Ito, Yoshifumi Development of \(L^p\)-calculus. (English) Zbl 1382.46027 J. Math., Tokushima Univ. 50, 91-111 (2016). 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)]. Reviewer: Dirk Werner (Berlin) 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 Keywords:\(L^p\)-differentiability; Sobolev functions; difference quotient Citations:Zbl 1266.26010 PDFBibTeX XMLCite \textit{Y. Ito}, J. Math., Tokushima Univ. 50, 91--111 (2016; Zbl 1382.46027)