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Alexander Volberg

Publications and source records attributed to Alexander Volberg.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial Bohnenblust--Hille bounds for product of cyclic groups

Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants governed by interaction order grow polynomially: if $p_d=2d/(d+1)$ and \[ \gamma_2=\frac12, \qquad \gamma_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3), \] then the $\ell^{p_d}$ norm of Fourier coefficients $\{\hat f(\alpha)\}$ is bounded by $L^\infty$ norm of $f$ multiplied by $C(K) d^{4\gamma_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.

math.CA↗
arXiv2026

Tightness of and counterexamples to several quantum estimates

We prove here several tightness results for such quantum inequalities as the comparison of operator norm and product norm of $d$-local hamiltonians, Bohnenblust--Hille inequality for $d$-local hamiltonians and for quantum Fourier entropy-influence conjecture, we also discuss the quantum Aaronson--Ambainis conjecture in a special case of anti-commuting Pauli strings.

math.AP↗
arXiv2026

Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(\Omega;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(\Omega)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $\Omega=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $\Omega=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $\Omega$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

math.FA↗
arXiv2026

Small moments of the sensitivity of polynomial threshold functions

In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^{\eta}]$ where $\eta < 1/2$. It is interesting to note the phase transition that happens at $\eta=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.

math.PR↗
arXiv2026

The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W]_{\mathbf{A}_2}^{3/2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3/2$ power persists if we replace the classical matrix $A_2$ characteristic by the "fattened", larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.

math.CA↗
arXiv2026

The Boolean surface area of polynomial threshold functions

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the weaker bound \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}). \]

cs.CC↗
arXiv2025

Approximating the operator norm of local Hamiltonians via few quantum states

Consider a Hermitian operator $A$ acting on a complex Hilbert space of dimension $2^n$. We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian, its operator norm can be approximated independently of $n$ by maximizing $|\braket{\psi|A|\psi}|$ over a small collection $\mathbf{X}_n$ of product states $\ket{\psi}\in (\mathbf{C}^{2})^{\otimes n}$. More precisely, we show that whenever $A$ is $d$-local, \textit{i.e.,} $\deg(A)\le d$, we have the following discretization-type inequality: \[ \|A\|\le C(d)\max_{\psi\in \mathbf{X}_n}|\braket{\psi|A|\psi}|. \] The constant $C(d)$ depends only on $d$. This collection $\mathbf{X}_n$ of $\psi$'s, termed a \emph{quantum norm design}, is independent of $A$, and consists of product states, and can have cardinality as small as $(1+\eps)^n$, which is essentially tight. Previously, norm designs were known only for homogeneous $d$-localHamiltonians $A$ \cite{L,BGKT,ACKK}, and for non-homogeneous $2$-local traceless $A$ \cite{BGKT}. Several other results, such as boundedness of Rademacher projections for all levels and estimates of operator norms of random Hamiltonians, are also given.

quant-ph↗
arXiv2024

Fourier growth of degree $2$ polynomials

We prove bounds for the absolute sum of all level-$k$ Fourier coefficients for $(-1)^{p(x)}$, where polynomial $p:\mathbf{F}_2^n \to \mathbf{F}_2$ is of degree $1$ or degree $2$.

math.NT↗
arXiv2024

Noncommutative Bohnenblust--Hille Inequality for qudit systems

Previous noncommutative Bohnenblust--Hille (BH) inequalities addressed operator decompositions in the tensor-product space $M_2(\mathbb{C})^{\otimes n}$; \emph{i.e.,} for systems of qubits \cite{HCP22,VZ23}. Here we prove noncommutative BH inequalities for operators decomposed in tensor-product spaces of arbitrary local dimension, \emph{i.e.,} $M_K(\mathbb{C})^{\otimes n}$ for any $K\geq2$ or on systems of $K$-level qudits. We treat operator decompositions in both the Gell-Mann and Heisenberg--Weyl basis, reducing to the recently-proved commutative hypercube BH \cite{DMP} and cyclic group BH \cite{SVZ} inequalities respectively. As an application we discuss learning qudit quantum observables.

math.FA↗
arXiv2023

Dimension-free discretizations of the uniform norm by small product sets

Let $f$ be an analytic polynomial of degree at most $K-1$. A classical inequality of Bernstein compares the supremum norm of $f$ over the unit circle to its supremum norm over the sampling set of the $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz-Zygmund-type inequalities for $L^p,1\le p\leq \infty$ norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of $\deg(f)$ and is instead governed by the maximum individual degree of $f$; i.e., the largest degree of $f$ when viewed as a univariate polynomial in any coordinate. For example, we find that for $n$-variate analytic polynomials $f$ of degree at most $d$ and individual degree at most $K-1$, $\|f\|_{L^\infty(\mathbf{D}^n)}\leq C(X)^d\|f\|_{L^\infty(X^n)}$ for any fixed $X$ in the unit disc $\mathbf{D}$ with $|X|=K$. The dependence on $d$ in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust-Hille inequality with an explicit constant $O(\log K)^{2d}$.

math.FA↗
arXiv2023

Stability and the equality case in the B-theorem

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}.

math.MG↗
arXiv2023

A dimension-free discrete Remez-type inequality on the polytorus

Consider $f:\Omega^n_K \to \mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$. In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $\Omega_K^n$ compare. We prove the following \emph{discretization of the uniform norm} for low-degree polynomials: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{\Omega_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups. Key to our argument is a special class of Fourier multipliers on $\Omega_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity.

math.CA↗
arXiv2023

Bohnenblust--Hille inequality for cyclic groups

For any $K>2$ and the multiplicative cyclic group $\Omega_K$ of order $K$, consider any function $f:\Omega_K^n\to\mathbf{C}$ and its Fourier expansion $f(z)=\sum_{\alpha\in\{0,1,\ldots,K-1\}^n}a_\alpha z^\alpha$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial. We prove a Bohnenblust--Hille (BH) inequality in this setting: the $\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$. This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus ($K =\infty$) and the hypercube ($K=2$) but those extreme cases of $K$ have special properties whose absence for intermediate $K$ prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of $\Omega_K^n$. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the $K \times K$ complex matrix algebra (or in the language of quantum mechanics, systems of $K$-level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators.

math.FA↗
arXiv2023

Quantum and classical low-degree learning via a dimension-free Remez inequality

Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice $\binom{[n]}{k}$, the hypergrid $[K]^n$, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $\{\exp(2\pi i k/K)\}_{k=1}^K$ controls the supremum of $f$ over the entire polytorus $(\{z\in\mathbf{C}:|z|=1\}^n)$, with multiplicative constant $C$ depending on $K$ and $\text{deg}(f)$ only. This Remez-type inequality appears to be the first such estimate that is dimension-free (i.e., $C$ does not depend on $n$). This dimension-free Remez-type inequality removes the main technical barrier to giving $\mathcal{O}(\log n)$ sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-$K$ qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust--Hille-type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work \cite{EI22, CHP, VZ22} that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-approximated by their low-degree truncations -- a phenomenon that greatly extends the reach of low-degree learning in quantum science \cite{CHP}.

math.AP↗
arXiv2022

Noncommutative Bohnenblust--Hille inequalities

Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree \cite{defant2019fourier}. Such inequalities have found great applications in learning low-degree Boolean functions \cite{eskenazis2022learning}. Motivated by learning quantum observables, a qubit analogue of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in \cite{RWZ22}. The conjecture was resolved in \cite{CHP}. In this paper, we give a new proof of these Bohnenblust--Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr's radius phenomenon on quantum Boolean cubes.

math.FA↗
arXiv2022

Pisier type inequalities for $K$-convex spaces

We generalize several theorems of Hyt\"onen-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 $\{\Delta^{-1} D_i\}_{i=1}^n$ in $L^p(\Omega_n, X)$.

math.AP↗