Mario Milman

Extrapolation theory: new results and applications

Extending earlier work by Jawerth and Milman, we develop in detail @S^(^p^) and @D^(^p^) methods of extrapolation. As an application we prove general forms of Yanos extrapolation theorem. Applications to logarithmic Sobolev inequalities, integrability of maps of finite distortion and logarithmic Sobolev spaces are given.

ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

Let Ω be an open domain in ℝn, let k∈ℕ,

Sobolev inequalities: Symmetrization and self-improvement via truncation☆

p\le \frac{n}{k}

POINTWISE SYMMETRIZATION INEQUALITIES FOR SOBOLEV FUNCTIONS AND APPLICATIONS

. Using a natural extension of the L(p, q) spaces and a new form of the Polya–Szego symmetrization principle, we extend the sharp version of the Sobolev embedding theorem:

Symmetrization inequalities and Sobolev embeddings

W_0^{k, p} (\Omega)\subset L (\frac{np}{n -kp}, p) to the limiting value

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

p =\frac{n}{k}

On the fundamental lemma of interpolation theory

. This result extends a recent result in [3] for the case k=1. More generally, if Y is a r.i. space satisfying some mild conditions, it is shown that

Commutators for the maximal and sharp functions

W_0^{k, Y} (\Omega)\subset Y_n (\infty, k) =\{f: t^{-k/n}(f^{\ast\ast} (t)-f^\ast (t))\in Y\}

A distance between orbits that controls commutator estimates and invertibility of operators

. Moreover Yn(∞,k) is not larger (and in many cases essentially smaller) than any r.i. space X(Ω) such that

On a limit class of approximation spaces

W_0^{k, Y} (\Omega)\subset X (\Omega)