Bogdan Bojarski

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class Wlocm,p (ℝn) or Wm,p(Q) for some cube Q ⊂ ℝn if and only if the formal (m − 1)-Taylor remainder Rm−1f(x,y) of f satisfies the pointwise inequality |Rm−1f(x,y)| ≤ |x − y|m [a(x) + a(y)] for some a ε Lp(Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝn.

On existence and representation of solutions for general degenerate Beltrami equations

We study the general degenerate Beltrami equations in the unit disk in Given an arbitrary analytic function with isolated singularities in we find criteria for the existence of a solution of the form where stands for a regular himeomorphism of onto itself.

Higher Order Sobolev-Type Spaces on the Real Line

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.

The geometry of Kato Grassmannians

We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.