Volterra operators on Hardy spaces of Dirichlet series

Abstract

ABSTRACT. For a Dirichlet series symbol g (s) = n≥1 b n n −s , the associated Volterra operator T g acting on a Dirichlet series f (s) = n≥1 a n n −s is defined by the integral f → − +∞ s f (w)g ′ (w) d w. We show that T g is a bounded operator on the Hardy space H p of Dirichlet series with 0 < p < ∞ if and only if the symbol g satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of BMOA(D). A further analogy with classical BMO is that exp(c|g |) is integrable (on the infinite polytorus) for some c > 0 whenever T g is bounded.In particular, such g belong to H p for every p < ∞. We relate the boundedness of T g to several other BMO type spaces: BMOA in halfplanes, the dual of H 1 , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of T g on reproducing kernels for appropriate sequences of subspaces of H 2 . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g .

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

“…Carleson measures for are poorly understood except in the special case that corresponds, under the Bohr lift (see Section 5), to a measure of compact support in the half plane . See also , where Carleson measures on are used to characterize the boundedness of Volterra operators.…”

Volterra operators and Hankel forms on Bergman spaces of Dirichlet series

“…As in the case of H 2 [13], we obtain sufficient/necessary conditions for T g to be bounded on the Hilbert spaces H 2 w . However, due to the lack of information of the behavior of the symbols in the strip 0 < s < 1/2, it seems difficult to get an " if and only if" condition.…”

“…Moreover, it is known that σ b = σ u [11] , and σ a − σ u ≤ 1 2 . Volterra operators (1.2) on the spaces H p have been investigated in [13]. Our aim is to study similar questions for the spaces H 2 w , associated to specific weights w in the class W defined below.…”

“…on the weak product H 2 w H 2 w . As in the case of H 2 [13], we only get partial results. For Dirichlet series involving d primes, we have…”

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

“…A positive answer is only known for even integers. Additionally, the boundedness of positive definite Hankel forms (1) can be formulated in terms of Carleson measures on D ∞ [36], and the same is true for the Volterra operators studied in [10]. From [15], it is known that a measure µ on D d , for d < ∞, is a H p -Carleson measure for one 0 < p < ∞ if and only if it is a Carleson measure for every 0 < p < ∞.…”