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2000 | 140 | 2 | 141-162
Tytuł artykułu

Linear extension operators for restrictions of function spaces to irregular open sets

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let an open set $Ω ⊂ ℝ^n$ satisfy for some 0≤d≤n and ε > 0 the condition: the $d$-Hausdorff content $H_d(Ω∩B) ≥ ε|B|^{d/n}$ for any ball B centered in Ω of volume |B|≤1. Let $H_p^s$ denote the Bessel potential space on $ℝ^n$ 1 < p < ∞,s > 0, and let $H_p^s[Ω]$ be the linear space of restrictions of elements of $H_p^s$ to Ω endowed with the quotient space norm. We find sufficient conditions for the existence of a linear extension operator for $H_p^s[Ω]$, i.e., a bounded linear operator $H_p^s[Ω]→H_p^s$ such that $ext⨍|_Ω}=⨍$ for all ⨍. The main result is that such an operator exists if (i) d > n-1 and s > (n-d)/min(p,2), or (ii) d≤n-1 and s-[s] > (n-d)/min(p,2). It is an open problem whether these assumptions are sharp.
Czasopismo
Rocznik
Tom
140
Numer
2
Strony
141-162
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-02-08
poprawiono
2000-01-24
Twórcy
  • Mathematisches Institut, Friedrich-Schiller-Universität Jena, Germany, rytchkov@math.princeton.edu
  • Mathematics Department, Princeton University, Princeton, NJ 08544, U.S.A.
Bibliografia
  • [1] D. R. Adams, The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces, in: Approximation and Function Spaces, Banach Center Publ. 22, PWN-Polish Sci. Publ., Warszawa, 1989, 9-24.
  • [2] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
  • [3] Yu. A. Brudnyĭ, Spaces defined by means of local approximation, Trudy Moskov. Mat. Obshch. 24 (1971), 69-132 (in Russian).
  • [4] H.-Q. Bui, M. Paluszyński, and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), 219-246.
  • [5] P. Bylund, Besov spaces and measures on arbitrary closed sets, Univ. of Umeå, Dept. of Math., Doctoral Thesis No 8, 1994.
  • [6] J. R. Dorronsoro, On the differentiability of Lipschitz-Besov functions, Trans. Amer. Math. Soc. 303 (1987), 229-240.
  • [7] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.
  • [8] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.
  • [9] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
  • [11] O. Frostman, Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Medd. Lunds Univ. Mat. Sem. 3 (1935), 1-118.
  • [12] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88.
  • [13] A. Jonsson, Besov spaces on closed sets by means of atomic decompositions, Univ. of Umeå, Dept. of Math., Research Report No 7, 1993.
  • [14] A. Jonsson and H. Wallin, Function Spaces on Subsets of $ℝ^n$, Math. Reports 2, Part 1, Harwood, New York, 1984.
  • [15] G. A. Kalyabin, The intrinsic norming of the retractions of Sobolev spaces onto plain domains with the points of sharpness, in: Abstracts of Conference on Functional Spaces, Approximation Theory, Nonlinear Analysis in Honor of S. M. Nikol'ski(Moscow, 1995), p. 330 (in Russian).
  • [16] Yu. V. Netrusov, Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel type, Trudy Mat. Inst. Steklov. 187 (1989), 162-177 (in Russian); English transl.: Proc. Steklov Inst. Math. 187 (1990), 185-203.
  • [17] J. Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123-130.
  • [18] C. A. Rogers, Hausdorff Measures, Cambridge Univ. Press, Cambridge, 1970.
  • [19] V. S. Rychkov, On a theorem of Bui, Paluszyński, and Taibleson, Trudy Mat. Inst. Steklov. (to appear).
  • [20] A. Seeger, A note on Triebel-Lizorkin spaces, in: Approximation and Function Spaces, Banach Center Publ. 22, PWN-Polish Sci. Publ., Warszawa, 1989, 391-400.
  • [21] P. A. Shvartsman, Extension theorems with preservation of local polynomial approximations, Yaroslavl Univ., 1986; VINITI Publ. 8.08.1986, No. 6457 (in Russian).
  • [22] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971.
  • [23] R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060.
  • [24] H. Triebel, Theory of Function Spaces, Monogr. Math. 78, Birkhäuser, Basel, 1983.
  • [25] H. Triebel, Theory of Function Spaces II, Monogr. Math. 84, Birkhäuser, Basel, 1992.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv140i2p141bwm
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