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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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

$\displaystyle \mathop {\lim \sup }\limits_{{\text{diam}}(S) \to 0} \;{({\text{d... ...rt S{\vert^{ - 1}}\int {\vert f(y) - {P_x}(y){\vert^p}dy} } \right\}^{1/p}} = 0$

where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0422645-7
PII: S 0002-9947(1977)0422645-7
Article copyright: © Copyright 1977 American Mathematical Society