ID 110916
Author
Keywords
Lp-calculus
Lp-function
Lp-differentiability
Lp-derivative
partial Lp-derivative
Lploc-function
Lploc-differentiability
Lploc-derivative
partial Lploc-derivative
Content Type
Departmental Bulletin Paper
Description
In this paper, we define the derivative or the partial derivative of a Lp-function in the sense of Lp-convergence. We also define the derivative and the partial derivative of a Lploc-function in the sense of Lploc-convergence. 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 Lp-convergence and Lploc-convergence is the Lp-calculus.
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 Lp-calculus, the derivative or the partial derivatives of a Lp-function are the derivative or the partial derivatives of the function calculated in the sense of L1loc-topology which are the Lp-functions for each p, (1 < p ≤ ∞) respectively.
For Lploc-functions, we have the similar results.
Especially, the L1-derivative or the partial L1-derivatives of a L1-function are the L1loc-derivative or the partial L1loc-derivatives 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