Pointwise interpolation inequalities for derivatives and their applications
by
T. Shaposhnikova
Department of Mathematics, University of Linkoeping
Coauthors: Vladimir Maz'ya

We present new pointwise interpolation inequalities involving the gradient of a function u in C¹(Rⁿ), the continuity modulus \omega of the gradient and the maximal function M^\diamond u defined by M^\diamondu(x)=sup_{r > 0}|B_r|^-1|\int_Br(x)[(y-x)/(|y-x|)]u(y)dy|. In particular, for \omega(r)=r^\alpha, \alpha > 0, we proved that |Ñu(x)|^\alpha+1 <= C(M^\diamondu(x))^\alphasup_{r > 0}r^-\alpha|Ñu(x)-|\partialB_r|^-1\int_Br(x)Ñu(y)ds_y|

where

C=  \alphan (n+\alpha)(n+\alpha+1) ((n+1)  \alpha+1 \alpha \alpha+1

is the best constant. For n=1, \alpha = 1 this implies the Landau type inequality |u'(x)|² <= ⁸/₃M^\diamondu(x)M^\diamondu''(x) with the best constant 8/3.

Some other pointwise interpolation inequalities without sharp constants are obtained and applied to the theory of function spaces. For example, we use them to give an elementary proof of the recent theorem of Brezis and Mironescu on the continuity of the composition operator W^{s, p}(Rⁿ) \cap W^{1, sp}(Rⁿ) \owns u --> f(u) in W^{s, p}(rⁿ), where s is noninteger.

Date received: August 7, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-50.