The Bellman functions of dyadic-like maximal operators and related inequalities

Melas, Antonios D.

doi:10.1016/j.aim.2004.04.013

Cited by 80 publications

(221 citation statements)

References 5 publications

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“…Fix an arbitrary constant K > 1 and a positive parameter δ (which will eventually be sent to 0). To provide the construction of an appropriate function (or rather a functional sequence), we will need the following lemma, which can be found in [3]. First we introduce a certain sequence (A n ) n≥0 of subsets of X.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Sharp weighted logarithmic bound for maximal operators

Oşekowski

2016

Arch. Math.

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Abstract. The paper contains the study of sharp weighted logarithmic estimates for maximal operators on probability spaces equipped with a tree-like structure. These inequalities can be regarded as LlogL versions of the classical estimates of Fefferman and Stein. The proof exploits the existence of a certain special function, enjoying appropriate majorization and concavity conditions. Mathematics Subject Classification. Primary 42B25; Secondary 46E30, 60G42.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Sharp weighted logarithmic bound for maximal operators

Oşekowski

2016

Arch. Math.

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“…Such a type of problems appears in many places in the literature, in the study of other classical operators and objects in harmonic analysis. See e.g., Melas [12], Melas et al [13] and Slavin et al [19] for related problems concerning the dyadic maximal operators; consult Burkholder [3] for related results for martingale transforms and the Haar system on [0, 1]; Vasyunin [22] studied similar questions for A p -weights on the real line; Slavin and Vasyunin [20] investigated similar problems for BMO functions on R; and more. Coming back to (1.3) and the restriction (1.4), we easily see that if Y = |X|, then the lower bound for ||F || L p (T) can be improved.…”

Section: Introductionmentioning

confidence: 99%

Sharp L p Bound for Holomorphic Functions on the Unit Disc

Oşekowski

2014

Mediterr. J. Math.

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Abstract. For any 1 < p < ∞ and any X, Y ∈ R satisfying |X| ≤ Y , we determine the optimal constant Cp(X, Y ) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReThis can be regarded as a reverse version of the classical estimates of Riesz and Essén. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.Mathematics Subject Classification. Primary 31B05; Secondary 30J99.

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“…The contribution of this paper is the sharp upper bound for M L p (w)→L p (w) both in terms of p and [w] Ap . Actually, we will work in the more general context of probability spaces equipped with a tree-like structure [4]. Here is the precise definition.…”

Section: Introductionmentioning

confidence: 99%

“…See e.g. [4,5,6,7,8,13,14] and the monograph [3], consult also references therein. The primary goal of the present paper is to establish a sharp weighted version of (1.2).…”

Section: Introductionmentioning

confidence: 99%