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Pavel Mozolyako

Publications and source records attributed to Pavel Mozolyako.

14 recordsLinked to original sources
arXiv2026

Discrete Gaussian Free Field via Hadamard's formula

We present a novel way of constructing the Gaussian Free Field on a weighted graph via a dynamical expansion of the Green function along an expanding family of subgraphs. Along the way we obtain the discrete analogue of the classical Hadamard variational formula regarding the variation of the Green function under infinitesimal variations of the domain. In order to develop necessary machinery we construct expanding bases of the naturally associated energy spaces. An interesting observation is that both our discrete Hadamard variation formula and and the related construction of the discrete Gaussian Free Field are completely dimension-free and do not require smoothness of any kind. The graph model contains geometric information via the edges which supply the discrete topological information, and by conductances which give metric information. Going to a continuum limit, we would then obtain continuous version of the Hadamard variational formula and the associated Hadamard operator in e.g. fractal geometries of arbitrary dimension.

math.PR↗
arXiv2023

Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

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.

math.CV↗
arXiv2022

Two-weight dyadic Hardy's inequalities

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope 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 based on a new 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's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.

math.CA↗
arXiv2021

Differences between the potential theories on a tree and on a bi-tree

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that partial energy estimate always valid on a usual tree by a trivial reason (and with constant $C=1$) cannot be valid in general on bi-tree with any $C$ whatsoever. On the other hand, a weaker partial energy estimate called surrogate maximum principle: $\int_{T^2} V^ν_\varepsilon \, dν\le C_τ\varepsilon^{1-τ} {\mathcal E}[ν]^τ |ν|^{1-τ}$ is valid on bi-tree with any $τ>0$. We show that unlike the estimate on a simple tree, one cannot make $τ=0$ on bi-tree. On tri-tree we know that the previous estimate (the surrogate maximum principle) is valid with $τ=2/3$. We do not know any such estimate with any $τ<1$ on four-tree. The third counterexample disproves the estimate $\int_{T^2} V^ν_x \, dν\le F(x)$ for any function $F$ whatsoever for some probabilistic $ν$ on bi-tree $T^2$. On a simple tree $F(x)=x$ would always suffice to make this inequality to hold.

math.AP↗
arXiv2021

Improved surrogate bi-parameter maximum principle

Logarithmic potentials and many other potentials satisfy maximum principle. The dyadic version of logarithmic potential can be easily introduced, it lives on dyadic tree and also satisfies maximum principle. But its analog on bi-tree does not have this property. We prove here that "on average" we can still have something like maximum principle on bi-tree. We use the surrogate maximum principle to prove embedding theorems of Carleson type on bi-disc.

math.AP↗
arXiv2020

Bi-parameter Carleson embeddings with product weights

Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc 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 of Dirichlet spaces includes the classical 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. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.

math.AP↗
arXiv2020

Carleson embedding on tri-tree and on tri-disc

We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves the embedding theorem of those Dirichlet spaces of holomorphic function on bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes embedding result of \cite{AMPVZ} from bi-tree to tri-tree. One of our embedding description is similar to Carleson--Chang--Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature of \cite{AMPVZ} was that embedding on bi-tree turned out to be equivalent to one box Carleson condition. This is in striking difference to works of Chang--Fefferman and well known Carleson quilt counterexample. We prove here the same unexpected result for the tri-tree. Finally, we explain the obstacle that prevents us from proving our results on polydiscs of dimension four and higher.

math.AP↗
arXiv2019

Bi-parameter embedding and measures with restriction energy condition

Nicola 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 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 a pervious paper of the authors) that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.

math.CA↗
arXiv2019

Oscillation of Functions in the Hölder class

We study the size of the set of points where the $α$-divided difference of a function in the Hölder class $Λ_α$ is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.

math.CA↗
arXiv2018

Bellman function sitting on a tree

In this note we give a proof-by-formula of certain important embedding inequalities on dyadic tree. This is done with the help of Bellman function. We also consider the case of a bi-tree, where a different approach is explained.

math.CA↗
arXiv2018

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In 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.

math.CV↗
arXiv2015

Boundary oscillations of harmonic functions in Lipschitz domains

Let $u(x,y)$ be a harmonic function in the halfspace $\mathbb{R}^n\times\mathbb{R}_+$ that grows near the boundary not faster than some fixed majorant $w(y)$. Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the Law of Iterated Logarithm (LIL). We extend this result to a class of Lipschitz domains in $\mathbb{R}^{n+1}$. In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight $w$. The growth rate of such Bloch function depends on $w$ and, for slowly increasing $w$, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.

math.CA↗