Strong differentiability properties of Bessel potentials
Authors:
Daniel J. Deignan and William P. Ziemer
Journal:
Trans. Amer. Math. Soc. 225 (1977), 113-122
MSC:
Primary 31B15
DOI:
https://doi.org/10.1090/S0002-9947-1977-0422645-7
MathSciNet review:
0422645
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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.
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© Copyright 1977
American Mathematical Society