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Alexander Volberg

Publications and source records attributed to Alexander Volberg.

At least 73 records · Page 4Linked to original sources
arXiv2017

Covering and separation of Chebyshev points for non-integrable Riesz potentials

For Riesz $s$-potentials $K(x,y)=|x-y|^{-s}$, $s>0$, we investigate separation and covering properties of $N$-point configurations $ω^*_N=\{x_1, \ldots, x_N\}$ on a $d$-dimensional compact set $A\subset \mathbb{R}^\ell$ for which the minimum of $\sum_{j=1}^N K(x, x_j)$ is maximal. Such configurations are called $N$-point optimal Riesz $s$-polarization (or Chebyshev) configurations. For a large class of $d$-dimensional sets $A$ we show that for $s>d$ the configurations $ω^*_N$ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as $N\to \infty$ of the best covering constant. For these purposes we compare best-covering configurations with optimal Riesz $s$-polarization configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty$) of the maximal $s$-polarization constants. In addition, we introduce the notion of "weak separation" for point configurations and prove this property for optimal Riesz $s$-polarization configurations on $A$ for $s>\text{dim}(A)$, and for $d-1\leqslant s < d$ on the sphere $\mathbb{S}^d$.

math.CA↗
arXiv2017

Sparse domination on non-homogeneous spaces with an application to $A_p$ weights

We extend Lerner's recent approach to sparse domination of Calderón--Zygmund operators to upper doubling (but not necessarily doubling), geometrically doubling metric measure spaces. Our domination theorem is different from the one obtained recently by Conde-Alonso and Parcet and yields a weighted estimate with the sharp power $\max(1,1/(p-1))$ of the $A_p$ characteristic of the weight.

math.CA↗
arXiv2016

Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$

We obtain a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.

math.CV↗
arXiv2016

On Tsirelson's theorem about triple points for harmonic measure

A theorem of Tsirelson from 1997 asserts that given three disjoint domains in $\mathbb R^{n+1}$, the set of triple points belonging to the intersection of the three boundaries where the three corresponding harmonic measures are mutually absolutely continuous has null harmonic measure. The original proof by Tsirelson is based on the fine analysis of filtrations for Brownian and Walsh-Brownian motions and can not be translated into potential theory arguments. In the present paper we give a purely analytical proof of the same result.

math.CA↗
arXiv2016

On weak weighted estimates of martingale transform

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheeden's conjecture was disproved by Reguera--Thiele.

math.AP↗
arXiv2016

Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach

The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequality, Beckner's inequality, Bobkov's functional isoperimetric inequality and several other inequalities) we use our method to generate new isoperimetric inequalities, in particular, we found the sharpening of Beckner--Sobolev inequalities with Gaussian measure. Key words: log-Sobolev inequality, Poincaré inequality, Bobkov's inequality, Gaussian isoperimetry, semigroups, maximum principle, Monge--Ampère equation with drift, exterior differential systems, backwards heat equation, (B) theorem

math.AP↗
arXiv2016

On a two-phase problem for harmonic measure in general domains

We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.

math.CA↗
arXiv2016

Poincaré inequality 3/2 on the Hamming cube

For any $n \geq 1$, and any $f :\{-1,1\}^{n} \to \mathbb{R}$ we have $ \Re\, \mathbb{E}\, (f + i\, |\nabla f|)^{3/2} \leq \Re\, (\mathbb{E}f)^{3/2}, $ where $z^{3/2}$ for $z=x+iy$ is taken with principal branch and $\Re$ denotes the real part.

math.AP↗
arXiv2016

Improving Beckner's bound via Hermite functions

We obtain an improvement of the Beckner's inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1,2)$, and moreover, we find the natural extension of the inequality for any real $p$.

math.AP↗
arXiv2015

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.

math.CA↗
arXiv2015

Bellman partial differential equation and the hill property for classical isoperimetric problems

The goal of this note is to have a systematic approach to generating isoperimetric inequalities from two concrete type of PDEs. We call these PDEs Bellman type because a totally analogous equations happen to rule many sharp estimates for singular integrals in harmonic analysis, and such estimates were obtained with the use of Hamilton--Jacobi--Bellman PDE. We show how classical inequalities of Brascamp--Lieb, Prekopa--Leindler, Ehrhard are particular case of this scheme, which allows us to augment the stock of such inequalities. We approach the isoperimetric inequalities as a maximum (minimum) principle for special types of functions. These functions are compositions of "Bellman function" and an appropriate flow built on test functions. Then the existence of maximum (minimum) principle for such compositions can be reduced to the requirement that Bellman function satisfies a concrete class of nonlinear PDE (written down below). We are left to solve this nonlinear PDE (sometimes a possible task) to enjoy isoperimetric inequalities. The nonlinear PDE that we will describe in this article can be reduced sometimes to solving Laplacian eigenvalue problem, $\bar{\partial}$-equation of certain type or just the linear heat equation.

math.AP↗
arXiv2014

Entropy conditions in two weight inequalities for singular integral operators

The new type of "bumping" of the Muckenhoupt $A_2$ condition on weights is introduced. It is based on bumping the entropy integral of the weights. In particular, one gets (assuming mild regularity conditions on the corresponding Young functions) the bump conjecture, proved earlier by A. Lerner and independently by Nazarov--Reznikov--Treil--Volberg, as a corollary of entropy bumping. But our entropy bumps cannot be reduced to the bumping with Orlicz norms in the solution of bump conjecture, they are effectively smaller. Henceforth we get somewhat stronger result than the one that solves the bump conjecture. New results concerning one sided bumping conjecture are obtained. All the results hold in the general non-homogeneous situation.

math.CA↗
arXiv2014

Mean type of functions of bounded characteristic and Martin functions in Denjoy domains

Functions of bounded characteristic in simply connected domains have a classical factorization to Blaschke, outer and singular inner parts. The latter has a singular measure on the boundary assigned to it. The exponential speed of change of a function when approaching a point of a boundary (mean type) corresponds to a point mass at this point. In this paper we consider the analogous relation for functions in arbitrary infinitely connected (Denjoy) domains. The factorization result holds of course with one important addition: all functions involved become multiple valued even though the initial function was single valued. The mean type now can be measured by using the Martin function of the domain. But this result does not follow from the lifting to the universal covering of the domain because of the simple (but interesting) reason that the mean types of the original and the lifted functions can be completely different.

math.CV↗
arXiv2014

A Bellman function proof of the $L^2$ bump conjecture

We approach the problem of finding the sharp sufficient condition of the boundedness of all two weight Calderon--Zygmund operators. We solve this problem in $L^2$ by writing a formula for a Bellman function of the problem.

math.CA↗
arXiv2014

Almost-additivity of analytic capacity and Cauchy independent measures

We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of arbitrary subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. We show a sort of converse to this geometric statement as well. However, we need that the discs in question would be separated, and it is not clear whether the separation condition is essential or not. We apply this result to find families $\{μ_j\}$ of measures in $\mathbb{C}$ with the following property. If the Cauchy integral operators $\mathcal{C}_{μ_j}$ from $L^2(μ_j)$ to itself are bounded uniformly in $j$, then $\mathcal{C}_μ$, $μ=\sumμ_j$, is also bounded from $L^2(μ)$ to itself.

math.AP↗
arXiv2014

Bellman approach to the one-sided bumping for weighted estimates of Calderón--Zygmund operators

We give again (see also arXiv:1112.0676) a proof of weighted estimate of any Calderón-Zygmund operator. This is under a universal sharp sufficient condition that is weaker than the so-called bump condition. Bump conjecture was recently solved independently and simultaneously by A. Lerner (arXiv:1202.1860) and Nazarov--Reznikov-Treil-Volberg (arXiv:1202.2406). The latter paper uses the Bellman approach. Immediately a very natural and seemingly simple question arises how to strengthen the bump conjecture result by weakening its assumptions in a natural symmetric way. This is what we are dealing with here. However we meet an unexpected and, in our opinion, deep obstacle, that allows us to achieve only a partial result. The result of the present article is slightly stronger than the one in arXiv:1112.0676. Our proof consists of two main parts: reduction to a simple model operator, construction of Bellman function for estimating this simple operator. The newer feature is that the domain of definition of our Bellman function is infinitely dimensional.

math.CA↗