| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 100 === 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.
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