Martingale Transforms

Burkholder, D. L.

doi:10.1214/aoms/1177699141

Cited by 346 publications

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“…This inequality was established by Burkholder in [2], actually in a slightly more general setting of martingales. In the boundary case p = 1, the inequality does not hold with any finite constant.…”

Section: Introductionmentioning

confidence: 80%

Weighted square function inequalities

Oşekowski¹

2018

Publicacions Matemàtiques

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Section: Introductionmentioning

confidence: 80%

Weighted square function inequalities

Oşekowski¹

2018

Publicacions Matemàtiques

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“…2 Together these results show that if u 0 is bounded away from 0 and σ is sublinear, then the solution to (1.1) is "weakly intermittent" [that is, highly peaked for large t]. Rather than describe why this is a noteworthy property, we refer the interested reader to the extensive bibliography of [17], which contains several pointers to the literature in mathematical physics that motivate [weak] intermittency.…”

Section: Introductionmentioning

confidence: 91%

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus

Khoshnevisan

2010

Probab. Theory Relat. Fields

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We study the stochastic heat equation ∂ t u = Lu + σ(u)Ẇ in (1 + 1) dimensions, whereẆ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a symmetric Lévy process that has finite exponential moments, and u 0 has exponential decay at ±∞. We prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting. [See, however,[19,20] for the analysis of the location of the peaks in a different model.]Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

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“…[1], [2]) that every general Haar system is an unconditional basis in L p ([0, 1]), 1 < p < ∞, and (2.10)…”

Section: Definition and Basic Properties Of Generalmentioning

confidence: 99%

“…This is checked by induction on d. It is clear that it holds for d = 1 with C λ,1 = 1. Now, let d > 1 and suppose that m ∈ N admits a representation m = s λ,ν 1 …”

Section: Then There Is a Constant C λD > 0 (Depending Only On λ And mentioning

confidence: 99%

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General Haar systems and greedy approximation

Kamont

2001

Studia Math.

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Abstract. We show that each general Haar system is permutatively equivalent in L p ([0, 1]), 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in L p ([0, 1]), 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in eachThis is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases inWe also note that the above-mentioned general Haar system is not permutatively equivalent to the whole dyadic Haar system in any1. Introduction. By a general Haar system corresponding to a dense sequence T = {t n : n ≥ 0} ⊂ [0, 1], we mean a sequence of orthonormal (in L 2 ([0, 1])) functions which are constant on intervals generated by the points of the sequence T -it is constructed analogously to the classical dyadic Haar system, but with the sequence T used instead of the sequence of dyadic points. (For a more detailed description of general Haar functions, see Section 2.2.) It has been shown by L. E. Dor and E. Odell [3] (cf. also the monograph [9]) that there are pairs of general Haar systems which are not equivalent in any L p ([0, 1]), 1 < p < ∞, p = 2. They have also asked whether there exist general Haar systems which are not permutatively equivalent in these spaces. (Recall that two basic sequences in a Banach space are called permutatively equivalent if one of them is equivalent to some permutation of the other.) In the present paper, we prove that each general Haar system is permutatively equivalent in L p ([0, 1]), 1 < p < ∞, to some subsequence of the dyadic Haar system (Theorem 3.2). We also give an example of a general Haar system

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Martingale Transforms

Cited by 346 publications

References 0 publications

Weighted square function inequalities

Weighted square function inequalities

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

General Haar systems and greedy approximation

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