Intrinsic characterizations of distribution spaces on domains

• Autorzy: V. S. Rychkov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Besov spaces; Triebel-Lizorkin spaces; spaces on domains; intrinsic characterizations; local means; maximal functions

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 127
• Numer: 3
• Strony: 277-298

Daty

wydano

1998

otrzymano

1997-02-03

poprawiono

1997-09-29

Twórcy

autor

V. S. Rychkov

• Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-4, 07743 Jena, Germany,

Bibliografia

• [Bes1] O. V. Besov, On a family of function spaces. Embedding theorems and extensions, Dokl. Akad. Nauk SSSR 126 (1959), 1163-1165 (in Russian).
• [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).
• [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).
• [Bes4] O. V. Besov, Embeddings of Sobolev-Liouville and Lizorkin-Triebel spaces on a domain, Dokl. Akad. Nauk 331 (1993), 538-540 (in Russian).
• [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).
• [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.
• [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.
• [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.
• [FeS] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.
• [FrR] J. Franke and T. Runst, Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces, Math. Nachr. 174 (1995), 113-149.
• [FrJ] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
• [Hei] N. J. H. Heideman, Duality and fractional integration in Lipschitz spaces, Studia Math. 50 (1974), 65-85.
• [Hes] M. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183-192.
• [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).
• [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).
• [Kal3] G. A. Kalyabin, personal communication, 1994.
• [Liz1] P. I. Lizorkin, Operators connected with fractional derivatives and classes of differentiable functions, Trudy Mat. Inst. Steklov. 117 (1972), 212-243 (in Russian).
• [Liz2] P. I. Lizorkin, Properties of functions of the spaces $Λ^r_pθ$, ibid. 131 (1974), 158-181 (in Russian).
• [Miy] A. Miyachi, $H^p$ spaces over open subsets of $ℝ^n$, Studia Math. 95 (1990), 205-228.
• [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.
• [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.
• [P2] J. Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123-130.
• [Rud] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.
• [Ry] V. S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, preprint.
• [Sch] T. Schott, Function spaces with exponential weights I, Math. Nachr., to appear.
• [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.
• [StT] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.
• [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.
• [Tri1] H. Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, Ark. Mat. 11 (1973), 13-64.
• [Tri2] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
• [Tri3] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
• [Tri4] H. Triebel, Local approximation spaces, Z. Anal. Anwendungen 8 (1989), 261-288.
• [TrW] H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673.