336 research outputs found
Gaussian heat kernel upper bounds via Phragm\'en-Lindel\"of theorem
We prove that in presence of Gaussian estimates, so-called
Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal
Gaussian upper bounds for the kernels of analytic families of operators on
metric measure spaces
Riesz meets Sobolev
We show that the boundedness, , of the Riesz transform on a
complete non-compact Riemannian manifold with upper and lower Gaussian heat
kernel estimates is equivalent to a certain form of Sobolev inequality. We also
characterize in such terms the heat kernel gradient upper estimate on manifolds
with polynomial growth
Sobolev algebras through heat kernel estimates
On a doubling metric measure space endowed with a "carr\'e du
champ", let be the associated Markov generator and the corresponding homogeneous Sobolev space of
order in , , with norm
. We give sufficient conditions on
the heat semigroup for the spaces to be algebras for the
pointwise product. Two approaches are developed, one using paraproducts
(relying on extrapolation to prove their boundedness) and a second one through
geometrical square functionals (relying on sharp estimates involving
oscillations). A chain rule and a paralinearisation result are also given. In
comparison with previous results ([29,11]), the main improvements consist in
the fact that we neither require any Poincar\'e inequalities nor
-boundedness of Riesz transforms, but only -boundedness of the
gradient of the semigroup. As a consequence, in the range , the
Sobolev algebra property is shown under Gaussian upper estimates of the heat
kernel only.Comment: 62 page
A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces
On doubling metric measure spaces endowed with a strongly local regular
Dirichlet form, we show some characterisations of pointwise upper bounds of the
heat kernel in terms of global scale-invariant inequalities that correspond
respectively to the Nash inequality and to a Gagliardo-Nirenberg type
inequality when the volume growth is polynomial. This yields a new proof and a
generalisation of the well-known equivalence between classical heat kernel
upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash
inequalities. We are able to treat more general pointwise estimates, where the
heat kernel rate of decay is not necessarily governed by the volume growth. A
crucial role is played by the finite propagation speed property for the
associated wave equation, and our main result holds for an abstract semigroup
of operators satisfying the Davies-Gaffney estimates
Heat kernel generated frames in the setting of Dirichlet spaces
Wavelet bases and frames consisting of band limited functions of nearly
exponential localization on Rd are a powerful tool in harmonic analysis by
making various spaces of functions and distributions more accessible for study
and utilization, and providing sparse representation of natural function spaces
(e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in
more general homogeneous spaces, on the interval and ball. The purpose of this
article is to develop band limited well-localized frames in the general setting
of Dirichlet spaces with doubling measure and a local scale-invariant
Poincar\'e inequality which lead to heat kernels with small time Gaussian
bounds and H\"older continuity. As an application of this construction, band
limited frames are developed in the context of Lie groups or homogeneous spaces
with polynomial volume growth, complete Riemannian manifolds with Ricci
curvature bounded from below and satisfying the volume doubling property, and
other settings. The new frames are used for decomposition of Besov spaces in
this general setting
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