508 research outputs found
On a quadratic estimate related to the Kato conjecture and boundary value problems
We provide a direct proof of a quadratic estimate that plays a central role
in the determination of domains of square roots of elliptic operators and, as
shown more recently, in some boundary value problems with boundary data.
We develop the application to the Kato conjecture and to a Neumann problem.
This quadratic estimate enjoys some equivalent forms in various settings. This
gives new results in the functional calculus of Dirac type operators on forms.Comment: Text of the lectures given at the El Escorial 2008 conference.
Revised after the suggestions of the referee. Some historical material added.
A short proof of the main result added under a further assumption. To appear
in the Proceeding
Second order elliptic operators with complex bounded measurable coefficients in , Sobolev and Hardy spaces
Let be a second order divergence form elliptic operator with complex
bounded measurable coefficients. The operators arising in connection with ,
such as the heat semigroup and Riesz transform, are not, in general, of
Calder\'on-Zygmund type and exhibit behavior different from their counterparts
built upon the Laplacian. The current paper aims at a thorough description of
the properties of such operators in , Sobolev, and some new Hardy spaces
naturally associated to .
First, we show that the known ranges of boundedness in for the heat
semigroup and Riesz transform of , are sharp. In particular, the heat
semigroup need not be bounded in if . Then we provide a complete description of {\it all}
Sobolev spaces in which admits a bounded functional calculus, in
particular, where is bounded.
Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces
associated to , that serves the range of beyond .
It includes, in particular, characterizations by the sharp maximal function and
the Riesz transform (for certain ranges of ), as well as the molecular
decomposition and duality and interpolation theorems
Kato's square root problem in Banach spaces
Let be an elliptic differential operator with bounded measurable
coefficients, acting in Bochner spaces of -valued functions
on . We characterize Kato's square root estimates and the -functional calculus of in
terms of R-boundedness properties of the resolvent of , when is a Banach
function lattice with the UMD property, or a noncommutative space. To
do so, we develop various vector-valued analogues of classical objects in
Harmonic Analysis, including a maximal function for Bochner spaces. In the
special case , we get a new approach to the theory of square roots
of elliptic operators, as well as an version of Carleson's inequality.Comment: 44 page
Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L^p
Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as
investigated in the papers by Axelsson, Keith and the second author, provided
insight into the solution of the Kato square-root problem for elliptic
operators in spaces, and allowed for an extension of these estimates to
other systems with applications to non-smooth boundary value problems. In this
paper, we determine conditions under which such operators satisfy conical
square function estimates in a range of spaces, thus allowing us to apply
the theory of Hardy spaces associated with an operator, to prove that they have
a bounded holomorphic functional calculus in those spaces. We also obtain
functional calculi results for restrictions to certain subspaces, for a larger
range of . This provides a framework for obtaining results on
perturbed Hodge Laplacians, generalising known Riesz transform bounds for an
elliptic operator with bounded measurable coefficients, one Sobolev
exponent below the Hodge exponent, and bounds on the square-root of
by the gradient, two Sobolev exponents below the Hodge exponent. Our proof
shows that the heart of the harmonic analysis in extends to for all
, while the restrictions in come from the
operator-theoretic part of the proof. In the course of our work, we
obtain some results of independent interest about singular integral operators
on tent spaces, and about the relationship between conical and vertical square
functions.Comment: 45 pages; mistake correcte
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