Carleson measures, multipliers and integration operators for spaces of Dirichlet type

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Abstract

For 0<p< and α>1, we let Dαp denote the space of those functions f which are analytic in the unit disc D={zC:|z|<1} and satisfy D(1|z|2)α|f(z)|pdxdy<. In this paper we characterize the positive Borel measures μ in D such that DαpLq(dμ), 0<p<q<. We also characterize the pointwise multipliers from Dαp to Dβq (0<p<q<) if p2<α<p. In particular, we prove that if (2+α)p(β+2)q>0 the only pointwise multiplier from Dαp to Dβq (0<p<q<) is the trivial one. This is not longer true for (2+α)p(β+2)q0 and we give a number of explicit examples of functions which are multipliers from Dαp to Dβq for this range of values.

Keywords

Spaces of Dirichlet type
Bergman spaces
Carleson measures
Multipliers
Integration operators
Inner functions

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