Kōmura's theorem explained

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 ^{t} _{0} \varphi(s) d s,

is differentiable at t for almost every 0 < t < T when φ : [0,&nbsp;''T''] → R lies in the L^p space L¹([0,&nbsp;''T'']; R).

Statement

Let (X, || ||) be a reflexive Banach space and let φ : [0,&nbsp;''T''] → X be absolutely continuous. Then φ is (strongly) differentiable almost everywhere, the derivative φ′ lies in the Bochner space L¹([0,&nbsp;''T'']; X), and, for all 0 ≤ t ≤ T,

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

References

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