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Development of \(L^p\)-calculus. (English) Zbl 1382.46027

The author defines a function \(f: \mathbb{R}\to \mathbb{R}\) to be \(L^p\)-differentiable with derivative \(f'\) if the difference quotients, \((\Delta_h f)(x)= (f(x+h)-f(x))/h\), converge to \(f'\) in the norm of \(L^p\), i.e., \(\|\Delta_h f -f'\|_p \to 0\) as \(h\to 0\); and likewise in the multivariate case. Some properties of this derivative are mentioned. No proofs are given.
Reviewer’s remarks. (1) The paper contains erroneous statements like “a sequence in \(L^p\) converges strongly if and only if it converges weakly.” (2) \(L^p\)-differentiable functions in the sense of the author are exactly those in the Sobolev space \(W^{1,p}(\mathbb{R}^d)\). (3) Much of the material in this paper can also be found in the author’s paper [J. Math., Univ. Tokushima 45, 49–66 (2011; Zbl 1266.26010)].

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26D05 Inequalities for trigonometric functions and polynomials

Citations:

Zbl 1266.26010
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