Browsing by Subject "Bergman metric"

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Open Access
Carleson measures for Besov spaces on the ball with applications
(Academic Press, 2007) Kaptanoǧlu, Hakkı Turgay
Carleson and vanishing Carleson measures for Besov spaces on the unit ball of CN are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type.
Open Access
Toeplitz operators on arveson and dirichlet spaces
(Birkhaeuser Science, 2007) Alpay, D.; Kaptanoǧlu, H. T.
We define Toeplitz operators on all Dirichlet spaces on the unit ball of CN and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. © Birkhäuser Verlag Basel/Switzerland 2007.