We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory. IndroductionNotation. We denote by D the unit disc D = {z ∈ C : |z| < 1} in the complex plane and by ∂D its boundary. We write A B (A B) if there is a constant independent on the variables on which A and B depend (which might be numbers, variables, sets...) such that A ≤ CB (CA ≥ B respectively), and A ≈ B, if A B and A B.In 1979, Alice Chang [15], extending a foundational result of Carleson [13] in one variable, characterized the Carleson measures for the Hardy space of the bidisc, that is, those measures µ on D 2 such that the identity operator boundedly maps H 2 (D) ⊗ H 2 (D) into L 2 (µ). At the same time, Stegenga [25] characterized the Carleson measures for the holomorphic Dirichlet space on the unit disc. Following standard use in complex function theory, we say that a measure µ is a Carleson measure for the Hilbert function space H if H continuously embeds into L 2 (µ).Carleson measures proved to be a central notion in the analysis of holomorphic spaces, as they intervene in the characterization of multipliers, interpolating sequences, and Hankel-type forms, in Corona-type problems, in the characterization of exceptional sets at the boundary, and more. In this article we characterize the Carleson measures for the Dirichlet space on the bidisc, and we obtain as a consequence a characterization of its multiplier space.As the Dirichlet space is defined by a Sobolev norm, it is not surprising that Stegenga's characterization is given in terms of a potential theoretic object, set capacity, and that the proof relies on deep results from Potential Theory, such as the Strong Capacitary Inequality. The main effort in this article is developing a bi-parameter potential theory which is rich enough to state and prove the characterization theorem. There are obstruction to doing so, which we will illustrate below.Other approaches to similar problems have been suggested in the past. The closest result is Eric Sawyer's characterization of the weighted inequalities for the bi-parameter Hardy operator [23]. Sawyer's extremely clever combinatorial-geometric argument does not seem to work in our context, or at least we were not able to 2010 Mathematics Subject Classification. 31B15, 31C20, 32A07, 46E35. Key words and phrases. Dirichlet space on the bidisc, trace inequality, Carleson measures, strong capacitary inequality, bitree.The results of Section 3.3 were obtained in the frameworks of the project 17-11-01064 by the Russian Science Foundation. N. Arcozzi is partially supported by the grants INDAM-GNAMPA 2017 "Operatori e disuguaglianze integrali in spazi con simmetrie" and PRIN 2018 "Variet...
We show that a (weighted) Carleson embedding from the bi-torus to the bi-disc is equivalent to a simple "box" condition, for product weights on the bidisc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale includes the usual Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the "box" condition does not suffice for the embedding to hold. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.
Nicola Arcozzi, Pavel Mozolyako, Giulia Sarfatti [3] recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on bi-tree. In this note we give one more proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in [2] that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.2010 Mathematics Subject Classification. 42B20, 42B35, 47A30. Key words and phrases. Carleson embedding on dyadic tree, bi-parameter Carleson embedding, Bellman function, capacity on dyadic tree and bi-tree.Theorem 3.
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