arXiv subjects

Emil Vuorinen

Publications and source records attributed to Emil Vuorinen.

23 records · Page 2Linked to original sources

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.

math.CA↗

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.

math.CA↗

arXiv2017

Two weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function

We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. It is shown that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the measures holds, then a family of dyadic shifts satisfies the two weight estimate in an $\mathcal{R}$-bounded sense if and only if it satisfies the direct- and the dual quadratic testing condition. In the case $p=q=2$ this reduces to the result by T. Hytönen, C. Pérez, S. Treil and A. Volberg. The dyadic square function satisfies the two weight estimate if and only if it satisfies the quadratic testing condition and the quadratic $\mathscr{A}_{p,q}$-condition holds. Again in the case $p=q=2$ we recover the result by F. Nazarov, S. Treil and A. Volberg. An example shows that in general the quadratic $\mathscr{A}_{p,q}$-condition is stronger than the Muckenhoupt type $A_{p,q}$-condition.

math.CA↗

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.

math.CA↗

arXiv2016

L^{p}(μ)-L^{q}(ν) characterization for well localized operators

We consider a two weight $L^{p}(μ) \to L^{q}(ν)$-inequality for well localized operators as defined and studied by F. Nazarov, S. Treil and A. Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct analogue of these results fails for for $p=q\not=2$. Here a new square function testing condition is introduced and applied to characterize the two weight norm inequality. The use of the square function testing condition is also demonstrated in connection with certain positive dyadic operators.

math.CA↗