ID  110916 
Author 
Ito, Yoshifumi
The University of Tokushima
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Keywords  Lpcalculus
Lpfunction
Lpdifferentiability
Lpderivative
partial Lpderivative
Lplocfunction
Lplocdifferentiability
Lplocderivative
partial Lplocderivative

Content Type 
Departmental Bulletin Paper

Description  In this paper, we define the derivative or the partial derivative of a Lpfunction in the sense of Lpconvergence. We also define the derivative and the partial derivative of a Lplocfunction in the sense of Lplocconvergence. Then we study their fundamental properties. Here assume that 1 ≤ p ≤ ∞ holds.
We say that the branch of analysis on the bases of the concepts of Lpconvergence and Lplocconvergence is the Lpcalculus. As the results, we have the following conclusions for the differential calculus of classical functions. Assume that 1 ≤ p ≤ ∞. Then we have the inclusion relations Lp ⊂ Lploc ⊂ L1loc. In the Lpcalculus, the derivative or the partial derivatives of a Lpfunction are the derivative or the partial derivatives of the function calculated in the sense of L1loctopology which are the Lpfunctions for each p, (1 < p ≤ ∞) respectively. For Lplocfunctions, we have the similar results. Especially, the L1derivative or the partial L1derivatives of a L1function are the L1locderivative or the partial L1locderivatives in the above sense, respectively. But the inverse facts are not necessarily true. 
Journal Title 
Journal of Mathematics

ISSN  13467387

NCID  AA11595324

Volume  50

Start Page  91

End Page  111

Sort Key  91

Published Date  2016

FullText File  
language 
eng

TextVersion 
Publisher

departments 
Science and Technology
