Open Access
October 2017 Fourier multiplier theorems on Besov spaces under type and cotype conditions
Jan Rozendaal, Mark Veraar
Banach J. Math. Anal. 11(4): 713-743 (October 2017). DOI: 10.1215/17358787-2017-0011


In this article, we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents p and q, which depend on the type p and cotype q of the underlying Banach spaces. In a previous article, we considered Lp-Lq multiplier theorems. In the current article, we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the Lp-Lq setting as well.

We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties, then the boundedness of the multiplier operator extrapolates to other values of p and q for which 1p1q remains constant.


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Jan Rozendaal. Mark Veraar. "Fourier multiplier theorems on Besov spaces under type and cotype conditions." Banach J. Math. Anal. 11 (4) 713 - 743, October 2017.


Received: 10 June 2016; Accepted: 12 October 2016; Published: October 2017
First available in Project Euclid: 18 May 2017

zbMATH: 06841251
MathSciNet: MR3708526
Digital Object Identifier: 10.1215/17358787-2017-0011

Primary: 42B15
Secondary: 42B35 , 46B20 , 46E40 , 47B38

Keywords: Besov spaces , extrapolation , Fourier type , operator-valued Fourier multipliers , type and cotype

Rights: Copyright © 2017 Tusi Mathematical Research Group

Vol.11 • No. 4 • October 2017
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