A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

Michał Wojciechowski

Studia Mathematica (2000)

  • Volume: 140, Issue: 3, page 273-287
  • ISSN: 0039-3223

Abstract

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It is proved that if m : ℝ d → ℂ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the H 1 space on the product domain ℝ d 1 × . . . × ℝ d k . This implies an estimate of the norm N ( m , L p ( ℝ d ) of the multiplier transformation of m on L p ( ℝ d ) as p→1. Precisely we get N ( m , L p ( ℝ d ) ) ≲ ( p - 1 ) - k . This bound is the best possible in general.

How to cite

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Wojciechowski, Michał. "A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains." Studia Mathematica 140.3 (2000): 273-287. <http://eudml.org/doc/216767>.

@article{Wojciechowski2000,
abstract = {It is proved that if $m : ℝ^d → ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^1$ space on the product domain $ℝ^\{d_1\}×...×ℝ^\{d_k\}$. This implies an estimate of the norm $N(m,L^p(ℝ^d)$ of the multiplier transformation of m on $L^p(ℝ^d)$ as p→1. Precisely we get $N(m,L^p(ℝ^d))≲(p-1)^\{-k\}$. This bound is the best possible in general.},
author = {Wojciechowski, Michał},
journal = {Studia Mathematica},
keywords = {Fourier multiplier; Hardy space on product domain; Marcinkiewicz-Hörmander condition},
language = {eng},
number = {3},
pages = {273-287},
title = {A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains},
url = {http://eudml.org/doc/216767},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Wojciechowski, Michał
TI - A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 3
SP - 273
EP - 287
AB - It is proved that if $m : ℝ^d → ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^1$ space on the product domain $ℝ^{d_1}×...×ℝ^{d_k}$. This implies an estimate of the norm $N(m,L^p(ℝ^d)$ of the multiplier transformation of m on $L^p(ℝ^d)$ as p→1. Precisely we get $N(m,L^p(ℝ^d))≲(p-1)^{-k}$. This bound is the best possible in general.
LA - eng
KW - Fourier multiplier; Hardy space on product domain; Marcinkiewicz-Hörmander condition
UR - http://eudml.org/doc/216767
ER -

References

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