Mixed-norm spaces and interpolation

• Autorzy: Joaquín M. Ortega , Joan Fàbrega
• Języki publikacji: EN

Abstrakty

EN

Let D be a bounded strictly pseudoconvex domain of $ℂ^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A^{p,q}_{δ,k}(D)$ of holomorphic functions with norm $∥f∥_{p,q,δ,k} = (∑_{|α|≤k} ʃ_{0}^{r_0} (ʃ_{∂D_{r}} |D^{α} f|^p dσ_{r})^{q/p} r^{δq/p-1} dr)^{1/q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A^{p}_{δ,k}(D)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A^{p,q}_{δ,k}(D)$ is the intersection of a Besov space $B^{p,q}_{s}(D)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.

Słowa kluczowe

EN

analytic functions; mixed-norm spaces; real interpolation; complex interpolation; Besov spaces of holomorphic functions

Kategorie tematyczne

• 32A99: None of the above, but in this section
• 32A10: Holomorphic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 109
• Numer: 3
• Strony: 233-254

Daty

wydano

1994

otrzymano

1993-01-26

poprawiono

1993-09-13

Twórcy

autor

Joaquín M. Ortega

• Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, E-08071 Barcelona, Spain.,

autor

Joan Fàbrega

• Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, E-08071 Barcelona, Spain.,

Bibliografia

• [AM] E. Amar, Suites d'interpolation pour les classes de Bergman de la boule et du polydisque de $ℂ^n$, Canad. J. Math. 30 (1978), 711-737.
• [BEA] F. Beatrous, Estimates for derivatives of holomorphic functions in pseudoconvex domains, Math. Z. 191 (1986), 91-116.
• [BEA-BU] F. Beatrous and J. Burbea, Sobolev spaces of holomorphic functions in the ball, Dissertationes Math. 276 (1989).
• [BER-LO] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
• [BE-CE] A. Bernal and J. Cerdà, Complex interpolation of quasi-Banach spaces with an A-convex containing space, Ark. Mat. 29 (1991), 183-201.
• [B-AN] B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32 (3) (1983), 91-110.
• [BR-OR1] J. Bruna and J. M. Ortega, Traces on curves of Sobolev spaces of holomorphic functions, Ark. Mat. 29 (1991), 25-49.
• [BR-OR2] J. Bruna and J. M. Ortega, Interpolation in Hardy-Sobolev spaces, preprint.
• [CO-RO] R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, Astérisque 77 (1980), 1-65.
• [CO-WE] R. Coifman et G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
• [COU] B. Coupet, Décomposition atomique des espaces de Bergman, Indiana Univ. Math. J. 38 (1990), 917-941.
• [DU] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
• [FOR] J. E. Fornæss, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), 529-569.
• [GA] S. Gadbois, Mixed-norm generalizations of Bergman spaces and duality, Proc. Amer. Math. Soc. 104 (1988), 1171-1180.
• [HA-LIT] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals II, Math. Z. 28 (1932), 612-634.
• [HO] T. Holmstedt, Interpolation of quasi-Banach spaces, Math. Scand. 26 (1970), 177-199.
• [LI1] E. Ligocka, On the space of Bloch harmonic functions and interpolation of spaces of harmonic and holomorphic functions, Studia Math. 87 (1987), 223-238.
• [LI2] E. Ligocka, On duality and interpolation for spaces of polyharmonic functions, ibid. 88 (1988), 139-163.
• [LU] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), 319-336.
• [OR] J. M. Ortega, The Gleason problem in Bergman-Sobolev spaces, Complex Variables 20 (1992), 157-170.
• [OR-FA] J. M. Ortega and J. Fàbrega, Division and extension in weighted Bergman-Sobolev spaces, Publ. Mat. 36 (1992), 837-859.
• [RO] R. Rochberg, Interpolation by functions in Bergman spaces, Michigan Math. J. 29 (1982), 229-236.
• [SH] J. H. Shi, Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $ℂ^n$, Trans. Amer. Math. Soc. 328 (1991), 619-637.
• [ST] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
• [STR] E. Straube, Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains, Trans. Amer. Math. Soc. 316 (1989), 653-671.
• [TR1] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
• [TR2] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
• [VA] N. Varopoulus, BMO functions and ∂̅ equation, Pacific J. Math. 71 (1977), 221-273.