representation formulas for locally defined operators mapping C m (D) into C 0 (D) and into C 1 (D) are given.

Transcription

1 "!$# %'& "*,+ - # %'./%0*,1 2 3 %0*,# 45% /%0:<; "=?> %0*,%7@A=?B"C 2! DE GFH IJ JLKM ONPI FH IJ JLKMQRNPE S T UP V ONP WUPXUPY ZNP [\U] ^ _a`<bdc e fhg c i Let D be a regular closed set in the open subspace G R n and C m D be the space of functions f D such that f C m G. The representation formulas for locally defined operators mapping C m D into C 0 D and into C 1 D are given. jlknm oapdqhr0s0t0uhpdv r0o For a real interval I R and a nonnegative integer m, we denote by C m I the set of all m-times continuously differentiable functions ϕ: I R. An operator K: C m I C 0 I or C m I C 1 I is said to be locally defined if for every two functions ϕ, ψ C m I and for every open subinterval J I the relation ϕ J = ψ J implies that Kϕ J = Kψ J. Answering a question posed by F. Neuman, the authors of [1] gave a representation formula for locally defined operators K: C m I C 0 I. Namely, they proved that: every locally defined operator K: C m I C 0 I must be of the form Kϕx = hx, ϕx, ϕ x,..., ϕ m x for a certain function h: I R m+1 R. Moreover, they proved that every locally defined operator K: C m I C 1 I must be of the form Kϕx = hx, ϕx,..., ϕ m 1 x. In this paper we generalize this result showing that analogous representation theorems hold true for locally defined operators K: C m D C 0 D and C m D into C 1 D, where D is a regular closed set in the open subspace G R n and C m D is the space of functions f D such that f C m G. The proofs of our theorems are similar in spirit to the proofs of Theorems 2 and 3 in [1]. AMS 2000 Subject Classification: 47H30.

2 wxy"zh{ ~}? ~zh z ƒ? ~ˆ~ Š kn qhœ0 v Ž\v o0 Lqhv Œ0 Let N 0 be a set of nonnegative integers and let N n 0 := n i=1 N 0 for n N. In this paper, for k = k 1,..., k n N n 0 and i = i 1,..., i n N n 0, we put k := k k n, k! := k 1!... k n!, k + i := k 1 + i 1,..., k n + i n, k i := k 1 i 1,..., k n i n for all i k, where the notation i k means that i s k s for every s {1,..., n}. Moreover, for i = i 1,..., i n N n 0 and x = x 1,..., x n R n, we put x i := x i xin n and x := n x 2 i. As a consequence of the Whitney Extension Theorem cf. [2] we get the following lemma. Lemma 1 Let B R n be a compact set with only one cluster point z R n. Suppose that m N 0 and {f k f k : B R, k N n 0, k m} where f 0,...,0 = f is a family of functions satisfying the condition f k x i m k i=1 f k+i z x z i = o x z m k as x z 1 i! for all x B, k m, k N n 0. If for some α > 0, x y = x y α max{ x z, y z }, x, y B, then there exists a function g of the class C m on R n satisfying the condition k g x k xkn n x = f k x for all x B, k N n 0 0kn ar0ua L s0œa dv o0œ0s r0 0Œ0qh pdr0qh Ž\ L 0 0v o0 C 1 D and k m. 2 C m D v oapdr C 0 D Lo0s v oapdr Let G be a nonempty and open set in the Euclidean space R n. By C m G we denote the space of m-times continuously differentiable functions on G.

3 š< œ< z?ž Ÿ~ y"zh l}~ž Ÿ~ y ˆ~}? ˆ~ w Definition 1 Let G be an open set in the Euclidean space R n and let D G be a regular closed set in the subspace G, i.e., D = G cl int D. A function f: D R is said to be of the class C m on D if there exists a function g C m G such that g D = f, i.e., C m D = {f D : f C m G}. Let J i R, i = 1,..., n, be open closed intervals. A set J R n, J = n J i, i=1 the Cartesian product of the intervals J i, will be called an open closed interval in R n. Now, we introduce the definition of locally defined operators of the type K: C m D C k D. Definition 2 Let m, k N 0 and let D be a regular closed set in the open subspace G R n. An operator K: C m D C k D is said to be locally defined if for every two functions ϕ, ψ C m D and for every open interval J R n ϕ D J = ψ D J = Kϕ D J = Kψ D J. We shall need the following lemma. Lemma 2 cf. [3], Theorem Let m, k N 0 and a closed interval D R n be fixed and let K: C m D C k D be a locally defined operator. Then for every x o D, ϕ, ψ C m D, if then j ϕ x j xjn n i Kϕ x i xin n x o = j ψ x j xjn n x o = i Kψ x i xin n x o for all j N n 0, j m, x o for all i N n 0, i k. Before formulating the main theorems we have to introduce the following notation. Let m N 0 be fixed. Then Sk := m k s=0 n + s 1 denotes the cardinality of the set of all partial derivatives of m k times continuously differentiable function ϕ: R n R. s

4 w y"zh{ ~}? ~zh z ƒ? ~ˆ~ Theorem 1 Let m N 0, n N and let D be a regular closed set in the open subspace G R n. If an operator K: C m D C 0 D is locally defined, then there exists a unique function h: D R S0 R such that Kφx = h for all φ C m D and x D. x, φx, φ x,..., φ x n x,..., m φ x m 1 x,..., m φ x m x n Proof. The proof is based on the concept of Theorem 2 in [1]. In order to define a function h: D R S0 R let us fix arbitrarily z = z 1,..., z n D and y j1,...,j n R such that j 1,..., j n {0,..., m}, j m. Let us take a polynomial P z1,...,z n,y 0,...,0,...,y 0,...,m x 1,..., x n m := j 1,...,jn=0 j jn m y j1,...,j n j 1!... j n! x 1 z 1 j1... x n z n jn, x 1,..., x n R n and put hz 1,..., z n, y 0,...,0,..., y 0,...,m := KP z1,...,z n,y 0,...,0,...,y 0,...,m z 1,..., z n. For any φ C m D, j N n 0 and j m j φ x j xjn n z 1,..., z n = Hence, by Lemma 2 for i = 0, we obtain j P z1,...,z n,φz, φ z,..., m φ x j xjm n x m n z z 1,..., z n. Kφz 1,..., z n = K P z1,...,z n,φz, φ z,..., m φ x m n z z1,..., z n and therefore Kφz 1,..., z n = h z 1,..., z n, φz, φ z,..., m φ z. Now, we prove the uniqueness of h. Let h 1 : D R S0 R be a function such that Kφz 1,..., z n = h 1 z 1,..., z n, φz, φ z,..., m φ z for all φ C m D and z = z 1,..., z n D. In order to show that h = h 1, let us fix an arbitrary z 1,..., z n D and y j1,...,j n R, j 1,..., j n {0,..., m}, j m. x m n x m n

5 š< œ< z?ž Ÿ~ y"zh l}~ž Ÿ~ y ˆ~}? ˆ~ w According to the definitions of h 1 and h, we have h 1 z 1,..., z n, y 0,...,0,..., y 0,...,m = KP z1,...,z n,y 0,...,0,...,y 0,...,m z 1,... z n which completes the proof. = hz 1,..., z n, y 0,...,0,..., y 0,...,m, Corollary 1 Let m N 0, n N and an open set G R n be fixed. If an operator K: C m G C 0 G is locally defined, then there exists a unique function h: G R S0 R such that Kφx = h for all φ C m G and x G. x, φx, φ x,..., φ x n x,..., m φ x m 1 x,..., m φ x m x n The following result may be proved in much the same way as Theorem 3 in [1]. Theorem 2 Let m, n N and let D be a regular closed set in the open subspace G R n. If an operator K: C m D C 1 D is locally defined, then there exists a unique function h: D R S1 R such that Kφx = h x, φx,..., m 1 φ x,..., m 1 φ x x1 m 1 for all φ C m D and x = x 1,..., x n D. xn m 1 Proof. By Theorem 1 there exists a unique function h: D R S0 R such that for all φ C m D and x 1,..., x n D Kφx 1,..., x n = h x 1,..., x n, φx 1,..., x n,..., m φ x m 1 x 1,..., x n,..., m φ x m x 1,..., x m. n In order to prove this theorem it is enough to show that for all i N n 0 such that i = m we have h y i x 1,..., x n, y 0,...,0,..., y m,0,...,0,..., y 0,...,0,m = 0. 3 Let us fix x o D and y i R where i N n 0, i m and let us choose an arbitrary i 0, i 0 = m, and a real sequence y i0,n N=0 such that y i0,0 = y i0 ; y i0,n y i0, N N; lim N y i 0,N = y i0,0.

6 0 y"zh{ ~}? ~zh z ƒ? ~ˆ~ Let φ N, for every N N 0, denotes the polynomial φ N x := y r x x o r + y i 0,N x x o i0, x D. i 0! r m r i 0 Fix an ε > 0. Since all functions Kφ N are continuous, for all N N there exists δ N > 0 such that x x o < δ N Kφ N x Kφ N x o < ε y i0,n y i0,0, x D. 4 Take an arbitrary α > 0 and choose a set B = {x N : N N 0 } D satisfying all the conditions listed in Lemma 1 with z = x o and such that and x N x o < δ N, N N 5 y i0,n y i0,0 lim =. 6 N x N x o Now define functions f i : B R, i N n 0, i m, by the formula f i x N := φ i N x N, N N 0. First we show that the family {f i f i : B R, i N n 0, i m} fulfills 1 for all i N n 0 such that i i 0. Since for all N N 0 f i x N = y i+r x N x o r + y i 0,N i 0 i! x N x o i0 i, and we infer that f i x N r i 0 i f i+r x o x N x o r = r i 0 i f i+r x o x N x o r = y i+r x N x o r + y i 0,0 i 0 i! x N x o i0 i, y i0,n y i0,0 x N x o i0 i i 0 i! = y i 0,N y i0,0 i 0 i! x N x o i0 i y i 0,N y i0,0 x N x o i0 i i 0 i! = y i 0,N y i0,0 x N x o m i i 0 i! = o x N x o m i.

7 š< œ< z?ž Ÿ~ y"zh l}~ž Ÿ~ y ˆ~}? ˆ~ w In the second case, when i N n 0 is such that i m does not satisfy the inequality i i 0, we have f i x N = y i+r x N x o r = f i+r x o x N x o r and therefore f i x N f i+r x o x N x o r = 0. Thus the family {f i f i : B R, i N n 0, i m} fulfills 1 and according to Lemma 1 there exists a function g C m R n such that i g x i xin n x N = i φ N x i xin n x N, N N 0, i N n 0, i m. 7 Hence and by 4, 5, 7 and Lemma 2 we have hx o, y 0,...,0,..., y i0,n,..., y 0,...,m hx o, y 0,...,0,..., y i0,0,..., y 0,...,m y i0,n y i0,0 = Kφ N x o Kφ 0 x o y i0,n y i0,0 Kφ N x N Kφ N x o y i0,n y i0,0 + Kφ N x N Kφ 0 x o y i0,n y i0,0 ε + Kgx N Kgx o y i0,n y i0,0 = ε + Kgx N Kgx o x N x o x N x o y i0,n y. i0,0 Since Kg C 1 D, we conclude that Kgx N Kgx o lim <. N x N x o Hence and by 6 we obtain 3 for i = i 0 N n 0 such that i 0 = m and the proof is completed. Corollary 2 Let m, n N and an open set G R n be fixed. If an operator K: C m G C 1 G is locally defined, then there exists a unique function h: G R S1 R such that Kφx = h for all φ C m G and x G. x, φx,..., m 1 φ x,..., m 1 φ x1 m 1 xn m 1 x

8 0 y"zh{ ~}? ~zh z ƒ? ~ˆ~ ªua«5o0r Œ0s0 0Œ0Ž\Œ0oapd I would like to thank the referee for helpful comments and suggestions. ~±$ ³², ³ ³µ~ ³ [1] K. Lichawski, J. Matkowski, J. Miś, Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math , no. 1-6, [2] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc , no. 1, [3] M. Wróbel, Some consequences of locally defined operators, in: Proceedings of the XII th Czech Polish Slovak Mathematical School, Faculty of Education of University J.E. Purkyně in Ústi nad Labem, 2005, Institute of Mathematics and Informatics Jan Dlugosz University Armii Krajowej 13/15 PL Częstochowa Poland m.wrobel@ajd.czest.pl Received: 20 November 2006; final version: 30 May 2007; available online: 9 November 2007.