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Abstract. We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari anc C. Fefferman are proved.Key words and phrases: Hardy space on the quaternionic ball; functions of a quaternionic variable;Hankel operators.Mathematics Subject Classification: 30G35, 46E22, 47B35. Notation. The symbol H denotes the set of the quaternions qwhere the x j 's are real numbers and the imaginary units i, j, k are subject to the rules i j = k, jk = i, ki = j and i 2 = j 2 = k 2 = −1. We identify the quaternions q whose imaginary part vanishes, Im(q) = 0, with real numbers, Re(q) ∈ R; and, similarly, we let I = Ri+R j+Rk be the set of the imaginary quaternions. The norm |q| ≥ 0 of q is |q| =

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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 find Riemannian metrics on the unit ball of the quaternions, which are naturally associated with reproducing kernel Hilbert spaces. We study the metric arising from the Hardy space in detail. We show that, in contrast to the one-complex variable case, no Riemannian metric is invariant under regular self-maps of the quaternionic ball. KEY WORDS AND PHRASES: Hardy space on the quaternionic ball; functions of a quaternionic variable; invariant Riemannian metric. MATHEMATICS SUBJECT CLASSIFICATION: 30G35, 46E22, 58B20. Notation. The symbol H denotes the set of the quaternionswhere the x j 's are real numbers and the imaginary units i, j, k are subject to the rules ij = k, jk = i, ki = j and i 2 = j 2 = k 2 = −1. We identify the quaternions q whose imaginary part vanishes, Im(q) = 0, with real numbers, Re(q) ∈ R; and, similarly, we let I = Ri + Rj + Rk be the set of the imaginary quaternions. The normThe open unit ball B in H contains the quaternions q such that |q| < 1. The boundary of B in H is denoted by ∂B. By the symbol S we denote the unit sphere of the imaginary quaternions: q ∈ I belongs to S if |q| = 1. For I in S, the slice L I = L −I in H contains all quaternions having the form q = x + yI, with x, y in R. If f is a real differentiable function on a domain Ω ⊆ H, we denote its real differential at a point w ∈ Ω by the symbol f * [w].