Sharp Martingale and Semimartingale Inequalities

Oşekowski, Adam

doi:10.1007/978-3-0348-0370-0

Cited by 133 publications

(118 citation statements)

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“…The symbol [X, Y ] will stand for the quadratic covariance process of X and Y , see e.g., Dellacherie and Meyer [7] The differential subordination implies many interesting inequalities comparing the sizes of X and Y . The literature on this subject is quite extensive, we refer the interested reader to the survey [5] by Burkholder, the paper of Wang [23] and the monograph [18]. Here we only mention one result, due to Bañuelos and Wang [23], which will be needed in our further considerations.…”

Section: A Martingale Inequalitymentioning

confidence: 99%

Sharp weak type estimates for Riesz transforms

Oşekowski

2014

Monatsh Math

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Let d be a given positive integer and let {R j } d j=1 denote the collection of Riesz transforms on R d . For 1 < p < ∞, we determine the best constant C p such that the following holds. For any locally integrable function f on R d and any j ∈ {1, 2, . . . , d},A related statement for Riesz transforms on spheres is also established. The proofs exploit Gundy-Varopoulos representation of Riesz transforms and appropriate inequality for orthogonal martingales.

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Section: A Martingale Inequalitymentioning

confidence: 99%

Sharp weak type estimates for Riesz transforms

Oşekowski

2014

Monatsh Math

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“…This technique originates from the theory of stochastic optimal control, and its connection with other areas of mathematics was firstly observed by Burkholder in [1], who studied certain sharp inequalities for martingale transforms. Since then, the method has been intensively developed in the subsequent works of Burkholder and his students (a convenient reference on the subject is the monograph [12] by the author). Furthermore, in the late 1990s, Nazarov, Treil and Volberg [10,11] showed that the method can be exploited in a much wider analytic context.…”

Section: Vol 78 (2014)mentioning

confidence: 99%

Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Oşekowski

2013

Integr. Equ. Oper. Theory

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Abstract. Suppose that d ≥ 1 is an integer, α ∈ (0, d) is a fixed parameter and let Iα be the fractional integral operator associated with d-dimensional Walsh-Fourier series on [0, 1)d . The paper contains the proof of the sharp weak-type estimateThe proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.Mathematics Subject Classification (2010). Primary 42B25, 42B30; Secondary 42B35.

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“…The literature on this subject is quite extensive and it is not possible to review it here; we refer the reader to the survey [4] by Burkholder, the paper of Wang [21] and the monograph [17] by the author. Here we only mention one result, of Bañuelos and Wang [21], to be needed later.…”

Section: Martingale Inequalitiesmentioning

confidence: 99%