158 research outputs found
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators
This is the third part of a series of four articles on weighted norm
inequalities, off-diagonal estimates and elliptic operators. For in some
class of elliptic operators, we study weighted norm inequalities for
singular 'non-integral' operators arising from ; those are the operators
for bounded holomorphic functions , the Riesz transforms
(or ) and its inverse
, some quadratic functionals and of
Littlewood-Paley-Stein type and also some vector-valued inequalities such as
the ones involved for maximal -regularity. For each, we obtain sharp or
nearly sharp ranges of using the general theory for boundedness of Part I
and the off-diagonal estimates of Part II. We also obtain commutator results
with BMO functions.Comment: 38 pages. Third of 4 paper
Weighted norm inequalities for fractional operators]
We prove weighted norm inequalities for fractional powers of elliptic
operators together with their commutators with BMO functions, encompassing what
is known for the classical Riesz potentials and elliptic operators with
Gaussian domination by the classical heat operator. The method relies upon a
good- method that does not use any size or smoothness estimates for
the kernels.Comment: accepted in Indiana University Mathematicla Journal. A thorough
reorganisation has been done on suggestions from the refere
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type
This is the second part of a series of four articles on weighted norm
inequalities, off-diagonal estimates and elliptic operators. We consider a
substitute to the notion of pointwise bounds for kernels of operators which
usually is a measure of decay. This substitute is that of off-diagonal
estimates expressed in terms of local and scale invariant estimates.
We propose a definition in spaces of homogeneous type that is stable under
composition. It is particularly well suited to semigroups. We study the case of
semigroups generated by elliptic operators.Comment: 40 pages. Second of 4 papers. Can be read independentl
Vertical versus conical square functions
We study the difference between vertical and conical square functions in the
abstract and also in the specific case where the square functions come from an
elliptic operator.Comment: 21 page
Sharp weighted estimates for approximating dyadic operators
We give a new proof of the sharp weighted inequality ||T||_{L^2(w)}
\leq c [w]_{A_2} where is the Hilbert transform, a Riesz transform, the
Beurling-Ahlfors operator or any operator that can be approximated by Haar
shift operators. Our proof avoids the Bellman function technique and two weight
norm inequalities. We use instead a recent result due to A. Lerner to estimate
the oscillation of dyadic operators.Comment: To appear in the Electronic Research Announcements in Mathematical
Science
Weighted norm inequalities, off-diagonal estimates and elliptic operators
We give an overview of the generalized Calder\'on-Zygmund theory for
"non-integral" singular operators, that is, operators without kernels bounds
but appropriate off-diagonal estimates. This theory is powerful enough to
obtain weighted estimates for such operators and their commutators with \BMO
functions. off-diagonal estimates when play an important
role and we present them. They are particularly well suited to the semigroups
generated by second order elliptic operators and the range of exponents
rules the theory for many operators constructed from the semigroup and
its gradient. Such applications are summarized.Comment: survey for the El Escorial 2008 proceeding
Lp self improvement of generalized Poincaré inequalities on spaces of homogeneous type
International audienceIn this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincaré or Sobolev-Poincaré inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup with being the Laplace-Beltrami operator. We obtain generalized Poincaré inequalities with oscillations that involve the semigroup and with right hand sides containing either or
Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem
It is well-known that quantitative, scale invariant absolute continuity (more
precisely, the weak- property) of harmonic measure with respect to
surface measure, on the boundary of an open set with Ahlfors-David regular boundary, is equivalent to the
solvability of the Dirichlet problem in , with data in
for some . In this paper, we give a geometric
characterization of the weak- property, of harmonic measure, and
hence of solvability of the Dirichlet problem for some finite . This
characterization is obtained under background hypotheses (an interior corkscrew
condition, along with Ahlfors-David regularity of the boundary) that are
natural, and in a certain sense optimal: we provide counter-examples in the
absence of either of them (or even one of the two, upper or lower,
Ahlfors-David bounds); moreover, the examples show that the upper and lower
Ahlfors-David bounds are each quantitatively sharp.Comment: This paper is a combination of arXiv:1712.03696 and arXiv:1803.07975
To appear in Invent. Mat
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