Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms

Adam Osękowski

Open Mathematics (2014)

  • Volume: 12, Issue: 8, page 1198-1213
  • ISSN: 2391-5455

Abstract

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We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.

How to cite

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Adam Osękowski. "Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms." Open Mathematics 12.8 (2014): 1198-1213. <http://eudml.org/doc/269645>.

@article{AdamOsękowski2014,
abstract = {We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.},
author = {Adam Osękowski},
journal = {Open Mathematics},
keywords = {Fourier multiplier; Singular integral; Martingale; singular integral; martingale},
language = {eng},
number = {8},
pages = {1198-1213},
title = {Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms},
url = {http://eudml.org/doc/269645},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Adam Osękowski
TI - Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms
JO - Open Mathematics
PY - 2014
VL - 12
IS - 8
SP - 1198
EP - 1213
AB - We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.
LA - eng
KW - Fourier multiplier; Singular integral; Martingale; singular integral; martingale
UR - http://eudml.org/doc/269645
ER -

References

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