Differential Calculus for L^p-functions and L_loc^p-functions Revisited

I studied the concepts of differentiability，derivatives and partial derivatives
as the fundamental concepts of differential calculus in Ito [4]，[5] .
ln this paper, we study the fundamental properties of derivatives and partal
derivatives of classical functions such as L^p-functions and L_loc^p-functions in
the sense of L^p-convergence and L_loc^p-convergence respectively.
Here we assume that p is a real number such that 1≤p<∞ holds.
ln the calculation of such derivatives and partial derivatives，we do
not need the theory of distributions except the case p = 1.
Thereby，I give the new characterization of Soboley spaces and give
the new meaning of Stone's Theorem.
Especially，in the cases of L2-functions and L_loc^2-functions，these
results have the essential role in the study of Schrödinger equations.