各式弱性微分性質與函數的量度性質

Abstract

依循W. Stepanoff、H Whitney及H. Federer的工作，我們研究函數與各種弱性微分有關的量度性質。綜合他們的工作，可知以下四個敘述的等價性： (1)u在D上幾乎處處幾近可微(approximately differentiable)； (2)給定ε>0，存在一個定義在R^n上的連續可微函數v，使得u與v相異點所成的集合的測度小於ε； (3)u的一次差分的幾近上極限(approximate limsup)在D上幾乎處處有限； (4)u的一階幾近偏導數在D上幾乎處處存在。 接著，W. S. Tai與F. C. Liu把這些結果推廣到更高階(非負整數)的弱性微分性質。我們更進一步地將其推廣到一般階(不限定為非負整數)，證明了以下定理： 主要定理. 對γ>0，以下敘述是等價的： (1)u在D上擁有γ階Lusin性質； (2)u在D上幾乎處處γ階Lipschitz連續； (3)u在D上幾乎處處γ階偏Lipschitz連續。 對於證明主要定理的重要工具─Whitney擴張定理，我們也做了仔細的研究，附加上範數的估計，將定理重新敘述成更容易應用的型式。

Metrical properties of measurable functions in terms of various forms of weak differentiability are studied along a line suggested by works of W. Stepanoff, H. Whitney, and H. Federer which can be summarily described as stating that the following four statements are equivalent: (1) u is approximately differentiable a.e. on D. (2) Given epsilon > 0, there is a C^1 function v on R^n such that |{x in D : u(x) does not equal v(x)}| < epsilon. (3) ap-limsup_{y tends to x}|u(y)-u(x)|/|y-x|< ∞ for almost all x ∈ D. (4) First order approximate partial derivatives of u exist a.e. on D. W. S. Tai and F. C. Liu then generalize the results to the situation involving higher (integral) order of weak differentiability. For a further generalization to fractional order, we prove the following theorem: Main Theorem. For gamma > 0, the following statements are equivalent: (1) u has Lusin property of order gamma on D. (2) u is approximately Lipschitz continuous of order gamma at almost every point of D. (3) u is partially approximately Lipschitz continuous of order gamma at almost all point of D. Whitney’s Extension Theorem, which is a main tool for the proof of the Main Theorem, is also given a detailed consideration and reformulated in a form with appropriate norm estimates. This form seems to be of a final touch and can be applied more effectively.