Carleson Measures for the Bloch Space

Abstract

Abstract. In this paper we study the positive Borel measures µ on the unit disc D in C for which the Bloch space B is continuously included in L p (dµ), 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure µ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for µ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator T µ , defined by T µ (f )(z) =

Reverse estimates in growth spaces

“…For example, see [13,29], where w(t) = 1/(1 − t) α , α > 0; see [14], where w(t) = log e 1−t . Multiplicative logarithmic perturbations of w(t) = 1/(1 − t) were considered in [12,18].…”

“…During the last years, several authors have been investigating the operators J g and V g ϕ (see, e.g., [2,14,15,18,23,29] and references therein).…”

“…The corresponding measures μ are called p-Carleson for X . The following characterization was proved in [14] for the case, where D = B 1 and w(t) = log e 1−t .…”

The Bloch space and the dual space of a Luecking-type subspace of

“…By results and methods of Girela, Peláez, Pérez-González and Rättyä [10], the proof given here is elementary.…”

Weighted composition operators on growth spaces

“…For X and Y under consideration we characterize in Section 3 the mappings g and ϕ for which the weighted composition operator C g ϕ : X → Y is bounded or compact. Observe that the recent article [8] includes the corresponding one-dimensional result for X = A − log (B 1 ) and Y = A p α (B 1 ), 0 < p < ∞, α > −1. Stević [9, Theorem 5.2] formulated a partial result for n ∈ N, X = A −β (B n ), β > 0, and Y = H(p, q; ω)(B n ); we define the space H(p, q; ω)(B n ) in Section 3.…”