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

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

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 of the theorem

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,

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

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. p. 105. ISBN 0-8218-0500-2. MR 1422252 (Theorem III.1.7)