Kōmura's theorem

In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ : [0, T] → R given by

${\displaystyle \Phi (t)=\int _{0}^{t}\varphi (s)\,\mathrm {d} s,}$

is differentiable at t for almost every 0 < t < T when φ : [0, T] → R lies in the L^p space L¹([0, T]; R).

Statement

Let (X, || ||) be a reflexive Banach space and let φ : [0, T] → X be absolutely continuous. Then φ is (strongly) differentiable almost everywhere, the derivative φ′ lies in the Bochner space L¹([0, T]; X), and, for all 0 ≤ t ≤ T,

${\displaystyle \varphi (t)=\varphi (0)+\int _{0}^{t}\varphi '(s)\,\mathrm {d} s.}$

References

• Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. pp. 105. ISBN 0-8218-0500-2. MR1422252 (Theorem III.1.7)