A Whitney extension theorem in $L^p$ and Besov spaces

Jonsson, Alf; Wallin, Hans

Annales de l'Institut Fourier, Tome 28 (1978) no. 1, pp. 139-192.

D’après le théorème de prolongement classique de Whitney on peut prolonger toute fonction dans Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ fermé, $k<\beta \le k+1$, $k$ un nombre entier non-négatif, à une fonction dans Lip$(\beta ,{\mathbf{R}}^n)$. Ici on désigne par Lip$(\beta ,F)$ l’espace des fonctions sur $F$ avec des dérivées partielles continues jusqu’à l’ordre $k$ qui satisfont certaines conditions de Lipschitz dans la norme supremum. Nous formons et montrons un théorème analogue dans la norme $L^p$.

Les restrictions à ${\mathbf{R}}^d$, $d<n$, des espaces potentiels besseliens dans ${\mathbf{R}}^n$ et les espaces de Besov ou les espaces de Lipschitz généralisés sont caractérisées par les travaux de plusieurs auteurs (O.V. Besov, E.M. Stein, et d’autres). Nous traitons, pour les espaces de Besov, le cas quand ${\mathbf{R}}^d$ est remplacé par un ensemble $F$ fermé d’une sorte beaucoup plus générale que les ensembles considérés précédemment. Notre méthode donne une démonstration nouvelle aussi dans le cas $F={\mathbf{R}}^d$. Elle donne aussi une contribution au problème de restriction et prolongement correspondant au cas $d=n$ avec $F$ égal à la fermeture d’un domaine dans ${\mathbf{R}}^n$.

The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ closed, $k<\beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\mathbf{R}}^n)$. Her Lip$(\beta ,F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^p$-norm.

The restrictions to ${\mathbf{R}}^d$, $d<n$, of the Bessel potential spaces in ${\mathbf{R}}^n$ and the Besov or generalized Lipschitz spaces in ${\mathbf{R}}^n$ have been characterized by the work of many authors (O.V. Besov, E.M. Stein, and others). We treat, for Besov spaces, the case when ${\mathbf{R}}^d$ is replaced by a closed set $F$ of a much more general kind than the sets which have been considered before. Our method of proof gives a new proof also in the case when $F={\mathbf{R}}^d$. It also gives a contribution to the restriction and extension problem corresponding to the case $d=n$ with $F$ equal to the closure of a domain in ${\mathbf{R}}^n$.

[1] D.R. Adams, Traces of potentials arising from translation invariant operators, Ann. Sc. Norm. Sup. Pisa, 25 (1971), 203-217. | Numdam | MR | Zbl

[2] D.R. Adams and N.G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Indiana Univ. Math. J., 22 (1973), 873-905. | Zbl

[3] R. Adams, N. Aronszajn, and K.T. Smith, Theory of Bessel potentials, Part II, Ann. Inst. Fourier, 17, 2 (1967), 1-135. | Numdam | MR | Zbl

[4] N. Aronszajn, Potentiels Besseliens, Ann. Inst. Fourier, 15, 1 (1965), 43-58. | Numdam | MR | Zbl

[5] N. Aronszajn, F. Mulla, and P. Szeptycki, On spaces of potentials connected with Lp-classes, Ann. Inst. Fourier, 13, 2 (1963), 211-306. | Numdam | MR | Zbl

[6] O.V. Besov, Investigation of a family of function spaces in connection with theorems of imbedding and extension (Russian), Trudy Mat. Inst. Steklov, 60 (1961), 42-81 ; Amer. Math. Soc. Transl., (2) 40 (1964), 85-126. | Zbl

[7] O.V. Besov, The behavior of differentiable functions on a non-smooth surface, Trudy Mat. Inst. Steklov, 117 (1972), 1-9. | MR | Zbl

[8] O.V. Besov, On traces on a nonsmooth surface of classes of differentiable functions, Trudy Mat. Inst. Steklov, 117 (1972), 11-24. | Zbl

[9] O.V. Besov, Estimates of moduli of smoothness on domains, and imbedding theorems, Trudy Mat. Inst. Steklov, 117 (1972), 25-53. | Zbl

[10] O.V. Besov, Continuation of functions beyond the boundary of a domain with preservation of differential-difference properties in Lp, (Russian), Mat. Sb, 66 (108) (1965), 80-96 ; Amer. Math. Soc. Transl., (2) 79 (1969), 33-52. | Zbl

[11] V.I. Burenkov, Imbedding and continuation for classes of differentiable functions of several variables defined on the whole space, Progress in Math., 2, pp. 73-161, New York, Plenum Press, 1968. | Zbl

[12] P.L. Butzer and H. Berens, Semi-groups of operators and approximation, Berlin, Springer-Verlag, 1967. | MR | Zbl

[13] A.P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49. | MR | Zbl

[14] H. Federer, Geometric measure theory, Berlin, Springer-Verlag, 1969. | MR | Zbl

[15] O. Frostman, Potentiels d'équilibre et capacité des ensembles, Thesis, Lund, 1935. | JFM | Zbl

[16] B. Fuglede, On generalized potentials of functions in the Lebesgue classes, Math. Scand., 8 (1960), 287-304. | MR | Zbl

[17] E. Gagliardo, Caratterizzazioni della trace sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Padoa, 27 (1957), 284-305. | Numdam | MR | Zbl

[18] A. Jonsson, Imbedding of Lipschitz continuous functions in potential spaces, Department of Math., Univ. of Umea, 3 (1973).

[19] P.I. Lizorkin, Characteristics of boundary values of functions of Lrp(En) on hyperplanes (Russian), Dokl. Akad. Nauk SSSR, 150 (1963), 986-989. | MR | Zbl

[20] J. Bergh and J. Lofstrom, Interpolation spaces, Berlin, Springer-Verlag, 1976. | MR | Zbl

[21] S.M. Nikol'Skii, Approximation of functions of several variables and imbedding theorems, Berlin, Springer-Verlag, 1975. | MR | Zbl

[22] S.M. Nikol'Skii, On imbedding, continuation and approximation theorems for differentiable functions of several variables (Russian), Usp. Mat. Nauk, 16, 5 (1961), 63-114 ; Russian Math. Surveys, 16, 5 (1961), 55-104. | MR | Zbl

[23] S.M. Nikol'Skii, On the solution of the polyharmonic equation by a variational method (Russian), Dokl. Akad. Nauk SSSR, 88 (1953), 409-411. | Zbl

[24] J. Peetre, On the trace of potentials, Ann. Scuola Norm. Sup. Pisa, 2,1 (1975), 33-43. | Numdam | MR | Zbl

[25] W. Rudin, Real and complex analysis, sec. ed. New York, McGraw-Hill, 1974. | MR | Zbl

[26] T. Sjödin, Bessel potentials and extension of continuous functions, Ark. Mat., 13,2 (1975), 263-271. | MR | Zbl

[27] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton, Princeton Univ. Press, 1970. | MR | Zbl

[28] E.M. Stein, The characterization of functions arising as potentials. II, Bull. Amer. Math. Soc., 68 (1962), 577-582. | MR | Zbl

[29] M.H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space, I, J. Math. Mech., 13 (1964), 407-480. | MR | Zbl

[30] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. | JFM | MR | Zbl

[31] H. Wallin, Continuous functions and potential theory, Ark. Mat., 5 (1963), 55-84. | MR | Zbl