172 research outputs found
Paley-Littlewood decomposition for sectorial operators and interpolation spaces
We prove Paley-Littlewood decompositions for the scales of fractional powers
of -sectorial operators on a Banach space which correspond to
Triebel-Lizorkin spaces and the scale of Besov spaces if is the classical
Laplace operator on We use the -calculus,
spectral multiplier theorems and generalized square functions on Banach spaces
and apply our results to Laplace-type operators on manifolds and graphs,
Schr\"odinger operators and Hermite expansion.We also give variants of these
results for bisectorial operators and for generators of groups with a bounded
-calculus on strips.Comment: 2nd version to appear in Mathematische Nachrichten, Mathematical News
/ Mathematische Nachrichten, Wiley-VCH Verlag, 201
Spectral multiplier theorems and averaged R-boundedness
Let be a -sectorial operator with a bounded
-calculus for some e.g. a
Laplace type operator on where is a
manifold or a graph. We show that has a H{\"o}rmander functional calculus
if and only if certain operator families derived from the resolvent the semigroup the wave operators or the
imaginary powers of are -bounded in an -averaged sense. If
is an space with -boundedness reduces
to well-known estimates of square sums.Comment: Error in the title correcte
The Daugavet equation for operators not fixing a copy of
We prove the norm identity , which is known as the
Daugavet equation, for operators on not fixing a copy of ,
where is a compact metric space without isolated points
Stochastic maximal -regularity
In this article we prove a maximal -regularity result for stochastic
convolutions, which extends Krylov's basic mixed -inequality for the
Laplace operator on to large classes of elliptic operators,
both on and on bounded domains in with
various boundary conditions. Our method of proof is based on McIntosh's
-functional calculus, -boundedness techniques and sharp
-square function estimates for stochastic integrals in -spaces.
Under an additional invertibility assumption on , a maximal space--time
-regularity result is obtained as well.Comment: Published in at http://dx.doi.org/10.1214/10-AOP626 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the R-boundedness of stochastic convolution operators
The -boundedness of certain families of vector-valued stochastic
convolution operators with scalar-valued square integrable kernels is the key
ingredient in the recent proof of stochastic maximal -regularity,
, for certain classes of sectorial operators acting on spaces
, . This paper presents a systematic study of
-boundedness of such families. Our main result generalises the
afore-mentioned -boundedness result to a larger class of Banach lattices
and relates it to the -boundedness of an associated class of
deterministic convolution operators. We also establish an intimate relationship
between the -boundedness of these operators and the boundedness of
the -valued maximal function. This analysis leads, quite surprisingly, to an
example showing that -boundedness of stochastic convolution operators fails
in certain UMD Banach lattices with type .Comment: to appear in Positivit
Embedding vector-valued Besov spaces into spaces of -radonifying operators
It is shown that a Banach space has type if and only for some (all)
the Besov space embeds into the
space \g(L^2(\R^d),E) of \g-radonifying operators . A
similar result characterizing cotype is obtained. These results may be
viewed as -valued extensions of the classical Sobolev embedding theorems.Comment: To appear in Mathematische Nachrichte
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