arXiv subjects

Tuomas P. Hytönen

Publications and source records attributed to Tuomas P. Hytönen.

At least 19 recordsLinked to original sources

arXiv2026

Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces

Given a complete doubling metric measure space $(X,ρ,μ)$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(μ)$ and the space of functions of bounded variation, achieving a full analogy in general Poincaré spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $γ\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(μ)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(μ)} \sim \|ρ^{-1}ϕ^{-γ}F\|_{L^{p,\infty}(ϕ^{γp}V^{-1})}, \qquad F\in\{Δf,m_f\},\quad ϕ\in\{ρ,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(μ)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=μ(B(x,ρ(x,y)))$ and $Δf(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,ρ(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(μ)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $γ\in(-\infty,-1)\cup(0,\infty)$ for $ϕ=ρ$ or $γ\in (-\infty,-\frac1d)\cup(0,\infty)$ for $ϕ=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.

math.FA↗

arXiv2026

Real-variable theory of function spaces with operator-valued $A_p$ weights in Banach spaces

While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective generalizations. However, we show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt $A_p$ weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the $φ$-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the $T(1)$ theorem and trace/extension theorems. A key tool is a version of the reverse Hölder inequality, which is weak enough to follow from the operator-valued $A_p$ condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical counterpart. In contrast to the established scalar and matrix-weighted theories, our approach cannot build on operator-weighted $L^p$ results, as these fail in a very definite way. We also strengthen the existing "no go" statements in Hilbert spaces, showing (among other counterexamples) that every infinite-dimensional Banach space has an operator-valued $A_p$ weight $V$ for which the Hilbert transform is unbounded on $L^p(V)$. This is nontrivial, since the lack of Hilbert space structure also complicates the construction of $A_p$ weights. Building on results from Banach space theory, we achieve this unboundedness by combining two distinct methods in two different classes of spaces (so-called $K$-convex ones and those that are {\em not} UMD), whose union covers all Banach spaces.

math.FA↗

arXiv2024

Reduced inequalities for vector-valued functions

Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product $\vec f(x)\cdot\vec g(y)$ instead of the product of lengths $|\vec f(x)| |\vec g(y)|$ that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of $L^p$ norms, also with $p<1$, that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method.

math.FA↗

arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces I: $A_p$-Dimensions of Matrix Weights and $φ$-Transform Characterizations

Let $s\in{\mathbb R}$, $q\in (0,\infty]$, and $τ\in[0,\infty)$. It is well known that Besov-type spaces $\dot B^{s,τ}_{p,q}$ with $p\in (0,\infty]$ and Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}$ with $p\in (0,\infty)$ when $τ\in [0,\infty)$ or with $p\in (0,\infty]$ when $τ=0$ on $\mathbb{R}^n$ consist of a general family of function spaces that cover not only the well-known Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}$ and $\dot F^{s}_{p,q}$ (when $τ=0$) but also several other function spaces of interest, such as Morrey spaces and $Q$ spaces. In three successive articles, the authors develop a complete real-variable theory of matrix-weighted Besov-type spaces $\dot B^{s,τ}_{p,q}(W)$ and matrix-weighted Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}(W)$ on $\mathbb{R}^n$, where $W$ is a matrix-valued Muckenhoupt $A_p$ weight. This article is the first one, whose main novelty exists in that the authors introduce the new concept, $A_p$-dimensions of matrix weights, and intensively study their properties, especially those elaborate properties expressed via reducing operators. The authors then introduce the spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$ and, using $A_p$-dimensions and their nice properties, the authors establish the $φ$-transform characterization of $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. The $A_p$-dimensions of matrix weights and their properties also enable the authors to obtain the sharp boundedness of almost diagonal operators on related sequence spaces in the subsequent second article and the optimal characterizations of molecules and wavelets, trace theorems, and the optimal boundedness of pseudo-differential operators and Calderón--Zygmund operators in the subsequent third article.

math.FA↗

arXiv2023

Some remarks on convex body domination

Convex body domination is an important elaboration of the technique of sparse domination that has seen significant development and applications over the past ten years. In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued functions, and yields matrix-weighted norm inequalities in this setting. We explore applications to "generalised commutators", obtaining new examples of bounded operators among linear combinations of compositions of pointwise multipliers and a singular integral operator.

math.FA↗

arXiv2021

The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator

Supplying the missing necessary conditions, we complete the characterisation of the $L^p\to L^q$ boundedness of commutators $[b,T]$ of pointwise multiplication and Calderón-Zygmund operators, for arbitrary pairs of $1 q$, our results are new even for special classical operators with smooth kernels. As an application, we show that every $f\in L^p(R^d)$ can be represented as a convergent series of normalised Jacobians $Ju=\det\nabla u$ of $u\in \dot W^{1,dp}(R^d)^d$. This extends, from $p=1$ to $p>1$, a result of Coifman, Lions, Meyer and Semmes about $J:\dot W^{1,d}(R^d)^d\to H^1(R^d)$, and supports a conjecture of Iwaniec about the solvability of the equation $Ju=f\in L^p(R^d)$.

math.CA↗

arXiv2019

Representation of singular integrals by dyadic operators, and the A_2 theorem

This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journé $T(1)$ theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.

math.CA↗

arXiv2019

The two-weight inequality for the Hilbert transform with general measures

The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $σ$ and $w$ on $\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $σ$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes'.

math.CA↗

arXiv2019

Restricted testing for positive operators

We prove that for certain positive operators $T$, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant $D>1$, depending only on the dimension $n$, such that the two weight norm inequality \begin{equation*} \int_{\mathbb{R}^{n}}T\left( fσ\right) ^{2}dω\leq C\int_{\mathbb{ R}^{n}}f^{2}dσ\end{equation*} holds for all $f\geq 0$ if and only if the (fractional) $A_{2}$ condition holds, and the restricted testing condition \begin{equation*} \int_{Q}T\left( 1_{Q}σ\right) ^{2}dω\leq C\left\ | Q\right\ |_{σ} \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{σ}\leq D\left\ | Q\right\ |_{σ}$. If $T$ is linear, we require as well that the dual restricted testing condition \begin{equation*} \int_{Q}T^{\ast }\left( 1_{Q}ω\right) ^{2}dσ\leq C\left\ | Q\right\ |_{ω} \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{ω}\leq D\left\ | Q\right\ |_{ω}$.

math.CA↗

arXiv2019

Of commutators and Jacobians

I discuss the prescribed Jacobian equation $Ju=\det\nabla u=f$ for an unknown vector-function $u$, and the connection of this problem to the boundedness of commutators of multiplication operators with singular integrals in general, and with the Beurling operator in particular. A conjecture of T. Iwaniec regarding the solvability for general datum $f\in L^p(R^d)$ remains open, but recent partial results in this direction will be presented. These are based on a complete characterisation of the $L^p$-to-$L^q$ boundedness of commutators, where the regime of exponents $p>q$, unexplored until recently, plays a key role. These results have been proved in general dimension $d\geq 2$ elsewhere, but I will here present a simplified approach to the important special case $d=2$, using a framework suggested by S. Lindberg.

math.AP↗

arXiv2018

Weighted Estimates for Operator-Valued Fourier Multipliers

We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.

math.CA↗

arXiv2018

Off-diagonal sharp two-weight estimates for sparse operators

For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and $A_{\infty}$-characteristics. These bounds are known to be sharp in many cases; as a new result, we prove their sharpness for fractional square functions.

math.CA↗

arXiv2017

Weak and strong $A_p$-$A_\infty$ estimates for square functions and related operators

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w]_{A_p}^{1/p}[w]_{A_\infty}^{1/2-1/p}\lesssim [w]_{A_p}^{1/2}$ for the weak type norm of square functions on $L^p(w)$ for $p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.

math.CA↗

arXiv2017

Sparse bounds for maximal rough singular integrals via the Fourier transform

We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally truncated singular integrals satisfying a sparse bound by $(1+\varepsilon,1+\varepsilon)$-averages for all $\varepsilon>0$, with linear growth in $\varepsilon^{-1}$. This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to maximally truncated singular integrals. Our results cover the rough homogeneous singular integrals $T_Ω$ on $\mathbb{R}^d$ with bounded angular part $Ω$ having vanishing integral on the $(d-1)$-dimensional sphere. Consequences of our sparse bound include novel quantitative weighted norm estimates as well as Fefferman-Stein type inequalities. In particular, we obtain that the $L^2(w)$ norm of the maximal truncation of $T_Ω$ depends quadratically on the Muckenhoupt constant $[w]_{A_2}$, extending a result originally by Roncal, Tapiola and the second author. A suitable convex-body valued version of the sparse bound is also deduced and employed towards novel matrix weighted norm inequalities for the maximal truncated rough homogeneous singular integrals. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.

math.CA↗

arXiv2016

The Holmes--Wick theorem on two-weight bounds for higher order commutators revisited

A sufficient condition for the two-weight boundedness of higher order commutators was recently obtained by Holmes and Wick in terms of an intersection of two BMO spaces. We provide an alternative proof, showing that the higher order case can be deduced by a classical Cauchy integral argument from the corresponding first order result of Holmes, Lacey and Wick.

math.CA↗

arXiv2015

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound that is quadratic in the $A_2$ constant $[w]_{A_2}$. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

math.CA↗

arXiv2015

Bounds for partial derivatives: necessity of UMD and sharp constants

We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of $L^p$ bounds between different partial derivatives $\partial^βu$ of $u\in C^\infty_c(\mathbb{R}^n,X)$. In particular, we show that the estimate $\|u_{xy}\|_p\leq K(\|u_{xx}\|_p+\|u_{yy}\|_p)$ characterizes the UMD property, and the best constant $K$ is equal to one half of the UMD constant. This precise value of $K$ seems to be new even for scalar-valued functions.

math.CA↗

arXiv2014

Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified approach to $p\leq q$ and $p>q$

We characterize the $L^p(σ)\to L^q(ω)$ boundedness of positive dyadic operators of the form $ T(fσ)=\sum_{Q\in\mathscr{D}}λ_Q\int_Q f\,\mathrm{d}σ\cdot 1_Q, $ and the $L^{p_1}(σ_1)\times L^{p_2}(σ_2)\to L^q(ω)$ boundedness of their bilinear analogues, for arbitrary locally finite measures $σ,σ_1,σ_2,ω$. In the linear case, we unify the existing "Sawyer testing" (for $p\leq q$) and "Wolff potential" (for $p>q$) characterizations into a new "sequential testing" characterization valid in all cases. We extend these ideas to the bilinear case, obtaining both sequential testing and potential type characterizations for the bilinear operator and all $p_1,p_2,q\in(1,\infty)$. Our characterization covers the previously unknown case $q<\frac{p_1p_2}{p_1+p_2}$, where we introduce a new two-measure Wolff potential.

math.CA↗