A norm on the holomorphic Besov space

Böe, Bjarte

doi:10.1090/S0002-9939-02-06529-2

A norm on the holomorphic Besov space
HTML articles powered by AMS MathViewer

by Bjarte Böe PDF

Proc. Amer. Math. Soc. 131 (2003), 235-241 Request permission

Abstract:

We obtain a description of the holomorphic Besov space that is valid for the indices $1 \leq p,q < \infty$, $0 < s < 1$. Applications to inner-outer factorisation, and to inner functions in particular, are provided.

References

Patrick Ahern, The mean modulus and the derivative of an inner function, Indiana Univ. Math. J. 28 (1979), no. 2, 311–347. MR 523107, DOI 10.1512/iumj.1979.28.28022
Alexandru Aleman, Hilbert spaces of analytic functions between the Hardy and the Dirichlet space, Proc. Amer. Math. Soc. 115 (1992), no. 1, 97–104. MR 1079693, DOI 10.1090/S0002-9939-1992-1079693-X
J. Arazy, S. D. Fisher, and J. Peetre, Besov norms of rational functions, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 125–129. MR 942262, DOI 10.1007/BFb0078868
I. A. Boricheva and E. M. Dyn′kin, A nonclassical problem of free interpolation, Algebra i Analiz 4 (1992), no. 5, 45–90 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 5, 871–908. MR 1202724
Lennart Carleson, A representation formula for the Dirichlet integral, Math. Z. 73 (1960), 190–196. MR 112958, DOI 10.1007/BF01162477
Konstantin M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math. 178 (1997), no. 2, 143–167. MR 1459259, DOI 10.1007/BF02392692
Konstantin M. Dyakonov, Besov spaces and outer functions, Michigan Math. J. 45 (1998), no. 1, 143–157. MR 1617421, DOI 10.1307/mmj/1030132088
Miroljub Jevtić, On Blaschke products in Besov spaces, J. Math. Anal. Appl. 149 (1990), no. 1, 86–95. MR 1054795, DOI 10.1016/0022-247X(90)90287-P
Miroslav Pavlović, On K. M. Dyakonov’s paper: “Equivalent norms on Lipschitz-type spaces of holomorphic functions” [Acta Math. 178 (1997), no. 2, 143–167; MR1459259 (98g:46029)], Acta Math. 183 (1999), no. 1, 141–143. MR 1719563, DOI 10.1007/BF02392949
Stefan Richter and Allen Shields, Bounded analytic functions in the Dirichlet space, Math. Z. 198 (1988), no. 2, 151–159. MR 939532, DOI 10.1007/BF01163287
Nikolai A. Shirokov, Analytic functions smooth up to the boundary, Lecture Notes in Mathematics, vol. 1312, Springer-Verlag, Berlin, 1988. MR 947146, DOI 10.1007/BFb0082810
N. A. Shirokov, Inner functions in O. V. Besov’s analytic classes, Algebra i Analiz 8 (1996), no. 4, 193–221 (Russian); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 675–694. MR 1418260
N. A. Shirokov, Outer functions in O. V. Besov’s analytic classes, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 217 (1994), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 22, 172–217, 222 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 85 (1997), no. 2, 1867–1897. MR 1327522, DOI 10.1007/BF02355296
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Manfred Stoll, Invariant potential theory in the unit ball of $\textbf {C}^n$, London Mathematical Society Lecture Note Series, vol. 199, Cambridge University Press, Cambridge, 1994. MR 1297545, DOI 10.1017/CBO9780511526183
I. E. VERBITSKI, Inner functions, Besov spaces and multipliers, Soviet Math. Dokl 29 (1984), no. 3.