Intrinsic characterizations of distribution spaces on domains

V. Rychkov

Studia Mathematica (1998)

  • Volume: 127, Issue: 3, page 277-298
  • ISSN: 0039-3223

Abstract

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We give characterizations of Besov and Triebel-Lizorkin spaces B p q s ( Ω ) and F p q s ( Ω ) in smooth domains Ω ⊂ ℝ n via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.

How to cite

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Rychkov, V.. "Intrinsic characterizations of distribution spaces on domains." Studia Mathematica 127.3 (1998): 277-298. <http://eudml.org/doc/216472>.

@article{Rychkov1998,
abstract = {We give characterizations of Besov and Triebel-Lizorkin spaces $B_\{pq\}^\{s\}(Ω)$ and $F_\{pq\}^s(Ω)$ in smooth domains $Ω ⊂ ℝ^n$ via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.},
author = {Rychkov, V.},
journal = {Studia Mathematica},
keywords = {Besov spaces; Triebel-Lizorkin spaces; spaces on domains; intrinsic characterizations; local means; maximal functions; characterizations of Besov and Triebel-Lizorkin spaces; convolutions with compactly supported smooth kernels; moment conditions; maximal function of a distribution},
language = {eng},
number = {3},
pages = {277-298},
title = {Intrinsic characterizations of distribution spaces on domains},
url = {http://eudml.org/doc/216472},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Rychkov, V.
TI - Intrinsic characterizations of distribution spaces on domains
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 3
SP - 277
EP - 298
AB - We give characterizations of Besov and Triebel-Lizorkin spaces $B_{pq}^{s}(Ω)$ and $F_{pq}^s(Ω)$ in smooth domains $Ω ⊂ ℝ^n$ via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.
LA - eng
KW - Besov spaces; Triebel-Lizorkin spaces; spaces on domains; intrinsic characterizations; local means; maximal functions; characterizations of Besov and Triebel-Lizorkin spaces; convolutions with compactly supported smooth kernels; moment conditions; maximal function of a distribution
UR - http://eudml.org/doc/216472
ER -

References

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  1. [Bes1] O. V. Besov, On a family of function spaces. Embedding theorems and extensions, Dokl. Akad. Nauk SSSR 126 (1959), 1163-1165 (in Russian). Zbl0097.09701
  2. [Bes2] O. V. Besov, On a family of function spaces in connection with embeddings and extensions, Trudy Mat. Inst. Steklov. 60 (1961), 42-81 (in Russian). 
  3. [Bes3] O. V. Besov, Extension of functions beyond a domain with preservation of difference-differential properties in L p , Mat. Sb. 66 (1965), 80-96 (in Russian). 
  4. [Bes4] O. V. Besov, Embeddings of Sobolev-Liouville and Lizorkin-Triebel spaces on a domain, Dokl. Akad. Nauk 331 (1993), 538-540 (in Russian). Zbl0835.46024
  5. [BIN] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Embedding Theorems, 2nd ed., Nauka-Fizmatgiz, Moscow, 1996 (in Russian). 
  6. [BPT1] 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. Zbl0861.42009
  7. [BPT2] H.-Q. Bui, M. Paluszyński and M. H. Taibleson, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1, in: Proc. Conf. on Harmonic Analysis in Honor of M. de Guzmán (El Escorial, 1996), to appear. Zbl0861.42009
  8. [CKS] D.-C. Chang, S. G. Krantz and E. M. Stein, H p theory on a smooth domain in ℝ n and elliptic boundary value problems, J. Funct. Anal. 114 (1993), 286-347. Zbl0804.35027
  9. [FeS] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. Zbl0222.26019
  10. [FrR] J. Franke and T. Runst, Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces, Math. Nachr. 174 (1995), 113-149. Zbl0843.35026
  11. [FrJ] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. Zbl0716.46031
  12. [Hei] N. J. H. Heideman, Duality and fractional integration in Lipschitz spaces, Studia Math. 50 (1974), 65-85. Zbl0287.46050
  13. [Hes] M. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183-192. 
  14. [Kal1] G. A. Kalyabin, Function spaces of Lizorkin-Triebel type in domains with Lipschitz boundary, Dokl. Akad. Nauk SSSR 271 (1983), 795-798 (in Russian). 
  15. [Kal2] G. A. Kalyabin, Theorems on extensions, multipliers and diffeomorphisms for generalized Sobolev-Liouville classes in domains with Lipschitz boundary, Trudy Mat. Inst. Steklov. 172 (1985), 173-186 (in Russian). 
  16. [Kal3] G. A. Kalyabin, personal communication, 1994. 
  17. [Liz1] P. I. Lizorkin, Operators connected with fractional derivatives and classes of differentiable functions, Trudy Mat. Inst. Steklov. 117 (1972), 212-243 (in Russian). 
  18. [Liz2] P. I. Lizorkin, Properties of functions of the spaces Λ p r θ , ibid. 131 (1974), 158-181 (in Russian). 
  19. [Miy] A. Miyachi, H p spaces over open subsets of ℝ n , Studia Math. 95 (1990), 205-228. 
  20. [Mur] T. Muramatu, On Besov spaces and Sobolev spaces of generalized functions defined on a general region, Publ. Res. Inst. Math. Sci. Kyoto Univ. 9 (1974), 325-396. Zbl0287.46046
  21. [P1] J. Peetre, Remarques sur les espaces de Besov. Le cas 0 < p < 1, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), 947-950. 
  22. [P2] J. Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123-130. Zbl0302.46021
  23. [Rud] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991. 
  24. [Ry] V. S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, preprint. 
  25. [Sch] T. Schott, Function spaces with exponential weights I, Math. Nachr., to appear. Zbl0909.46036
  26. [See] 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. 
  27. [StT] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989. 
  28. [Tai] M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space I. Principal properties, J. Math. Mech. 13 (1964), 407-479. Zbl0132.09402
  29. [Tri1] H. Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, Ark. Mat. 11 (1973), 13-64. 
  30. [Tri2] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. 
  31. [Tri3] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. 
  32. [Tri4] H. Triebel, Local approximation spaces, Z. Anal. Anwendungen 8 (1989), 261-288. Zbl0695.46015
  33. [TrW] H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673. Zbl0843.46022

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