998 research outputs found
Square Function Estimates and Functional Calculi
In this paper the notion of an abstract square function (estimate) is
introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is
a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators.
By the seminal work of Kalton and Weis, this definition is a coherent
generalisation of the classical notion of square function appearing in the
theory of singular integrals. Given an abstract functional calculus (E, F, Phi)
on a Banach space X, where F (O) is an algebra of scalar-valued functions on a
set O, we define a square function Phi_gamma(f) for certain H-valued functions
f on O. The assignment f to Phi_gamma(f) then becomes a vectorial functional
calculus, and a "square function estimate" for f simply means the boundedness
of Phi_gamma(f). In this view, all results linking square function estimates
with the boundedness of a certain (usually the H-infinity) functional calculus
simply assert that certain square function estimates imply other square
function estimates. In the present paper several results of this type are
proved in an abstract setting, based on the principles of subordination,
integral representation, and a new boundedness concept for subsets of Hilbert
spaces, the so-called ell-1 -frame-boundedness. These abstract results are then
applied to the H-infinity calculus for sectorial and strip type operators. For
example, it is proved that any strip type operator with bounded scalar
H-infinity calculus on a strip over a Banach space with finite cotype has a
bounded vectorial H-infinity calculus on every larger strip.Comment: 49
The stochastic Weiss conjecture for bounded analytic semigroups
Suppose -A admits a bounded H-infinity calculus of angle less than pi/2 on a
Banach space E with Pisier's property (alpha), let B be a bounded linear
operator from a Hilbert space H into the extrapolation space E_{-1} of E with
respect to A, and let W_H denote an H-cylindrical Brownian motion. Let
gamma(H,E) denote the space of all gamma-radonifying operators from H to E. We
prove that the following assertions are equivalent:
(i) the stochastic Cauchy problem dU(t) = AU(t)dt + BdW_H(t) admits an
invariant measure on E;
(ii) (-A)^{-1/2} B belongs to gamma(H,E);
(iii) the Gaussian sum \sum_{n\in\mathbb{Z}} \gamma_n 2^{n/2} R(2^n,A)B
converges in gamma(H,E) in probability.
This solves the stochastic Weiss conjecture proposed recently by the second
and third named authors.Comment: 17 pages; submitted for publicatio
On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological
sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of
the phase space of classical General Relativity to spatially homogeneous
situations which is then quantized by the methods of LQG. Thus, LQC is a
quantum mechanical toy model (finite number of degrees of freedom) for LQG(a
genuine QFT with an infinite number of degrees of freedom) which provides
important consistency checks. However, it is a non trivial question whether the
predictions of LQC are robust after switching on the inhomogeneous fluctuations
present in full LQG. Two of the most spectacular findings of LQC are that 1.
the inverse scale factor is bounded from above on zero volume eigenstates which
hints at the avoidance of the local curvature singularity and 2. that the
Quantum Einstein Equations are non -- singular which hints at the avoidance of
the global initial singularity. We display the result of a calculation for LQG
which proves that the (analogon of the) inverse scale factor, while densely
defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in
full LQG, if curvature singularity avoidance is realized, then not in this
simple way. In fact, it turns out that the boundedness of the inverse scale
factor is neither necessary nor sufficient for curvature singularity avoidance
and that non -- singular evolution equations are neither necessary nor
sufficient for initial singularity avoidance because none of these criteria are
formulated in terms of observable quantities.After outlining what would be
required, we present the results of a calculation for LQG which could be a
first indication that our criteria at least for curvature singularity avoidance
are satisfied in LQG.Comment: 34 pages, 16 figure
Discrete maximal regularity of time-stepping schemes for fractional evolution equations
In this work, we establish the maximal -regularity for several time
stepping schemes for a fractional evolution model, which involves a fractional
derivative of order , , in time. These schemes
include convolution quadratures generated by backward Euler method and
second-order backward difference formula, the L1 scheme, explicit Euler method
and a fractional variant of the Crank-Nicolson method. The main tools for the
analysis include operator-valued Fourier multiplier theorem due to Weis [48]
and its discrete analogue due to Blunck [10]. These results generalize the
corresponding results for parabolic problems
- âŠ