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 "L"¹( [0, "T"] ; R).

tatement of the 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 "L"¹( [0, "T"] ; "X"), and, for all 0 ≤ "t" ≤ "T",

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

References

* cite book
last = Showalter
first = Ralph E.
title = Monotone operators in Banach space and nonlinear partial differential equations
series = Mathematical Surveys and Monographs 49
publisher = American Mathematical Society
location = Providence, RI
year = 1997
pages = 105
isbn = 0-8218-0500-2 MathSciNet|id=1422252 (Theorem III.1.7)