Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol

Petermichl, Stefanie

doi:10.1016/s0764-4442(00)00162-2

Cited by 80 publications

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“…In [5] it is shown that the Hilbert transform H on the real line R is contained in a multiple of the closed convex hull of the "dyadic shifts" {X α,r } α∈R, r>0 . Here, the dyadic shift X α,r is defined by…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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A version of Burkholder’s theorem for operator-weighted spaces

Petermichl

Pott

2003

Proc. Amer. Math. Soc.

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Abstract. Let W be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space H. We prove that if the dyadic martingale transforms are uniformly bounded on L 2 R (W ) for each dyadic grid in R, then the Hilbert transform is bounded on L 2 R (W ) as well, thus providing an analogue of Burkholder's theorem for operator-weighted L 2 -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into "dyadic shifts".

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…But the proof in [6] seems to be dimension dependent, while our method is dimension independent and very general. It relies on the decomposition of the Hilbert transform by the first author in [5].…”

Section: Introductionmentioning

confidence: 99%

A version of Burkholder’s theorem for operator-weighted spaces

Petermichl

Pott

2003

Proc. Amer. Math. Soc.

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“…We will reduce the problem to upper and lower bounds of certain square functions, using the averaging technique from [5].…”

Section: Formulation Of Resultsmentioning

confidence: 99%

“…Our proof uses a certain averaging technique introduced by the first author in [5]. The new bound for the Hilbert transform follows from upper and lower bounds for the square function in just one line.…”

Section: Introductionmentioning

confidence: 99%

An estimate for weighted Hilbert transform via square functions

Petermichl

Pott

2001

Trans. Amer. Math. Soc.

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“…We use the fact that the Hilbert transform can be represented as averages of dyadic shifts (see [7,12]). This allows us to write for b ∈ bmo([0, 1] 2 ) and…”

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confidence: 99%

Logarithmic Mean Oscillation on the Polydisc, Multi-Parameter Paraproducts and Iterated Commutators

Sehba

2014

J Fourier Anal Appl

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Abstract. We introduce another notion of bounded logarithmic mean oscillation in the N -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO, bmo d (T N ) to the dyadic product BMO space, BMO d (T N ). We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of bmo(R N ) consisting of functions with support in [0,1] N to BMO(R N ).

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Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol

Cited by 80 publications

References 0 publications

A version of Burkholder’s theorem for operator-weighted spaces

A version of Burkholder’s theorem for operator-weighted spaces

An estimate for weighted Hilbert transform via square functions

Logarithmic Mean Oscillation on the Polydisc, Multi-Parameter Paraproducts and Iterated Commutators

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