Lp theory for outer measures and two themes of Lennart Carleson united

Do, Yen; Thiele, Christoph

doi:10.48550/arxiv.1309.0945

Cited by 3 publications

(15 citation statements)

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“…Outer measure spaces give us a very convenient language for doing that. We present here some basic facts about such spaces: a reader interested in more details should consult the paper [1]. The outer measure µ * extends to arbitrary A ⊂ D by the usual recipe: one considers all collections K ⊂ D such that A ⊂ I∈K D(I) and put…”

Section: 2mentioning

confidence: 99%

“…We will need the following simple fact. It is a particular instance of the Radon-Nikodym property for outer measure spaces from [1].…”

Section: 2mentioning

confidence: 99%

“…where ∆x := x − x 0 , ∆y := y − y 0 . Define B[1](X) = (xy) 1/2 , B [2] (X) := −(xy) 2 , so B = 128Q 3/2 B [1] + B [2] .…”

Section: Investigation Of Bellman Functionsmentioning

confidence: 99%

“…are not necessarily bounded in L 2 (w) with w satisfying the A 2 condition, even in the simple model case of the standard dyadic lattice in R 2 with a general underlying measure ν. 1 Here by martingale transform we mean an operator which is diagonal in the basis of the martingale difference spaces; but martingale multiplier is a martingale transform with all blocks being just multiples of identity.…”

Section: Introductionmentioning

confidence: 99%

“…Also, interesting results about non-homogeneous case were obtained in [9], where a dyadic analogue of the Hilbert Transform was considered. 1 The case of dyadic lattice in R, with a general underlying measure ν is an exception: all Haar subspaces are one-dimensional, so any Haar transform is a Haar multiplier (all blocks are multiples of identity), and it is the main result of this paper that the bounded in the unweighted L 2 Haar multipliers act in L 2 (w) if w satisfies the A2 condition.…”

Section: Introductionmentioning

confidence: 99%

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Weighted martingale multipliers in the non-homogeneous setting and outer measure spaces

Thiele

Treil

Volberg³

2015

Advances in Mathematics

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We investigate the unconditional basis property of martingale differences in weighted L 2 spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity [w] A 2 = sup I w I w −1 I , defined through averages · I relative to the reference measure ν, implies that each martingale transform relative to ν is bounded in L 2 (w dν). Moreover, we prove the linear in [w] A 2 estimate of the unconditional basis constant of the Haar system.Even in the classical case of the standard dyadic lattice in R n , where the results about unconditional basis and linear in [w] A 2 estimates are known, our result gives something new, because all the estimates are independent of the dimension n.Our approach combines the technique of outer measure spaces with the Bellman function argument.2010 Mathematics Subject Classification. 42B20, 42B35, 47A30.

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Section: 2mentioning

confidence: 99%

“…We will need the following simple fact. It is a particular instance of the Radon-Nikodym property for outer measure spaces from [1].…”

Section: 2mentioning

confidence: 99%

“…where ∆x := x − x 0 , ∆y := y − y 0 . Define B[1](X) = (xy) 1/2 , B [2] (X) := −(xy) 2 , so B = 128Q 3/2 B [1] + B [2] .…”

Section: Investigation Of Bellman Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning