Survey Article: A user's guide to Bellman functions

Wittwer, Janine

doi:10.1216/rmj-2011-41-3-631

Cited by 13 publications

(8 citation statements)

References 10 publications

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“…The Bellman function method is a powerful technique which allows the study of various interesting estimates in probability and analysis. There are several papers which contain a detailed description of the general methodology as well as many examples and applications, see for example [9][10][11][13][14][15]. We present the appropriate modification of the technique so that it works for BLO functions.…”

Section: On the Methods Of Proofmentioning

confidence: 99%

Sharp Estimates for Functions of Bounded Lower Oscillation

Oşekowski¹

2012

Bull. Aust. Math. Soc.

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Let f : R → R be a locally integrable function of bounded lower oscillation. The paper contains the proofs of sharp strong-type, weak-type and exponential estimates for the mean oscillation of f . In particular, this yields the precise value of the norm of the embedding BLO ⊂ BMO p , 1 ≤ p < ∞. Higher-dimensional analogues for anisotropic BLO spaces are also established.2010 Mathematics subject classification: primary 42B37; secondary 49L20.

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Section: On the Methods Of Proofmentioning

confidence: 99%

Sharp Estimates for Functions of Bounded Lower Oscillation

Oşekowski¹

2012

Bull. Aust. Math. Soc.

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“…The same paper [53] explains the connection between the Nazarov-Treil-Volberg approach and the earlier work of Burkholder on martingale inequalities, see [16] and also [17,18]. If interested in the genesis of Bellman functions and the overview of the method, the reader is also referred to Volberg et al [53,64,51] and Wittwer [66]. The method has seen a whole series of applications, yet until recently (see [19,20]) mostly in Euclidean harmonic analysis.…”

Section: Power Functions and The Bellman Function Of Nazarov And Treilmentioning

confidence: 96%

Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

Carbonaro¹,

Dragičević²

2016

Preprint

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We introduce a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the L p theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) L p -contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindoš and Pipher established close ties between p-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The p-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.

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“…For an in-depth treatise on recent advances in martingale inequalities the reader is referred to [71]. If interested in the genesis of Bellman functions and the overview of the method, the reader is also referred to [68,81,82]. The method has seen a whole series of applications, yet until recently (see [10,11]) mostly in Euclidean harmonic analysis.…”

Section: Bounded H ∞ -Calculus Via Nazarov-treil Bellman Functionmentioning

confidence: 99%

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

Carbonaro¹,

Dragičević²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators L . We prove that if −L generates an analytic semigroup on L 2 (γ∞), then L has bounded holomorphic functional calculus on L r (γ∞), 1 < r < ∞, in any sector of angle ϑ > ϑ * r , where γ∞ is the associated invariant measure and ϑ * r the sectoriality angle of L on L r (γ∞). The angle ϑ * r is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.

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Survey Article: A user's guide to Bellman functions

Cited by 13 publications

References 10 publications

Sharp Estimates for Functions of Bounded Lower Oscillation

Sharp Estimates for Functions of Bounded Lower Oscillation

Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

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