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Henri Martikainen

Publications and source records attributed to Henri Martikainen.

At least 37 records · Page 2Linked to original sources
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.

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arXiv2019

Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators

Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then $$ \| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(μ) \to L^p(λ)} \lesssim_{[μ]_{A_p}, [λ]_{A_p}} \prod_{i=1}^k\|b_i\|_{\operatorname{bmo}(ν^{θ_i})} , $$ where $p \in (1,\infty)$, $θ_i \in [0,1]$, $\sum_{i=1}^kθ_i=1$, $μ, λ\in A_p$, $ν:= μ^{1/p}λ^{-1/p}$. Here $A_p$ stands for the bi-parameter weights in $\mathbb{R}^n \times \mathbb{R}^m$ and $\operatorname{bmo}(ν)$ is a suitable weighted little BMO space. We also simplify the proof of the known first order case.

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arXiv2019

A new approach to non-homogeneous local $Tb$ theorems

We develop a new general method to prove various non-doubling local Tb theorems. The method combines the non-homogeneous good lambda method of Tolsa, the big pieces Tb theorem of Nazarov-Treil-Volberg and a new change of measure argument based on stopping time techniques. We also improve known results and discuss some further applications.

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arXiv2018

Dyadic-probabilistic methods in bilinear analysis

We demonstrate and develop dyadic-probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view. The main result is a new global $Tb$ theorem for Calderón-Zygmund operators in this setting. Our main tools include maximal truncations, adapted Cotlar type inequalities and suppression and big piece methods. While proving our bilinear results we also advance and refine the linear theory of Calderón-Zygmund operators by improving techniques and results. For example, we simplify and make more efficient some non-homogeneous summing arguments appearing in $T1$ type proofs. As a byproduct, we can manage with ease quite general modulus of continuity in the kernel estimates. Our testing conditions are also quite general by virtue of the big piece method of proof.

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arXiv2018

Bilinear Calderón-Zygmund theory on product spaces

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$ estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like $[b_1,T]_1$ and $[b_2, [b_1, T]_1]_2$, where $b_1$ and $b_2$ are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.

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arXiv2018

Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts

We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $\|b\|_{\operatorname{bmo}(\mathbb{R}^{n+m})} = 1$ and $1 < p, q \le \infty$ and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \|[b, T]_1(f_1, f_2)\|_{L^r(\mathbb{R}^{n+m})} \lesssim \|f_1\|_{L^p(\mathbb{R}^{n+m})} \|f_2\|_{L^q(\mathbb{R}^{n+m})}. $$ Previously the range $r \le 1$ was proved only in the paraproduct free situation. The main novelty lies in the treatment of the so called partial paraproducts.

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arXiv2018

Commutators of bilinear bi-parameter singular integrals

We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b, T]_1 \colon L^p(\mathbb{R}^{n+m}) \times L^q(\mathbb{R}^{n+m}) \to L^r(\mathbb{R}^{n+m})$ in the full range $1 < p, q \le \infty$, $1/2 < r < \infty$ satisfying $1/p+1/q = 1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular integrals satisfying the $T1$ type conditions. In doing so we simplify the corresponding linear proof. Lastly, we prove analogous results for iterated commutators.

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arXiv2017

Multi-parameter estimates via operator-valued shifts

We prove new results for multi-parameter singular integrals. For example, we prove that bi-parameter singular integrals in $\mathbb{R}^{n+m}$ satisfying natural $T1$ type conditions map $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ to $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ for all $p,q \in (1,\infty)$ and UMD function lattices $E$. This result is shown to hold even in the $\mathcal{R}$-boundedness sense for all suitable families of bi-parameter singular integrals. On the technique side we demonstrate how many dyadic multi-parameter operators can be bounded by using, and further developing, the theory of operator-valued dyadic shifts. Even in the scalar-valued case this is an efficient way to bound the various so called partial paraproducts, which are key operators appearing in the multi-parameter representation theorems. Our proofs also entail verifying the $\mathcal{R}$-boundedness of various families of multi-parameter paraproducts.

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arXiv2017

Bilinear representation theorem

We represent a general bilinear Calderón-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse T1 theorem for bilinear singular integrals.

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arXiv2016

Non-homogeneous square functions on general sets: suppression and big pieces methods

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $Tb$ theorems. The setting is new: we consider conical square functions with cones $\{x \in \mathbb{R}^n \setminus E: |x-y| < 2 \operatorname{dist}(x,E)\}$, $y \in E$, defined on general closed subsets $E \subset \mathbb{R}^n$ supporting a non-homogeneous measure $μ$. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls $B \subset E$, which are doubling and of small boundary. Due to the general set $E$ we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local $Tb$ theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.

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arXiv2016

Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $μ$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1} β_{2,μ}^{2}(3Q)\frac{\ell(Q)}{μ(Q)} 1_{Q}(x) $$ is finite for $μ$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $ε> 0$, we construct a Radon probability measure on $[0,1]^{2} \subset \mathbb{R}^{2}$ with the properties that $J_{1}(x) \leq ε$ for all $x \in \operatorname{spt} μ$, but nevertheless the $1$-dimensional lower density of $μ$ vanishes almost everywhere. In particular, $μ$ is purely $1$-unrectifiable.

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arXiv2016

Characterising the big pieces of Lipschitz graphs property using projections

We characterise the big pieces of Lipschitz graphs property in terms of projections. Roughly speaking, we prove that if a large subset of an $n$-Ahlfors-David regular set $E \subset \mathbb{R}^d$ has plenty of projections in $L^{2}$, then a large part of $E$ is contained in a single Lipschitz graph. This is closely related to a question of G. David and S. Semmes.

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arXiv2015

Improved Cotlar's inequality in the context of local $Tb$ theorems

We prove in the context of local $Tb$ theorems with $L^p$ type testing conditions an improved version of Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen-Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen-Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures $μ$ in $\mathbb{R}^d$ satisfying $μ(B(x,r)) \le Cr^n$, $n \in (0, d]$. The range of exponents in the Cotlar type inequality depend on $n$. Without assuming buffer we get the full range of exponents $p,q \in (1,2]$ for measures with $n \le 1$, and in general we get $p, q \in [2-ε(n), 2]$, $ε(n) > 0$. Consequences for (non-homogeneous) local $Tb$ theorems are discussed.

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arXiv2014

Some obstacles in characterising the boundedness of bi-parameter singular integrals

The famous $T1$ theorem for classical Calderón-Zygmund operators is a characterisation for their boundedness in $L^{2}$. In the bi-parameter case, on the other hand, the current $T1$ theorem is merely a collection of sufficient conditions. This difference in mind, we study a particular dyadic bi-parameter singular integral operator, namely the full mixed bi-parameter paraproduct $P$, which is precisely the operator responsible for the outstanding problems in the bi-parameter theory. We make several remarks about $P$, the common theme of which is to demonstrate the delicacy of the problem of finding a completely satisfactory product $T1$ theorem. For example, $P$ need not be unconditionally bounded if it is conditionally bounded -- a major difference compared to the corresponding one-parameter model operators. Moreover, currently the theory even lacks a characterisation for the potentially easier unconditional boundedness. The product BMO condition is sufficient, but far from necessary: we show by example that unconditional boundedness does not even imply the weaker rectangular BMO condition.

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arXiv2014

Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1,2)$

We continue the study of local $Tb$ theorems for square functions defined in the upper half-space $(\mathbb{R}^{n+1}_+, μ\times dt/t)$. Here $μ$ is allowed to be a non-homogeneous measure in $\mathbb{R}^n$. In this paper we prove a boundedness result assuming local $L^q$ type testing conditions in the difficult range $q \in (1,2)$. Our theorem is a non-homogeneous version of a result of S. Hofmann valid for the Lebesgue measure. It is also an extension of the recent results of M. Lacey and the first named author where non-homogeneous local $L^2$ testing conditions have been considered.

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arXiv2013

Local $Tb$ theorem with $L^2$ testing conditions and general measures: Calderón-Zygmund operators

Local $Tb$ theorems with $L^p$ type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. Until very recently, local $Tb$ theorems in the non-homogeneous case had only been proved assuming scale invariant ($L^{\infty}$ or BMO) testing conditions. The combination of non-scale-invariance and general measures is a delicate issue. In a previous paper we overcame this obstacle in the model case of square functions defined using general measures. In this paper we finally tackle the very demanding case of Calderón-Zygmund operators. That is, we prove a non-homogeneous local $Tb$ theorem with $L^2$ type testing conditions for all Calderón-Zygmund operators. In doing so we prove general twisted martingale transform inequalities which turn out to be subtle in our general framework.

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arXiv2013

Local Tb theorem with L2 testing conditions and general measures: Square functions

Local Tb theorems with Lp type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant ($L^{\infty}$ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a non-homogeneous local Tb theorem with L2 type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. The proof uses various recent innovations including a Whitney averaging formula and the insertion of a Calderon-Zygmund stopping data of a fixed function in to the construction of the twisted martingale difference operators.

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