tailieunhanh - Đề tài " Whitney’s extension problem for Cm "

Let f be a real-valued function on a compact set in Rn , and let m be a positive integer. We show how to decide whether f extends to a Cm function on Rn . Introduction Continuing from [F2], we answer the following question (“Whitney’s extension problem”; see [hW2]). Question 1. Let ϕ be a real-valued function defined on a compact subset E of Rn . How can we tell whether there exists F ∈ C m (Rn ) with F = ϕ on E? Here, m ≥ 1 is given, and C m (Rn ) denotes the space. | Annals of Mathematics Whitney s extension problem for Cm By Charles Fefferman Annals of Mathematics 164 2006 313 359 Whitney s extension problem for Cm By Charles Fefferman Dedicated to Julie Abstract Let f be a real-valued function on a compact set in Rn and let m be a positive integer. We show how to decide whether f extends to a Cm function on Rn. Introduction Continuing from F2 we answer the following question Whitney s extension problem see hW2 . Question 1. Let p be a real-valued function defined on a compact subset E of Rn. How can we tell whether there exists F E Cm Rn with F p on E Here m 1 is given and Cm Rn denotes the space of real-valued functions on Rn whose derivatives through order m are continuous and bounded on Rn. We fix m n 1 throughout this paper. We write Rx for the ring of m-jets of functions at x E Rn and we write Jx F for the m-jet of the function F at x. As a vector space Rx is identified with P the vector space of real mth degree polynomials on Rn and Jx F is identified with the Taylor polynomial 52 F x y - J . Ị0Ị m We answer also the following refinement of Question 1. Question 2. Let p and E be as in Question 1. Fix x E E and P E Rx. How can we tell whether there exists F E Cm Rn with F p on E and Ji F P In particular we ask which m-jets at x can arise as the jet of a Cm function vanishing on E. This is equivalent to determining the Zariski paratangent space from Bierstone-Milman-Pawlucki BMP1 . Supported by Grant No. DMS-0245242. 314 CHARLES FEFFERMAN A variant of Question 1 replaces C R by Cm w Rra the space of Cm functions whose mth derivatives have a given modulus of continuity w. This variant is well-understood thanks to Brudnyi and Shvartsman B BS1 2 3 4 S1 2 3 and my own papers F1 2 4 . See also Zobin Z1 2 for a related problem. In particular F2 F4 broaden the issue by answering the following. Question 3. Suppose we are given a modulus of continuity w an arbitrary subset E c Rra and functions p E R ơ E 0 to . How can we tell .