Searcharxiv⌕ Search

arXiv subjects

Dmitriy Zanin

Publications and source records attributed to Dmitriy Zanin.

At least 19 recordsLinked to original sources
arXiv2026

Sharp Fractional Riesz Estimates on the Hypercube

Let $\Omega_{n}=\{-1,1\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\leq 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(\Omega_n;\ell_2^n)} \leq c_{{\rm abs}}(p-1)^{-2}\|\Delta^{1/p}f\|_{L_p(\Omega_n)}. \] The exponent $\frac1p$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\nabla e^{-t\Delta}$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.

math.FA↗
arXiv2026

Isomorphisms between symmetric spaces over infinite and finite von Neumann algebras

The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class {of}) symmetric spaces over the hyperfinite $II_1$ factor $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. Precisely, we show that for any symmetric function space $E(0,1)$ (in the sense of Lindenstrauss and Tzafriri) such that both $E(0,1)$ and its K\"othe dual have the Kruglov property, the symmetric operator space $E(\mathcal{R})$ is isomorphic to some symmetric space $Z_E^2(\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. This result establishes a noncommutative version of a well-known result due to Johnson, Maurey, Schechtman and Tzafriri, and answers the noncommutative version of a question due to Mityagin.

math.FA↗
arXiv2026

Weyl asymptotic formulas in the nilpotent Lie group setting

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.

math.FA↗
arXiv2026

Schur bounded patterns and submajorisation

We characterise the Schur bounded patterns of ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals $\mathcal{C}_p$ with $0<p<1.$ Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns.

math.FA↗
arXiv2025

Fuglede theorem for symmetric spaces of $\tau$-measurable operators

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of $\tau$-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of $L_p$-spaces for $1<p<\infty$. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.

math.FA↗
arXiv2025

Uniformity of Maximal Hypoellipticity on Graded Lie Groups: From Pointwise to Global

On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old "freeze-unfreeze" strategy into the hypoelliptic setting, and offers a transparent and flexible framework for lifting symbol-level hypoelliptic properties to global elliptic estimates, without relying on pseudodifferential calculus. In addition, we prove that symmetric operators of hypoelliptic type on a graded Lie group are self-adjoint.

math.AP↗
arXiv2025

Singular value asymptotics on compact smooth Riemaniann manifolds

Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $\Delta_{G}$, and let $\Pi_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by multiplication operators and Riesz transforms on $\mathbb{R}^{d}$. Denote ${\rm sym}_{X}$ the principal symbol mapping of $\Pi_{X}$. For any $S\in\Pi_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}\mu(t,S(1+\Delta_G)^{-\frac{d}{2p}}) =(2\pi\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}d\lambda)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $d\lambda$ is the Liouville measure on the cotangent bundle $T^{\ast}X$.

math.FA↗
arXiv2024

Spectral asymptotic formula of Bessel--Riesz commutator

Let $R_{\lambda,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $\lambda>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{\lambda,j}]$, the commutator of $R_{\lambda,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{\lambda,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.

math.FA↗
arXiv2024

A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds

From the viewpoint of $*$-homomorphism on $C^{*}$-algebras, we establish the principal symbol mapping for filtered manifolds which are locally isomorphic to stratified Lie groups. Let $\mathbb{G}$ be a stratified Lie group, and let $M$ be a filtered manifold with a $\mathbb{G}$-atlas and a smooth positive density $\nu$. For the $C^{*}$-algebra bundle $E_{hom}$ of $M$ constructed from quasi-Riesz transforms on $\mathbb{G}$, we show that there exists a surjective $*$-homomorphism $${\rm sym}_{M}:\Pi_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,\nu))\subset \Pi_{M}$$ where the domain $\Pi_{M}\subset\mathcal{B}(L_{2}(M,\nu))$ is a $C^{*}$-algebra and $C_{b}(E_{hom})$ is the $C^{*}$-algebra of bounded continuous sections of $E_{hom}$. Especially, we do not make any assumptions on the lattice of the osculating group of $M$ or the assumption of compactness on manifolds in \cite{DAO3,DAO4}.

math.OA↗
arXiv2024

Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras

Let $\mathcal{M}$ be a semifinite von Neumann algebra and let $E$ be a symmetric function space on $(0,\infty)$. Denote by $E(\mathcal{M})$ the non-commutative symmetric space of measurable operators affiliated with $\mathcal{M}$ and associated with $E.$ Suppose $n\in \mathbb{N}$ and $E\cap L_{\infty}\not\subset L_{n,1}$, where $L_{n,1}$ is the Lorentz function space with the fundamental function $\varphi(t)=t^{1/n}$. We prove that for every $\varepsilon>0$ and every commuting self-adjoint $n$-tuple $(\alpha(j))_{j=1}^n,$ where $\alpha(j)$ is affiliated with $\mathcal{M}$ for each $1\leq j\leq n,$ there exists a commuting $n$-tuple $(\delta(j))_{j=1}^n$ of diagonal operators affiliated with $\mathcal{M}$ such that $\max\{\|\alpha(j)-\delta(j)\|_{E(\mathcal{M})},\|\alpha(j)-\delta(j)\|_{\infty}\}<\varepsilon$ for each $1\le j\le n$. In the special case when $\mathcal{M}=B(H)$, our results yield the classical Kuroda and Bercovici-Voiculescu theorems.

math.OA↗
arXiv2024

Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

Let $E(\mathbb{T}^{d}_{\theta}),F(\mathbb{T}^{d}_{\theta})$ be two symmetric operator spaces on noncommutative torus $\mathbb{T}^{d}_{\theta}$ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo--Nirenberg interpolation inequality with respect to $\mathbb{T}^{d}_{\theta}$: if $G=E^{1-\frac{l}{k}}F^{\frac{l}{k}}$ with $ 0\leq l\leq k$ and if the Ces\`{a}ro operator is bounded on $E$ and $F$, then \begin{align*} \|\nabla^lx\|_{G(\mathbb{T}^{d}_{\theta})}\leq 2^{3\cdot 2^{k-2}-2}(k+1)^d\|C\|_{E\to E}^{1-\frac{l}{k}}\|C\|_{F\to F}^{\frac{l}{k}}\|x\|_{E(\mathbb{T}^{d}_{\theta})}^{1-\frac{l}{k}}\|\nabla^kx\|_{F(\mathbb{T}^{d}_{\theta})}^{\frac{l}{k}},\; x\in W^{k,1}(\mathbb{T}^{d}_{\theta}), \end{align*} where $W^{k,1}(\mathbb{T}^{d}_{\theta})$ is the Sobolev space on $\mathbb{T}^{d}_{\theta}$ of order $k\in\mathbb{N}$. Our method is different from the previous settings, which is of interest in its own right.

math.FA↗
arXiv2024

Endpoint Schatten class properties of commutators

We study trace ideal properties of the commutators $[(-\Delta)^{\frac{\epsilon}{2}},M_f]$ of a power of the Laplacian with the multiplication operator by a function $f$ on $\mathbb R^d$. For a certain range of $\epsilon\in\mathbb R$, we show that this commutator belongs to the weak Schatten class $\mathcal L_{\frac d{1-\epsilon},\infty}$ if and only if the distributional gradient of $f$ belongs to $L_{\frac d{1-\epsilon}}$. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.

math.FA↗
arXiv2024

Diagonality modulo symmetric spaces in semifinite von Neumann algebras

In the study on the diagonality of an $n$-tuple $\alpha=(\alpha(j))_{j=1}^n$ of commuting self-adjoint operators modulo a given $n$-tuple $\Phi=(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus $k_{\Phi}(\alpha)$ and proved that $\alpha$ is diagonal modulo $(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ if and only if $k_{\Phi}(\alpha)=0.$ We prove that the same assertion holds true when $B(H)$ is replaced with a $\sigma$-finite semifinite von Neumann algebra $\mathcal{M}$, and $\mathcal{J}_1,\ldots,\mathcal{J}_n$ are replaced with symmetric spaces $E_1(\mathcal{M}),\ldots,E_n(\mathcal{M})$ associated with $\mathcal{M}.$

math.OA↗
arXiv2023

Calder\'on's commutator on Stratified Lie groups

Motivated by the recent work of Gimperlein and Goffeng on Calder\'on's commutator on compact Heisenberg type manifolds and the related weak Schatten class estimates, we establish the characterisation of $L^p$ boundedness for Calderon's commutator on stratified Lie groups. We further study related weak Schatten class estimates for second order commutators on two step stratified Lie groups, which include the Heisenberg groups. This latter result is obtained using double operator integral techniques which are novel in this area.

math.FA↗
arXiv2023

Local invariants of conformally deformed non-commutative tori II: multiple operator integrals

We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative $d$-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order $k$ and in any dimension $d$. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for $d\in\{2,4\}$ and $k\in\{0,2,4\}$. We exemplify this by writing down some known ($k=2$, $d=2$) and some novel ($k=2$, $d\geq 3$) formulas in the modular operator.

math.OA↗
arXiv2022

Asymptotics of singular values for quantum derivatives

We obtain Weyl type asymptotics for the quantised derivative $\dbar f$ of a function $f$ from the homgeneous Sobolev space $\dot{W}^1_d(\mathbb{R}^d)$ on $\mathbb{R}^d.$ The asymptotic coefficient $\|\nabla f\|_{L_d(\mathbb R^d)}$ is equivalent to the norm of $\dbar f$ in the principal ideal $\mathcal{L}_{d,\infty},$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dbar f.$ Our methods are based on the $C^{\ast}$-algebraic notion of the principal symbol mapping on $\mathbb{R}^d$, as developed recently by the last two authors and collaborators.

math.SP↗
arXiv2022

Optimality of Cwikel-Solomyak estimates in Orlicz spaces

We discuss the optimality of Cwikel-Solomyak estimates for the uniform operator norm and establish optimality of M.Z Solomyak's results \cite{Solomyak1995} within the class of Orlicz spaces. Our methods are based on finding the optimal version of the Sobolev embedding theorem.

math.FA↗