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

Oşekowski, Adam

doi:10.1007/s00020-013-2116-7

Cited by 7 publications

(3 citation statements)

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“…The proofs in [17], [11], as well as in the current paper rely on Bellman functions adapted to trees. Similar nested structures have also been used in Bellmanfunction contexts by Melas [6] and Melas, Nikolidakis, and Stavropoulos [9]; see also the works of Bañuelos and Osȩkowski [1] and Osȩkowski [12]. The important distinction is that the trees used by those authors were homogeneous, meaning every element of the tree had the same number of offspring, all of the same measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $α$-trees

Osȩkowski,

Slavin,

Vasyunin

2019

Preprint

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We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from BMO(R n ) into BLO(R n ). As a consequence, we show that the BMO → BLO norm of the natural operator equals 1 for all n, and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called α -trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on α, but also allows a majorant independent of α, hence the dimension-free norm constant. We also describe the decay of the norm with respect to the difference between the average of a function on a cube and the infimum of its maximal function on that cube. An explicit norm-optimizing sequence is constructed.

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Section: Introduction and Main Resultsmentioning

confidence: 87%

The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $α$-trees

Osȩkowski,

Slavin,

Vasyunin

2019

Preprint

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“…martingales for which only one martingale difference ∆F n is non-zero) that without the regularity assumption the operator I α may not be continuous as an L p → L q operator (see Remark 2 below). On the other hand, the papers [3] and [10] provide results about the sharp constants in weak-type inequalities for operators similar to I α on uniform filtrations (i.e. each atom in A(F n ) is split into m atoms of equal probability in A(F n+1 ); here m is independent of w and n); the corresponding constants appear to be uniformly bounded with respect to m. This hints there must be a reasonable (i.e.…”

Section: Martingale Fractional Integrationmentioning

confidence: 99%

Fractional integration for irregular martingales

Stolyarov¹,

Yarcev²

2020

Preprint

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“…The Bellman function arising in our reasonings seems to be interesting in itself (though we will not find the sharp Bellman function, only provide a supersolution in Theorem 3.1). Very little is known about Bellman functions related to fractional operators, however, see [2] and [20] for two sharp supersolutions.…”

Section: Introductionmentioning

confidence: 99%