in this paper, we show that in each nite dimensional hilbert space, a frame of subspaces is an ultra bessel sequence of subspaces. we also show that every frame of subspaces in a nite dimensional hilbert space has frameness bound.

Applying an upper bound estimate for small L2 ball probability for fractional Brownian motion (fBm), we prove the non-degeneracy of Sobolev pseudo-norms of fBm.

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients a are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume...

Mixing is relevant to many areas of science and engineering, including the pharmaceutical and food industries, oceanography, atmospheric sciences, and civil engineering. In all these situations one goal is to quantify and often then to improve the degree of homogenisation of a substance being stirred, referred to as a passive scalar or tracer. A classical measure of mixing is the variance of th...

We show that every admissible representation of a real reductive group has a canonical system of Sobolev norms parametrized by positive characters of a minimal parabolic subgroup. These norms are compatible with morphisms of representations. Similar statement also holds for representations of reductive p-adic groups. 1991 Mathematics Subject Classification: 22E46 43A70 46E39

In trying to improve Weinstock's results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the ∂-operator on polycylinders for (γ, q)-forms. This generalizes our results for the same operator on poly-cylinders previously obtained, and can be applied to a number of other problems such as the Corona problem. 1. Introduction. Had we...

The aim of this paper is to study Sobolev-type embeddings and their optimality. We work in the frame of rearrangement-invariant norms and unbounded domains. We establish the equivalence of a Sobolev embedding to the boundedness of a certain Hardy operator on some cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the ...

1. Interpolation inequalities. A classical problem in analysis is to understand how “smoothness” controls norms that measure the “size” of functions. Maz’ya recognized in his classic text on Sobolev spaces the intrinsic importance of inequalities that would refine both Hardy’s inequality and Sobolev embedding. Dilation invariance and group symmetry play an essential role in determining sharp co...