## Strong differentiability properties of Bessel potentials

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- by Daniel J. Deignan and William P. Ziemer
- Trans. Amer. Math. Soc.
**225**(1977), 113-122 - DOI: https://doi.org/10.1090/S0002-9947-1977-0422645-7
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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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## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- 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