Inequality for Burkholder’s martingale transform

Ivanisvili, Paata

doi:10.2140/apde.2015.8.765

Cited by 9 publications

(7 citation statements)

References 10 publications

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“…It can be derived by using the methods recently developed in works of N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy and P. Ivanisvili (see e.g. [7,8,6]). Appendices 4.2.…”

Section: Bellman Function Approachmentioning

confidence: 99%

Lower Bounds for Uncentered Maximal Functions in Any Dimension

Ivanisvili

Jaye

Nazarov

2016

Int Math Res Notices

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Abstract. In this paper we address the following question: given p ∈ (1, ∞), n ≥ 1, does there exists a constantwhere M f is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the question for the uncentered maximal function operator and the almost centered maximal function operator. In addition, we also present the Bellman function approach of Melas, Nikolidakis and Stavropoulos to maximal function operators defined over various types of families of sets, and in case of parallelepipeds we will show that

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“…It can be derived by using the methods recently developed in works of N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy and P. Ivanisvili (see e.g. [7,8,6]). Appendices 4.2.…”

Section: Bellman Function Approachmentioning

confidence: 99%

Lower Bounds for Uncentered Maximal Functions in Any Dimension

Ivanisvili

Jaye

Nazarov

2016

Int Math Res Notices

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“…Among various applications of four vertex theorems, we mention a paper of Berger and Calabi et al [52] on physics of floating bodies, and recent work of Bray and Jauregui [14] in general relativity. See also the works of Arnold [7,8] for relations with contact geometry, the book of Ovsienko and Tabachnikov [47] for projective geometric aspects, Angenent [6] for connections with mean curvature flow, which are also discussed in [23], and Ivanisvili et al [36,35] for applications to the study of Bellman functions. Other references and more background on four vertex theorems may be found in [18,65,22,48,26].…”

Section: Introductionmentioning

confidence: 99%

Boundary torsion and convex caps of locally convex surfaces

Ghomi

2017

J. Differential Geom.

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Abstract. We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4 vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.

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“…Consider the case p ∈ (0, 1) ∪ (2, ∞). Then it follows from Lemma 29 of Section 3.2 in [5] that locally, say for some δ ∈ (0, 1], there exists a function a(s) : [0, δ] → [−1, 0] such that a(0) = 0, a(s) is strictly decreasing, and the function B(u, w) defined parametrically by The case p ∈ (−∞, 0) ∪ (1, 2) is different because τ γ changes sign from + to −, and in this case an "angle" arises with vertex sitting around the point s = 0 (see Section 3 in [5]).…”

Section: Concluding Remarks On Envelopesmentioning

confidence: 89%

“…The above inequality is Theorem 1.3 from [3], with α := a a+b . We should mention that estimate (5) does not follow solely from Theorem 2.1. It follows from the fact that both inequalities (3) and (2) hold true and the fact that (3) is sharp in a sense of Remark 2.2.…”

Section: The Inequality Reverses Ifmentioning

confidence: 99%

Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

Ivanisvili,

Mooney

2019

Preprint

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Motivated by the inequality Carbery (2006) raised the question what is the "right" analogue of this estimate in L p for p = 2. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an L p version of this inequality by providing upper bounds for f + g p p in terms of the quantities f p p , g p p and f g p/2 p/2 when p ∈ (0, 1] ∪ [2, ∞), and lower bounds when p ∈ (−∞, 0) ∪ (1, 2), thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for f + g p p also when p ∈ (−∞, 0) ∪ (1, 2) and lower bounds when p ∈ (0, 1] ∪ [2, ∞). For p ∈ [1, 2] we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of f + g p p for p ∈ R, p = 0, are the best possible in terms of the quantities f p p , g p p and f g p/2 p/2 , and we characterize the equality cases.

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Inequality for Burkholder’s martingale transform

Cited by 9 publications

References 10 publications

Lower Bounds for Uncentered Maximal Functions in Any Dimension

Lower Bounds for Uncentered Maximal Functions in Any Dimension

Boundary torsion and convex caps of locally convex surfaces

Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

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