Searcharxiv⌕ Search

arXiv subjects

Tuomas Hytönen

Publications and source records attributed to Tuomas Hytönen.

At least 19 recordsLinked to original sources
arXiv2026

Schatten--Lorentz properties of commutators with matrix weights

We study the membership of commutators $[b,T]$ of Calderón--Zygmund operators in the Schatten--Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type $(X,ρ,μ)$ with a matrix $A_2$ weight $W$, we characterise, for $p\in(1,\infty)$ and $q\in[1,\infty)$, the Schatten--Lorentz norm $\|[b,T]\|_{S^{p,q}(L^2(W))}$ in terms of an oscillation (Besov-type) norm $\|b\|_{\mathrm{Osc}^{p,q}}$ of the symbol: for all $p\in(1,\infty)$, finiteness of $\|b\|_{\mathrm{Osc}^{p,q}}$ is necessary; and for $p>p(η)$, where $p(η)$ depends on the regularity of the kernel of $T$ and a dimensional parameter of the space, the two-sided estimate $\|[b,T]\|_{S^{p,q}(L^2(W))}\approx\|b\|_{\mathrm{Osc}^{p,q}}$ holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.

math.CA↗
arXiv2026

Rough spectral asymptotics for commutators of general singular integrals

Recently, Schatten class membership of commutators arising from Connes' quantised calculus has been characterised in a general framework, which covers multiple concrete situations of interest. In contrast to this, related results dealing with the asymptotic behaviour of the singular values of these commutators have been restricted to a relatively short list of specific examples only. In this work, we show that a version of the spectral asymptotic formula, involving comparable lower and upper limits in place of an exact limit, remains valid in great generality of singular integrals over metric measure spaces. A key proof ingredient of independent interest is a new asymptotic version of the Marcinkiewicz interpolation theorem. We also present a new approach to commutator upper bounds at the critical index; while not quite as general as more elaborate methods, it is general enough to reproduce the known upper bounds in Heisenberg and Carnot groups in a much simpler way.

math.FA↗
arXiv2026

Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón--Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-Δ+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.

math.FA↗
arXiv2026

Schatten properties of commutators on metric spaces

We characterise the Schatten class $S^p$ properties of commutators $[b,T]$ of singular integrals and pointwise multipliers in a general framework of (quasi-)metric measure spaces. This covers, unifies, and extends a range of previous results in different special cases. As in the classical results on $\mathbb R^d$, the characterisation has three parts: (1) For $p>d$, we have $[b,T]\in S^p$ if and only if $b$ is in a suitable Besov (or fractional Sobolev) space. (2) For $p\leq d$, we have $[b,T]\in S^p$ if and only if $b$ is constant. (3) For $p=d$, we have $[b,T]\in S^{d,\infty}$ (a weak-type Schatten class) if and only if $b$ is in a first-order Sobolev space. Result (1) extends to all spaces of homogeneous type as long as there are appropriate singular integrals, but for the more delicate properties (2) and (3), we assume a complete doubling metric space supporting a suitable Poincaré inequality, which is still very general. These latter results depend on new characterisations of constant functions and Sobolev spaces over such spaces obtained in a companion paper of the author with R. Korte. Even when specialised to various concrete domains considered earlier, the present results extend ones available in the literature by covering a larger class of operators with minimal kernel assumptions, removing a-priori assumptions on the pointwise multiplier $b$, and allowing Schatten classes on the weighted spaces $L^2(w)$ with an arbitrary Muckenhoupt weight $w\in A_2$. Even on $\mathbb R^d$, such weighted results were previously known for a few special operators $T$ only, and on all other domains, they are completely new.

math.FA↗
arXiv2026

$A_\infty$-invariance of oscillatory norms, and Schatten characterisations of commutators

Schatten class properties of commutators $[b,T]$ of pointwise multipliers $b$ and singular integral operators $T$ have been characterised in a variety of settings. An abstract framework, covering many of these results as special cases, was proposed by the author [arXiv:2411.02613]. However, recent results about commutators of the concrete Bessel-Riesz transforms by Fan-Li-Sukochev-Zanin [arXiv:2411.14928] are beyond this abstract setting. In this work, we present an extension of the framework of [arXiv:2411.02613], introducing two measures $μ$ and $ν$ that are $A_\infty$-equivalent to each other. The commutators act on a given space $L^2(μ)$, but the characterising function space norms of the multiplier $b$ are taken with respect to another measure $ν$. In this way, assumptions like Ahlfors regularity and Poincaré inequality on the original measure $μ$ may be relaxed, as long as there is an $A_\infty$-equivalent measure $ν$ that satisfies these assumptions. In the Bessel example, the original $μ$ fails to be Ahlfors regular, but $ν$ is simply the Lebesgue measure. Within this framework, the Schatten norm characterisations of commutators of the Bessel-Riesz transforms at the critical-index by Fan-Li-Sukochev-Zanin [op cit.] are recovered by a completely different argument, replacing non-commutative techniques by real-variable harmonic analysis and hardly using any specifics of the Bessel setting. As a by-product, we also obtain a simpler characterisation in the non-critical case, replacing an ad-hoc Besov space of Fan-Lacey-Li-Xiong [J. Funct. Anal. 2026] by a classical Besov space.

math.FA↗
arXiv2026

Real-variable theory of matrix-weighted multi-parameter Besov--Triebel--Lizorkin-type spaces

We develop a comprehensive theory for a general class of multi-parameter function spaces of Besov-Triebel-Lizorkin type, with a matrix weight. We prove the equivalence of different quasi-norms, the identification of function and sequence spaces via the $φ$-transform, the boundedness of almost diagonal operators and multi-parameter singular integrals under minimal assumptions, molecular and wavelet characterisations, and Sobolev-type embedding theorems. We identify matrix-weighted $L^p$ spaces, Sobolev spaces, and multi-parameter BMO spaces as examples of our general scale of spaces. Thus, our result on the boundedness of multi-parameter singular integrals on these spaces is seen as an extension, with a different method, of a recent theorem of Domelevo et al. [J. Math. Anal. Appl. 2024] on matrix-weighted $L^p$ spaces. For this theory, we develop several tools of independent interest. Many previous results were restricted to integrability exponents $p\in(1,\infty)$, while Besov-Triebel-Lizorkin spaces naturally involve the full range $p\in(0,\infty)$. We extend the definition of multi-parameter $A_p$ matrix weights to $p\in(0,1]$ and establish their basic properties, culminating in the $L^p$-boundedness of a matrix-weighted strong maximal operator (suitably rescaled when $p\in(0,1]$) for all $p\in(0,\infty)$. For $p\in(1,\infty)$, this is due to Vuorinen [Adv. Math. 2024] by convex-set-valued techniques of Bownik and Cruz-Uribe [arXiv 2022; Math. Ann. (to appear)]; the lack of convexity requires us to develop a new approach that works for all $p\in(0,\infty)$. We also need and prove a multi-parameter extension of Carleson-type embeddings from Frazier and Roudenko [Math. Ann. 2021] but attributed by them to F. Nazarov. We prove the necessity of the conditions of the new embedding using a nontrivial elaboration of Carleson's classical counterexample [Mittag-Leffler Rep. 1974].

math.FA↗
arXiv2026

Characterisations of Sobolev spaces and constant functions over metric spaces

In a doubling metric measure space $(X,ρ,μ)$ supporting a Poincaré inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of the finiteness of certain integrals through a new approach. As a key tool of independent potential, we introduce a novel ``macroscopic'' Poincaré inequality, whose right-hand side has oscillations of the same form as the left-hand side, but at a smaller macroscopic scale $r\in(0,R)$. Besides intrinsic interest, these results are motivated by applications to quantitative compactness properties of commutators $[f,T]$ of pointwise multipliers and singular integrals. With pivotal use of the present results, a characterisation of commutator mapping properties, over the same class of general domains $(X,ρ,μ)$, is obtained in a companion paper.

math.FA↗
arXiv2025

Schatten properties of commutators of fractional integrals on spaces of homogeneous type

Extending classical results of Janson and Peetre (1988) on the Schatten class $S^p$ membership of commutators of Riesz potentials on the Euclidean space, we obtain analogous results for commutators $[b,T]$, where $T\in\{T_\varepsilon,\widetilde T_α\}$ belongs to either one of two natural classes of fractional integral operators on a space of homogeneous type. Our approach is based on recent related work of Hytönen and Korte on singular (instead of fractional) integrals; working directly with the kernels, it differs from the Fourier analytic considerations of Janson and Peetre, covering new operators even when specialised to $\mathbb R^d$. The cleanest case of our characterization in spaces of lower dimension $d> 2$ and satisfying a $(1,2)$-Poincaré inequality is as follows. For a parameter $\varepsilon \in (0,\frac{1}{2}-\frac{1}{d})$ describing the order of the fractional integral $T_\varepsilon $, we have a dichotomy: If $\frac{d}{1+d\varepsilon }<p<\frac{1}{\varepsilon}$, then $[b,T_{\varepsilon}]\in S^p$ if and only if $b$ belongs to a suitable Besov (or fractional Sobolev) space. If $0<p\leq \frac{d}{1+d\varepsilon }$, then $[b,T_{\varepsilon}]\in S^p$ if and only if $b$ is constant. This is analogous to the result for singular integrals, where a similar cut-off happens at $p=d$, formally corresponding to fractional order $\varepsilon =0$. We also obtain results for other parameter values, including dimensions $0<d\leq 2$. As an application, these results are used to show Schatten properties of commutators of fractional Bessel operators, complementing recent related results of Fan, Lacey, Li, and Xiong (2025) on commutators of singular integrals in the Bessel setting.

math.FA↗
arXiv2025

Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_\infty$ Weights

Introduced by A. Volberg, matrix $A_{p,\infty}$ weights provide a suitable generalization of Muckenhoupt $A_\infty$ weights from the classical theory. In our previous work, we established new characterizations of these weights. Here, we use these results to study inhomogeneous Besov-type and Triebel--Lizorkin-type spaces with such weights. In particular, we characterize these spaces, in terms of the $φ$-transform, molecules, and wavelets, and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón--Zygmund operators on these spaces. This is the first systematic study of inhomogeneous Besov--Triebel--Lizorkin-type spaces with $A_{p,\infty}$-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.

math.FA↗
arXiv2024

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces II: Sharp Boundedness of Almost Diagonal Operators

This article is the second one of three successive articles of the authors on the matrix-weighted Besov-type and Triebel--Lizorkin-type spaces. In this article, we obtain the sharp boundedness of almost diagonal operators on matrix-weighted Besov-type and Triebel--Lizorkin-type sequence spaces. These results not only possess broad generality but also improve several existing related results in various special cases covered by this family of spaces. This improvement depends, on the one hand, on the notion of $A_p$-dimensions of matrix weights and their properties introduced in the first article of this series and, on the other hand, on a careful direct analysis of sequences of averages avoiding maximal operators. While a recent matrix-weighted extension of the Fefferman--Stein vector-valued maximal inequality would provide an alternative route to some of our results in the restricted range of function space parameters $p,q\in(1,\infty)$, our approach covers the full scale of exponents $p\in(0,\infty)$ and $q\in(0,\infty]$ that is relevant in the theory of function spaces.

math.FA↗
arXiv2024

Quantitative estimates for bounded holomorphic semigroups

In this paper we revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood--Paley--Stein theory for symmetric diffusion semigroups.

math.FA↗
arXiv2023

On the sense of convergence in the dyadic representation theorem

The dyadic representation of any singular integral operator, as an average of dyadic model operators, has found many applications. While for many purposes it is enough to have such a representation for a "suitable class" of test functions, we show that, under quite general assumptions (essentially minimal ones to make sense of the formula), the representation is actually valid for all pairs $(f,g)\in L^p(\mathbb R^d)\times L^{p'}(\mathbb R^d)$, not just test functions.

math.CA↗
arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces III: Characterizations of Molecules and Wavelets, Trace Theorems, and Boundedness of Pseudo-Differential Operators and Calderón--Zygmund Operators

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel--Lizorkin-type spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón--Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}(W)$ and $\dot F^{s}_{p,q}(W)$. In particular, even when reverting to the boundedness of Calderón--Zygmund operator on unweighted Triebel--Lizorkin spaces $\dot F^{s}_{p,q}$, these results are still better.

math.FA↗
arXiv2023

New Characterizations and Properties of Matrix $A_\infty$ Weights

We provide several new characterizations of $A_{p,\infty}$-matrix weights, originally introduced by A. Volberg as matrix-valued substitutes of the classical $A_\infty$ weights. In analogy with the notion of $A_p$-dimension of matrix weights introduced in our previous work, we introduce the concepts of the lower and the upper dimensions of $A_{p,\infty}$-matrix weights, which enable us to obtain sharp estimates related to their reducing operators. In a follow-up work, these results will play a key role in the study of function spaces with $A_{p,\infty}$-matrix weights, which extends earlier results in the more restricted class of $A_p$-matrix weights.

math.FA↗
arXiv2023

Singular integrals in uniformly convex spaces

We consider the action of finitely truncated singular integral operators on functions taking values in a Banach space. Such operators are bounded for any Banach space, but we show a quantitative improvement over the trivial bound in any space with an equivalent uniformly convex norm. This answers a question asked by Naor and the author, who previously proved the result in the important special case of the finite Hilbert transforms. The proof, which splits the operator into a cancellative part and two paraproducts, follows the broad outline of similar results for genuinely singular (non-truncated) operators in the narrower class of UMD spaces. Thus we revisit and survey the recent techniques behind such results, but our precise setting, the main theorem, and some aspects of its proof, are new. Curiously, it turns out that the paraproducts admit somewhat better bounds than the full operator. In a large class of spaces other than UMD, they remain bounded even without the finite truncations.

math.FA↗
arXiv2023

Fractional Bloom boundedness and compactness of commutators

Let $T$ be a non-degenerate Calderón-Zygmund operator and let $b:\mathbb{R}^d\to\mathbb{C}$ be locally integrable. Let $1<p\leq q<\infty$ and let $μ^p\in A_p$ and $λ^q\in A_q,$ where $A_{p}$ denotes the usual class of Muckenhoupt weights. We show that \begin{align*} \|[b,T]\|_{L^p_μ\to L^q_λ}\sim \|b\|_{\operatorname{BMO}_ν^α},\qquad [b,T]\in \mathcal{K}(L^p_μ, L^q_λ)\quad\mbox{iff}\quad b\in \operatorname{VMO}_ν^α, \end{align*} where $L^p_μ=L^p(μ^p)$ and $α/d = 1/p-1/q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\operatorname{BMO}_ν^α$ and $\operatorname{VMO}_ν^α$ spaces are defined through the following fractional oscillation and Bloom weight \begin{align*} \mathcal{O}_ν^α(b;Q) = ν^{-α/d}(Q)\Big(\frac{1}{ν(Q)}\int_Q |b-\langle b\rangle_Q|\Big),\qquad ν = \big(\fracμλ\big)^β,\quad β= (1+α/d)^{-1}. \end{align*} The key novelty is dealing with the off-diagonal range $p<q$, whereas the case $p=q$ was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions $b$, while other arguments based on the median of $b$ on a set are inherently real-valued.

math.CA↗
arXiv2022

Dyadic representation theorem using smooth wavelets with compact support

The representation of a general Calderón--Zygmund operator in terms of dyadic Haar shift operators first appeared as a tool to prove the $A_2$ theorem, and it has found a number of other applications. In this paper we prove a new dyadic representation theorem by using smooth compactly supported wavelets in place of Haar functions. A key advantage of this is that we achieve a faster decay of the expansion when the kernel of the general Calderón--Zygmund operator has additional smoothness.

math.CA↗