Function Spaces and Applications

Friday January 22 (S6)

Petr Kaplicky (Charles University)
Gehring type results for generalized (Navier-)Stokes system in 2D.

Gehring type results for generalized (Navier-)Stokes system in 2D. Abstract: We present results concerning regularity of the local weak solutions of the generalized Navier-Stokes system in two spatial variables. The generalization consists in the fact that we replace the Laplace operator by the general elliptic operator of the second order with bounded measurable coefficients. By means of the reverse Hölder inequality we show that the natural integrability of the solution can be improved. Consequently, the convective term can be considered as the right hand side, and also the integrability of the gradient of the solution can be increased by Meyers' approach.

This is joint work with Jörg Wolf from Humboldt University of Berlin.

Yitzhak Weit (University of Haifa)
The mean value property of harmonic functions and the Heat equation

A class of radial measures $\mu$ on Rn is defined so that integrable harmonic functions are characterized as the solutions of $f\mu = f$. These results are applied to the investigation of some periodic solutions of the Heat equation.

Salvador Rodriguez (Universitat de Barcelona)
A note on bilinear multipliers

The classical result of De Leeuw (Ann. Math., 1965) states that for $1<p<\infty$, given a continuous and bounded function such that defines a Fourier multiplier on Lp, the sequence defined by its restriction to the integers defines a Fourier multiplier in Lp in the periodic case.

A similar result for bilinear multipliers was obtained by D. Fan and S. Sato (J. Aust. Math. Soc., 2001).

We present an abstract version of a De Leeu's type result for bilinear multipliers that allows D. Fan and Sato's result to be recovered and to generalize G. Diestel and Grafakos' (Nagoya Math. Journal 2007) result also.

Gogatishvili Amiran (Academy of Sciences, Prague)
Calderon-type interpolation theorems and applications to sharp Sobolev embeddings

We present a new Calderon-type interpolation theorems for operators satisfying nonstandard endpoint estimates. Using this new interpolation theorems we give a characterization of sharp Sobolev embeddings for spaces based on rearrangement invariant spaces on domains with a sufficiently smooth boundary. We show that each optimal sobolev embedding can be obtained by the real interpolation of the well-known endpoint embeddings.