84 research outputs found
L p -estimates for the heat semigroup on differential forms, and related problems
We consider a complete non-compact Riemannian manifold satisfying the volume
doubling property and a Gaussian upper bound for its heat kernel (on
functions). Let -- k be the Hodge-de Rham Laplacian on
differential k-forms with k 1. By the Bochner decomposition formula --
k = * + R k. Under the assumption that the negative part
R -- k is in an enlarged Kato class, we prove that for all p [1,
], e --t -- k p--p C(t log t) D 4 (1-- 2
p) (for large t). This estimate can be improved if R -- k is strongly
sub-critical. In general, (e --t -- k) t>0 is not
uniformly bounded on L p for any p = 2. We also prove the gradient estimate e
--t p--p Ct -- 1 p , where is the Laplace-Beltrami
operator (acting on functions). Finally we discuss heat kernel bounds on forms
and the Riesz transform on L p for p > 2
Exact observability, square functions and spectral theory
In the first part of this article we introduce the notion of a
backward-forward conditioning (BFC) system that generalises the notion of
zero-class admissibiliy introduced in [Xu,Liu,Yung]. We can show that unless
the spectum contains a halfplane, the BFC property occurs only in siutations
where the underlying semigroup extends to a group. In a second part we present
a sufficient condition for exact observability in Banach spaces that is
designed for infinite-dimensional output spaces and general strongly continuous
semigroups. To obtain this we make use of certain weighted square function
estimates. Specialising to the Hilbert space situation we obtain a result for
contraction semigroups without an analyticity condition on the semigroup.Comment: 17 page
Bernstein inequalities via the heat semigroup
We extend the classical Bernstein inequality to a general setting including
Schr{\"o}dinger operators and divergence form elliptic operators on Riemannian
manifolds or domains. Moreover , we prove a new reverse inequality that can be
seen as the dual of the Bernstein inequality. The heat kernel will be the
backbone of our approach but we also develop new techniques such as
semi-classical Bernstein inequalities, weak factorization of smooth functions
{\`a} la Dixmier-Malliavin and BM O -- L multiplier results (in
contrast to the usual L -- BM O ones). Also, our approach reveals a
link between the L p-Bernstein inequality and the boundedness on L p of the
Riesz transform. The later being an important subject in harmonic analysis.
2010 Mathematics Subject Classifications: 35P20, 58J50, 42B37 and 47F05.Comment: Revised version, to appear in Math. An
Fractional anisotropic Calder\'on problem on complete Riemannian manifolds
We prove that the metric tensor of a complete Riemannian manifold is
uniquely determined, up to isometry, from the knowledge of a local
source-to-solution operator. This later is associated to a fractional power of
the Laplace-Belrami operator . Our result holds under the condition
that the metric tensor is known in an arbitrary small subdomain. We also
consider the case of closed manifolds and provide an improvement of the main
result in \cite{FGKU
A simple construction of the Anderson operator via its quadratic form in dimensions two and three
We provide a simple construction of the Anderson operator in dimensions two
and three. This is done through its quadratic form. We rely on an exponential
transform instead of the regularity structures or paracontrolled calculus which
are usually used for the construction of the operator. The knowledge of the
form is robust enough to deduce important properties such as positivity and
irreducibility of the corresponding semigroup. The latter property gives
existence of a spectral gap
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