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We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Apw with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>0, we show that the principle fails whenever 0

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Apw with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>0, we show that the principle fails whenever 0

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Apw with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>0, we show that the principle fails whenever 0

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Apw with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>0, we show that the principle fails whenever 0

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

[EN] We characterize the boundedness and compactness of Toeplitz operators T-a with radial symbols a in weighted H-infinity-spaces H(v)(infinity)on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc. We also construct a bounded harmonic symbol f such that T-f is not bounded in H-v(infinity) for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on H-v(infinity) . (C) 2020 Elsevier Inc. All rights reserved.

We prove that, for every α>−1, the pull-back measure φ(Aα) of the measure dAα(z)=(α+1)(1−|z|2)αdA(z), where A is the normalized area measure on the unit disk D, by every analytic self-map φ:D→D is not only an (α+2)-Carleson measure, but that the measure of the Carleson windows of size εhεh is controlled by εα+2 times the measure of the corresponding window of size h. This means that the property of being an (α+2)-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces.

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces Bα of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Mobius invariant differential operators. In the doubly connected case the three Eisenstein series E-2, E-4, E-6 are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these., QC 20130814

We prove a new sharp inequality for norms in weighted Bergman space. This inequality is then used to derive isoperimetric inequalities for geometric functionals which are closely related to the torsional rigidity of a simply connected domain.

We prove a new sharp inequality for norms in weighted Bergman space. This inequality is then used to derive isoperimetric inequalities for geometric functionals which are closely related to the torsional rigidity of a simply connected domain.