Search results for "Besov spaces"

On fractional smoothness and Lp-approximation on the Wiener space

2015

Interpolation and approximation in L2(γ)

AbstractAssume a standard Brownian motion W=(Wt)t∈[0,1], a Borel function f:R→R such that f(W1)∈L2, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space B2,qθ(γ)≔(L2(γ),D1,2(γ))θ,q, obtained via the real interpolation method, by the behavior of aX(f(X1);τ)≔∥f(W1)-PXτf(W1)∥L2, where τ=(ti)i=0n is a deterministic time net and PXτ:L2→L2 the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals x0+∑i=1nvi-1(Xti-Xti-1) with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers aX(f(X1);τ) can be used to descr…

The Besov capacity in metric spaces

2016

We study a capacity theory based on a definition of Haj{\l} asz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are $\gamma$-medians, for which we also prove a new version of a Poincar\'e type inequality.

Traces of weighted function spaces: dyadic norms and Whitney extensions

2017

The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.

Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs

2021

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivat…