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Equilibrium measures on trees
Full text linkWe give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For p= 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders
Ahlfors regular spaces have regular subspaces of any dimension
Full text linkWe characterize Q-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each 0 < alpha < Q, an alpha-regular subspace. As an application, we give an alternative simple proof of the existence of alpha-regular subspaces of a Q-dimensional complete Ahlfors regular metric space (X, rho), which was proved in [8]
Discrete hilbert transform à la gundy-Varopoulos
Full text linkWe show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy-Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting
Bi-parameter embedding and measures with restricted energy conditions
Full text linkNicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree
Bellman Function Sitting on a Tree
Full text linkIn this note we give a proof-by-formula of certain important embedding inequalities on a dyadic tree. We also consider the case of a bi-tree, where a different approach is explained
Two-weight dyadic Hardy inequalities
Full text linkWe present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research
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Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees
Full text linkIn this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature
Twist points of planar domains
No full textWe establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains
A potential theoretic approach to twisting
No full textTheta Series in Advanced Mathematics, vol. 4; D. Bakry et al., eds
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