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 of Mathematics
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