tailieunhanh - Đề tài " Cm extension by linear operators "

Introduction and statement of results Let E ⊂ Rn , and m ≥ 1. We write C m (E) for the Banach space of all real-valued functions ϕ on E such that ϕ = F on E for some F ∈ C m (Rn ). The natural norm on C m (E) is given by ϕ C m (E) = inf{ F C m (Rn ) : F ∈ C m (Rn ) and F = ϕ on E} . Here, as usual, C m (Rn ) is the space of real-valued functions on Rn with continuous and bounded derivatives through order m; and F C m (Rn ) = |β|≤m x∈Rn max sup |∂ β F (x)| . | Annals of Mathematics Cm extension by linear operators By Charles Fefferman Annals of Mathematics 166 2007 779 835 Cm extension by linear operators By Charles Fefferman 0. Introduction and statement of results Let E c Rn and m 1. We write Cm E for the Banach space of all real-valued functions V on E such that V F on E for some F E Cm Rn . The natural norm on Cm E is given by II V Ilơ- E inf F lie R F E Cm Rn and F V on E . Here as usual Cm Rn is the space of real-valued functions on Rn with continuous and bounded derivatives through order m and II F lie . -. max sup d@F x . @ m eR. The first main result of this paper is as follows. Theorem 1. For E c Rn and m 1 there exists a linear map T Cm E - Cm Rn such that A Tv V on E for each V E Cm E and B The norm of T is bounded by a constant depending only on m and n. This result was announced in 16 . To prove Theorem 1 it is enough to treat the case of compact E. In fact given an arbitrary E c Rn we may first pass to the closure of E without difficulty and then reduce matters to the compact case via a partition of unity. Theorem 1 is a special case of a theorem involving ideals of m-jets. To state that result we fix m n 1. For x E Rn we write Rx for the ring of m-jets at x of smooth realvalued functions on Rn. For F E Cm Rn we write Jx F for the m-jet of F at x. Our generalization of Theorem 1 is as follows. Partially supported by Grant Nos. DMS-0245242 DMS-0070692. 780 CHARLES FEFFERMAN Theorem 2. Let E c Rn be compact. For each x G E let I x be an ideal in Rx. Set J F G Cm Rn Jx F G I x for all x G E . Thus J is an ideal in Cm Rn and Cm Rn J is a Banach space. Let n Cm Rn Cm Rn J be the natural projection. Then there exists a linear map T Cm Rn J - Cm Rn such that A nT for all G Cm Rn J and B The norm of T is less than a constant depending only on m and n. Specializing to the case I x Jx F F 0 at x we recover Theorem 1. The study of Cm extension by linear operators goes back to Whitney 25 26 27 and Theorems 1 and 2 are

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