Pointwise differentiability and absolute continuity
Thomas Bagby,William P. Ziemer +1 more
TL;DR: In this paper, the relationship between Lp differentiability and Sobolev functions has been studied, and it is shown that if a function has an Lp derivative everywhere except for a set small in capacity and if these derivatives are in Lp, then the function is a Soboleve function.
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Abstract: This paper is concerned with the relationships between Lp differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in Lp, and 0 s I < k, then f has an Lp derivative of order 1 everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an Lp derivative everywhere except for a set small in capacity and if these derivatives are in Lp, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.
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TL;DR: In this article, the integral Bibliography List of symbols index is extended to include charges and BV functions, and the BV function can be used to integrate charges and charges.
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References
•Book
Singular Integrals and Differentiability Properties of Functions.
Elias M. Stein
- 01 Feb 1971
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
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Singular integrals and differentiability properties of functions
TL;DR: This paper explores singular integrals and their implications on differentiability properties of functions, examining the relationships between these concepts and their applications in various mathematical frameworks, particularly in harmonic analysis and partial differential equations.
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Local properties of solutions of elliptic partial differential equations
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