arXiv subjects
Publications and source records attributed to Alexander Volberg.
In this paper, we show the stability, and characterize the equality cases in the strong B-inequality of Cordero-Erasquin, Fradelizi and Maurey \cite{B-conj}. As an application, we establish uniqueness of Bobkov's maximal Gaussian measure position from \cite{Bobkov-Mpos}.
The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $σ$-finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each "piece" is found.
A Schwarz function on an open domain $Ω$ is a holomorphic function satisfying $S(ζ)=\overlineζ$ on $Γ$, which is part of the boundary of $Ω$. Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if $Ω$ is simply connected and $Γ=\partial Ω\cap D(ζ,r)$, then $Γ$ has to be regular real analytic. This paper is an attempt to describe $Γ$ when the boundary condition is slightly relaxed. In particular, three different scenarios over a simply connected domain $Ω$ are treated: when $f_1(ζ)=\overlineζf_2(ζ)$ on $Γ$ with $f_1,f_2$ holomorphic and continuous up to the boundary, when $\mathcal{U}/\mathcal{V}$ equals certain real analytic function on $Γ$ with $\mathcal{U},\mathcal{V}$ positive and harmonic on $Ω$ and vanishing on $Γ$, and when $S(ζ)=Φ(ζ,\overlineζ)$ on $Γ$ with $Φ$ a holomorphic function of two variables. It turns out that the boundary piece $Γ$ can be, respectively, anything from $C^\infty$ to merely $C^1$, regular except finitely many points, or regular except for a measure zero set.
This note contains some estimates for tail spaces on Hamming cube. We use analytic paraproduct operator for that purpose. We also show several types of Bernstein--Markov inequalities for Banach space valued functions on Hamming cube. Here the novelty is in getting rid of some irritating logarithms and in proving Bernstein--Markov inequalities for $|\nabla f|_X$ rather than for $Δ^{1/2} f$ for $X$ valued polynomials on Hamming cube.
We generalize several theorems of Hytönen-Naor \cite{HN} using the approach from \cite{IVHV}. In particular, we give yet another necessary and sufficient condition (see (3.2)) to be a $K$-convex space, where the sufficiency was proved by Naor--Schechtman \cite{NS}. This condition is in terms of the boundedness of the second order Riesz transforms $\{Δ^{-1} D_i\}_{i=1}^n$ in $L^p(Ω_n, X)$.
In this paper we are concerned with the Bohnenblust--Hille type inequalities for certain polynomials of bounded degree but of very large number of variables. As the polynomials will be defined on groups, one can think about the problem as the estimate of Sidon constants. In most cases the sharp constants are unknown. We estimate the universal constant concerning the Sidon type estimates of degree $d$ polynomials of $n$ variables $z_1, \dots, z_n$ with unimodular coefficients. For polynomials that have constant absolute value of coefficients, this allows us to improve the estimate from \cite{DGMS}. The main result is Theorem 1.7 below.
What can be said about the domain $\Om$ in $\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^δ$? What can we say about $\Om$ if the Boundary Harnack Principle holds in the form $u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the boundary also nice? We discuss these questions below and give answers in very special cases.
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1$-Poincaré inequality on Hamming cube, the previous results are due to Talagrand and Ben Efraim--Lust-Piquard.
In this paper we estimate the tail of distribution (i.e., the measure of the set $\{f\ge x\}$) for those functions $f$ whose dyadic square function is bounded by a given constant. In particular we get a bit better estimate than the estimate following from the Chang--Wilson--Wolf theorem. In the paper we investigate the Bellman function corresponding to the problem. A curious structure of this function is found: it has jumps of the first derivative at a dense subset of interval $[0,1]$ (where it is calculated exactly), but it is of $C^\infty$-class for $x>\sqrt3$ (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists in the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to computer.
In every state of a quantum particle, Wigner's quasidistribution is the unique quasidistribution on the phase space with the correct marginal distributions for position, momentum, and all their linear combinations.
We describe a program to construct a counterexample to the Deift conjecture, that is, an almost periodic function whose evolution under the KdV equation is not almost periodic in time. The approach is based on a dichotomy found by Volberg and Yuditskii in their solution of the Kotani problem, which states that there exists an analytic condition that distinguishes between almost periodic and non-almost periodic reflectionless potentials with resolvent set given by a Widom domain.
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.
Motivated by the questions in the theory of Fredholm stability in Banach space and Kato's strictly singular operators we answer several natural questions concerning ``orthogonality'' in normed spaces and the properties of metric projections. What the reader will see below might have benn known long ago, but we did not find it in the literature. Some open (for us) questions are formulated at the end of Sections 7 and 9.
We measure a sign interlacing phenomenon for Bessel sequences $ (u_{k})$ in $ L^{2}$ spaces in terms of the Kantorovich--Rubinstein mass moving norm $ \Vert u_{k}\Vert_{KR}$. Our main observation shows that, quantitatively, the rate of the decreasing $ \Vert u_{k}\Vert_{KR}\longrightarrow 0$ havily depends on S. Bernstein $ n$-widths of a compact of Lipschitz functions. In particular, it depends on the dimension of the measure space. We have sharp results on the worst and the best rate of convergence of Kantorovich--Rubinstein norms of frames on $d$-dimensional cube. Those rates are sharp.
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.
Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully analyze what goes wrong in the absence of this extra assumption. At the end of this note we also list a counterexample to the existing proof of characterization of bi-parameter repeated commutator with the Hilbert transforms. This is a counterexample to the proof, and it is not a counterexample to the statement of factorization result in bi-disc, or to Nehari's theorem in bi-disc. To the best of our knowledge Nehari's theorem on bi-disc is still open. Moreover its dyadic bi-parameter version considered in the present paper is also still open for general symbol without any extra restrictions.
Let $\bfT$ is a certain tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter commutator the following equivalence holds $ \|\bfT b-b \bfT \| \asymp \|b\|_{bmo^d}$. This result is well-known for many types of bi-parameter commutators, see \cite{FS}, \cite{DLWY} and \cite{DPSK} for more details.
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.