Besov spaces and Bergman projections on the ball

Kaptanoğlu, H.Turgay

Comptes Rendus. Mathématique, Tome 335 (2002) no. 9, pp. 729-732.

Nous étudions une class d'opérateurs différentiels radiaux conduisant à une classification naturelle des espaces de Besov diagonaux dans la boule unité de ${ℂ}^N$. Nous donnons les conditions précises pour la bornitude des projections de Bergman de certains espaces L^p sur des espaces de Besov. Nous déterminons aussi des inverses à droite pour ces projections. Nous présentons des applications à l'interpolation complexe.

A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of ${ℂ}^N$. Precise conditions are given for the boundedness of Bergman projections from certain L^p spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented.

[1] Agler, J.; McCarthy, J.E. Complete Nevanlinna–Pick kernels, J. Funct. Anal., Volume 175 (2000), pp. 111-124

[2] Alpay, D.; Kaptanoğlu, H.T. Integral formulas for a sub-Hardy Hilbert space on the ball with complete Nevanlinna–Pick reproducing kernel, C. R. Acad. Sci. Paris, Série I, Volume 333 (2001), pp. 285-290

[3] Alpay, D.; Kaptanoğlu, H.T. Some-finite dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem, Integral Equations Operator Theory, Volume 42 (2002), pp. 1-21

[4] D. Alpay, H.T. Kaptanoğlu, Gleason's problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball, J. Math. Anal. Appl., 2002, to appear

[5] Arazy, J. Boundedness and compactness of generalized Hankel operators on bounded symmetric domains, J. Funct. Anal., Volume 137 (1996), pp. 97-151

[6] Arazy, J.; Fisher, S.D.; Janson, S.; Peetre, J. Membership of Hankel operators on the ball in unitary ideals, J. London Math. Soc., Volume 43 (1991), pp. 485-508

[7] Arazy, J.; Upmeier, H. Invariant inner product in spaces of holomorphic functions on bounded symmetric domains, Doc. Math., Volume 2 (1997), pp. 213-261

[8] Ball, J.A.; Trent, T.; Vinnikov, V. Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Oper. Theory Adv. Appl., 122, Birkhäuser, Basel, 2001, pp. 89-138

[9] Beatrous, F.; Burbea, J. Holomorphic Sobolev spaces on the ball, Dissertationes Math., Volume 276 (1989)

[10] Hahn, K.T.; Youssfi, E.H. Möbius invariant Besov p-spaces and Hankel operators in the Bergman space on the ball in ${ℂ}^n$, Complex Variables Theory Appl., Volume 17 (1991), pp. 89-104

[11] Peloso, M.M. Möbius invariant spaces on the unit ball, Michigan Math. J., Volume 39 (1992), pp. 509-536

[12] Shaw, K.E. Tangential limits and exceptional sets for holomorphic Besov functions in the unit ball of ${ℂ}^n$, Illinois J. Math., Volume 37 (1993), pp. 171-185

[13] Yan, Z. Invariant differential operators and holomorphic function spaces, J. Lie Theory, Volume 10 (2000), pp. 1-31

[14] Zhu, K. Operator Theory in Function Spaces, Dekker, New York, 1990

[15] Zhu, K. Holomorphic Besov spaces on bounded symmetric domains, I, Quart. J. Math. Oxford, Volume 46 (1995), pp. 239-256

[16] Zhu, K. Holomorphic Besov spaces on bounded symmetric domains, II, Indiana Univ. Math. J., Volume 44 (1995), pp. 1017-1031