Strong differentiability properties of Bessel potentials

Abstract:

This paper is concerned with the “strong” ${L_p}$ differentiability properties of Bessel potentials of order $\alpha > 0$ of ${L_p}$ functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial ${P_x}(y)$ of degree $k \leqslant \alpha$ such that \[ \lim \sup \limits _{{\text {diam}}(S) \to 0} \;{({\text {diam}}\;S)^{ - k}}{\left \{ {|S{|^{ - 1}}\int {|f(y) - {P_x}(y){|^p}dy} } \right \}^{1/p}} = 0\] where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.