A Whitney extension theorem in L p and Besov spaces
Annales de l'Institut Fourier, Volume 28 (1978) no. 1, p. 139-192
The classical Whitney extension theorem states that every function in Lip(β,F), FR n , F closed, k<βk+1, k a non-negative integer, can be extended to a function in Lip(β,R n ). Her Lip(β,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 R d , d<n, of the Bessel potential spaces in R n and the Besov or generalized Lipschitz spaces in 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 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=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 R n .
D’après le théorème de prolongement classique de Whitney on peut prolonger toute fonction dans Lip(β,F), FR n , F fermé, k<βk+1, k un nombre entier non-négatif, à une fonction dans Lip(β,R n ). Ici on désigne par Lip(β,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 R d , d<n, des espaces potentiels besseliens dans 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 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=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 R n .
@article{AIF_1978__28_1_139_0,
     author = {Jonsson, Alf and Wallin, Hans},
     title = {A Whitney extension theorem in $L^p$ and Besov spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {28},
     number = {1},
     year = {1978},
     pages = {139-192},
     doi = {10.5802/aif.684},
     zbl = {0369.46031},
     mrnumber = {81c:46024},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1978__28_1_139_0}
}
Jonsson, Alf; Wallin, Hans. A Whitney extension theorem in $L^p$ and Besov spaces. Annales de l'Institut Fourier, Volume 28 (1978) no. 1, pp. 139-192. doi : 10.5802/aif.684. http://www.numdam.org/item/AIF_1978__28_1_139_0/

[1] D.R. Adams, Traces of potentials arising from translation invariant operators, Ann. Sc. Norm. Sup. Pisa, 25 (1971), 203-217. | Numdam | MR 44 #4508 | Zbl 0219.46027

[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 0285.31007

[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 37 #4281 | Zbl 0185.19703

[4] N. Aronszajn, Potentiels Besseliens, Ann. Inst. Fourier, 15, 1 (1965), 43-58. | Numdam | MR 32 #1549 | Zbl 0141.30403

[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 31 #5076 | Zbl 0121.09604

[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 0158.13901

[7] O.V. Besov, The behavior of differentiable functions on a non-smooth surface, Trudy Mat. Inst. Steklov, 117 (1972), 1-9. | MR 52 #6403 | Zbl 0279.46018

[8] O.V. Besov, On traces on a nonsmooth surface of classes of differentiable functions, Trudy Mat. Inst. Steklov, 117 (1972), 11-24. | Zbl 0279.46019

[9] O.V. Besov, Estimates of moduli of smoothness on domains, and imbedding theorems, Trudy Mat. Inst. Steklov, 117 (1972), 25-53. | Zbl 0279.46020

[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 0189.43305

[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 0185.20603

[12] P.L. Butzer and H. Berens, Semi-groups of operators and approximation, Berlin, Springer-Verlag, 1967. | MR 37 #5588 | Zbl 0164.43702

[13] A.P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49. | MR 26 #603 | Zbl 0195.41103

[14] H. Federer, Geometric measure theory, Berlin, Springer-Verlag, 1969. | MR 41 #1976 | Zbl 0176.00801

[15] O. Frostman, Potentiels d'équilibre et capacité des ensembles, Thesis, Lund, 1935. | JFM 61.1262.02 | Zbl 0013.06302

[16] B. Fuglede, On generalized potentials of functions in the Lebesgue classes, Math. Scand., 8 (1960), 287-304. | MR 28 #2241 | Zbl 0196.42002

[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 21 #1525 | Zbl 0087.10902

[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 27 #2853 | Zbl 0199.44401

[20] J. Bergh and J. Lofstrom, Interpolation spaces, Berlin, Springer-Verlag, 1976. | MR 58 #2349 | Zbl 0344.46071

[21] S.M. Nikol'Skii, Approximation of functions of several variables and imbedding theorems, Berlin, Springer-Verlag, 1975. | MR 51 #11073 | Zbl 0307.46024

[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 26 #6757 | Zbl 0117.29101

[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 0053.07403

[24] J. Peetre, On the trace of potentials, Ann. Scuola Norm. Sup. Pisa, 2,1 (1975), 33-43. | Numdam | MR 52 #8912 | Zbl 0308.46031

[25] W. Rudin, Real and complex analysis, sec. ed. New York, McGraw-Hill, 1974. | MR 49 #8783 | Zbl 0278.26001

[26] T. Sjödin, Bessel potentials and extension of continuous functions, Ark. Mat., 13,2 (1975), 263-271. | MR 53 #843 | Zbl 0314.31005

[27] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton, Princeton Univ. Press, 1970. | MR 44 #7280 | Zbl 0207.13501

[28] E.M. Stein, The characterization of functions arising as potentials. II, Bull. Amer. Math. Soc., 68 (1962), 577-582. | MR 26 #547 | Zbl 0127.32002

[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 29 #462 | Zbl 0132.09402

[30] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. | JFM 60.0217.01 | MR 1501735 | Zbl 0008.24902

[31] H. Wallin, Continuous functions and potential theory, Ark. Mat., 5 (1963), 55-84. | MR 29 #2425 | Zbl 0134.09404