Searcharxiv⌕ Search

arXiv subjects

Henri Martikainen

Publications and source records attributed to Henri Martikainen.

At least 19 recordsLinked to original sources
arXiv2026

The Zygmund conjecture and Rey's exponential integrability conjecture

We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\ldots,s_{m-1},ϕ(s_1,\ldots,s_{m-1}))$, where $ϕ$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L)^{m-2}$ estimate for every $m\geq3$. The estimate is new for $m>3$. Our approach is to prove Rey's recent conjecture on the exponential integrability of the overlap function $h_{\mathcal G}=\sum_{I\in\mathcal G}\mathbf{1}_I$, with exponent $1/(m-1)$, for sparse incomparable families $\mathcal G$ of $m$-parameter dyadic rectangles, $m\geq2$. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.

math.CA↗
arXiv2026

A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles

We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let $\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $Ω$, and write $e_i(I):=e_i(I;Ω)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\colon\mathbb N^2\to[0,\infty)$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|Ω| \] holds uniformly in $(\mathcal U,Ω)$ if and only if the following fixed-total condition holds: \[ \|w\|_{\mathrm{ft}}:=\sup_{N\geq0}\sum_{a=0}^Nw(a,N-a)<\infty. \] For instance, $\|(a+1)^{-s}(b+1)^{-(1-s)}\|_{\mathrm{ft}}\sim[s(1-s)]^{-1}$ for $0<s<1$, so this allows weights that are not summable in either variable, while one-depth theory holds precisely for summable weights. Exactly the same characterization holds for dyadic Zygmund rectangles $I=I^1\times I^2\times I^3$ with $\ell(I^3)=\ell(I^1)\ell(I^2)$. We use probabilistic packing methods to prove the sufficiency, and these techniques also yield sparse refinements of the preceding estimates, which self-improve to various forms of exponential integrability.

math.CA↗
arXiv2026

Curved commutators in the plane

We complete the $L^p$ boundedness theory of commutators of Hilbert transforms along monomial curves by providing the previously missing lower bounds. This optimal result now covers all monomial curves while the previous result assumed the curve to intersect adjacent quadrants of the plane. We also develop, under a qualitative $\mathrm{BMO}$ assumption of the symbol, the corresponding quantitative lower bound in the context of curves with non-vanishing torsion.

math.CA↗
arXiv2024

On entangled and multi-parameter commutators

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of commutators $[b,T]$ from $L^p \to L^q$. Previously, only the $p=q$ upper bound in terms of a Zygmund type little $\operatorname{BMO}$ space was known for general operators, and it appears that there has been some confusion about the corresponding lower bound in recent literature. We give complete characterizations whenever $p \le q$ for a general class of non-degenerate Zygmund type singular integrals. Some of the results are somewhat surprising in view of existing papers - for instance, compactness always forces $b$ to be constant. Even in the simpler situation of bi-parameter singular integrals it appears that this has not been observed previously.

math.CA↗
arXiv2024

Zygmund dilations: bilinear analysis and commutator estimates

We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $(x_1, x_2, x_3) \mapsto (δ_1 x_1, δ_2 x_2, δ_1 δ_2 x_3)$ in $\mathbb{R}^3$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates.

math.CA↗
arXiv2024

On the compactness of the bi-commutator

We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calderón-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm $L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then extend these to other exponents, including the diagonal $p_i = q_i$, with a new extrapolation argument. In particular, the natural product $\operatorname{VMO}$ condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality $p_i \le q_i$ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama.

math.CA↗
arXiv2022

Multiresolution analysis and Zygmund dilations

Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting - in $\mathbb{R}^3 = \mathbb{R} \times \mathbb{R} \times \mathbb{R}$ they are the dilations $(x_1, x_2, x_3) \mapsto (δ_1 x_1, δ_2 x_2, δ_1 δ_2 x_3)$. The dyadic multiresolution analysis and the related dyadic-probabilistic methods have been very impactful in the modern product singular integral theory. However, the multiresolution analysis has not been understood in the Zygmund dilation setting or in other modified product space settings. In this paper we develop this missing dyadic multiresolution analysis of Zygmund type, and justify its usefulness by bounding, on weighted spaces, a general class of singular integrals that are invariant under Zygmund dilations. We provide novel examples of Zygmund $A_p$ weights and Zygmund kernels showcasing the optimality of our kernel assumptions for weighted estimates.

math.CA↗
arXiv2022

Exotic Calderón-Zygmund operators

We study singular integral operators with kernels that are more singular than standard Calderón-Zygmund kernels, but less singular than bi-parameter product Calderón-Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calderón-Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.

math.CA↗
arXiv2021

Genuinely multilinear weighted estimates for singular integrals in product spaces

We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications -- for example, a free access to mixed-norm estimates in the full range of exponents.

math.CA↗
arXiv2020

Modern singular integral theory with mild kernel regularity

We present a framework based on modified dyadic shifts to prove multiple results of modern singular integral theory under mild kernel regularity. Using new optimized representation theorems we first revisit a result of Figiel concerning the UMD-extensions of linear Calderón-Zygmund operators with mild kernel regularity and extend our new proof to the multilinear setting improving recent UMD-valued estimates of multilinear singular integrals. Next, we develop the product space theory of the multilinear singular integrals with modified Dini-type assumptions, and use this theory to prove bi-parameter weighted estimates and two-weight commutator estimates.

math.CA↗
arXiv2020

Multilinear singular integrals on non-commutative $L^p$ spaces

We prove $L^p$ bounds for the extensions of standard multilinear Calderón-Zygmund operators to tuples of UMD spaces tied by a natural product structure. This can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space - in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative $L^p$ spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.

math.CA↗
arXiv2020

Multilinear operator-valued Calderón-Zygmund theory

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative $L^p$-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property $α$.

math.CA↗
arXiv2020

Off-diagonal estimates for bi-commutators

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón-Zygmund operators, and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.

math.CA↗
arXiv2020

Some new weighted estimates on product spaces

We complete our theory of weighted $L^p(w_1) \times L^q(w_2) \to L^r(w_1^{r/p} w_2^{r/q})$ estimates for bilinear bi-parameter Calderón--Zygmund operators under the assumption that $w_1 \in A_p$ and $w_2 \in A_q$ are bi-parameter weights. This is done by lifting a previous restriction on the class of singular integrals by extending a classical result of Muckenhoupt and Wheeden regarding weighted BMO spaces to the product BMO setting. We use this extension of the Muckenhoupt-Wheeden result also to generalise some two-weight commutator estimates from bi-parameter to multi-parameter. This gives a fully satisfactory Bloom type upper estimate for $[T_1, [T_2, \ldots [b, T_k]]]$, where each $T_i$ can be a completely general multi-parameter Calderón--Zygmund operator.

math.CA↗
arXiv2020

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ``finite'' case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

math.CA↗
arXiv2019

Banach-valued multilinear singular integrals with modulation invariance

We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinear Calderón-Zygmund operators recently obtained by the same authors.

math.CA↗
arXiv2019

Bloom type upper bounds in the product BMO setting

For a bounded singular integral $T_n$ in $\mathbb{R}^n$ and a bounded singular integral $T_m$ in $\mathbb{R}^m$ we prove that $$ \| [T_n^1, [b, T_m^2]] \|_{L^p(μ) \to L^p(λ)} \lesssim_{[μ]_{A_p}, [λ]_{A_p}} \|b\|_{\operatorname{BMO}_{\textrm{prod}}(ν)}, $$ where $p \in (1,\infty)$, $μ, λ\in A_p$ and $ν:= μ^{1/p}λ^{-1/p}$. Here $T_n^1$ is $T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\mathbb{R}^n \times \mathbb{R}^m$ and $\operatorname{BMO}_{\textrm{prod}}(ν)$ is a weighted product BMO space.

math.CA↗