Radial weights and mixed norm inequalities for the disc multiplier

“…This estimate for the disc multiplier was proved independently by A. Córdoba [3] (in a paper of the year 1989) and G. Mockenhaupt [7] (in a paper of the year 1990). A little bit later (in a paper of the year 1992) this result was also obtained as a corollary of the main theorem on weighted inequalities in L 2 for the disc multiplier by A. Carbery, F. Soria, and the author [1], a problem suggested by E. Stein [11] and solved there for the case of radial weights. Moreover, in [1] we also obtain extremal bounds for the operatorT between the spaces…”

Section: Introduction and Statement Of Theoremsmentioning

confidence: 85%

“…This fact is the basic step of A. Córdoba in the proof of the mixed norm estimate for the operator T (see [3]). This estimate was also obtained in [1] by different methods which allowed the authors to obtain the following extremal estimate for Q:…”

Section: Introduction and Statement Of Theoremsmentioning

confidence: 92%

“…We will only study the proof corresponding to m = 1 for p = p 0 , p 1 and will indicate what to do for the case m = 3.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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Extremal Vector Valued Inequalities for Hankel Transforms

Romera

2009

Rev. Mat. Complut.

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The disc multiplier may be seen as a vector valued operator when we consider its projections in terms of the spherical harmonics. In this form, it represents a vector valued Hankel transform. We know that, for radial functions, it is bounded on the spaces. Here we prove that there exist weak-type estimates for this operator for the extremal exponents, that is, it is bounded from L, p0 < q < p1, and we consider radial functions.

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Weighted mixed‐norm inequalities through extrapolation

Duoandikoetxea

Oruetxebarria

2019

Mathematische Nachrichten

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We prove that operators satistying weighted inequalities with radial weights are bounded in mixed-norm spaces of radial-angular type, even with a weight in the radial part. This is achieved by using the usual extrapolation methods, adapted to the radial setting. All the versions of the extrapolation theorem can be adapted to this setting, and in particular we get results in variable Lebesgue spaces and also for multilinear operators. Furthermore, quantitative estimates are obtained with this approach, but their sharpness remains an open question. K E Y W O R D S extrapolation of weights, mixed-norm spaces, Muckenhoupt weights, variable exponent M S C ( 2 0 1 0 ) 42B20, 42B25, 42B35 1482

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Radial weights and mixed norm inequalities for the disc multiplier

Cited by 39 publications

References 12 publications

WEAK BEHAVIOUR OF FOURIER-NEUMANN SERIES Research supported by grants of the DGI and UR.

WEAK BEHAVIOUR OF FOURIER-NEUMANN SERIES Research supported by grants of the DGI and UR.

Extremal Vector Valued Inequalities for Hankel Transforms

Weighted mixed‐norm inequalities through extrapolation

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