even in this setting. The other interesting feature of
our approach is that we are able to prove the separated bump results (which
always imply the corresponding double bump results) as a consequence of the
double bump theorem
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