In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between Besov defined by differences and Fourier-analytical decompositions as well Sobolev/Triebel-Lizorkin spaces; Various new characterizations for norms in terms different K-functionals. For instance, derive...

We give an example of a linear, time-dependent, Schrödinger operator with optimal growth Sobolev norms. The construction is explicit, and relies on comprehensive study the linear Lowest Landau Level equation time-dependent potential.

We investigate the relationship between compactness of embeddings Sobolev spaces built upon rearrangement-invariant into endowed with $d$-Ahlfors measures under certain restriction on speed their decay o

We prove higher order concentration bounds for functions on Stiefel and Grassmann manifolds equipped with the uniform distribution. This partially extends previous work unit sphere. Technically, our results are based logarithmic Sobolev techniques measures manifolds. Applications include Hanson–Wright type inequalities certain distance between subspaces of Rn.

We investigate properties of subspaces of L2 spanned by subsets of a finite orthonormal system bounded in the L∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L1 and the L2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L2 , complementary to...

We consider Hardy operators on the half-space, that is, ordinary and fractional Schrödinger with potentials given by appropriate power of distance to boundary. show scales homogeneous Sobolev spaces generated Laplacian are comparable each other when coupling constant is not too large in a quantitative sense. Our results extend those whole Euclidean space rely recent heat kernel bounds.

The aim of this paper is the interpolation between nullspaces of a fixed partial differential operator in different Sobolev spaces, for example the subspaces of divergence-free functions. We present several methods of proofs, which allow for handling various operators in different geometries, with or without boundary conditions. Our application is the optimal approximation of divergence-free fu...

We study the convergence of the finite element method with Lagrange multipliers for approximately solving the Dirichlet problem for a second-order elliptic equation in a plane domain with piecewise smooth boundary. Assuming mesh refinements around the corners, we construct families of boundary subspaces that are compatible with triangular Lagrange elements in the interior, and we carry out the ...

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function f ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on...