Banach Journal of Mathematical Analysis

Fourier multiplier theorems on Besov spaces under type and cotype conditions

Jan Rozendaal and Mark Veraar

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Abstract

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 $L^{p}$-$L^{q}$ 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 $L^{p}$-$L^{q}$ 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 $\frac{1}{p}-\frac{1}{q}$ remains constant.

Article information

Source
Banach J. Math. Anal. Volume 11, Number 4 (2017), 713-743.

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1495094417

Digital Object Identifier
doi:10.1215/17358787-2017-0011

Subjects
Primary: 42B15: Multipliers
Secondary: 42B35: Function spaces arising in harmonic analysis 46B20: Geometry and structure of normed linear spaces 46E40: Spaces of vector- and operator-valued functions 47B38: Operators on function spaces (general)

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

Citation

Rozendaal, Jan; Veraar, Mark. Fourier multiplier theorems on Besov spaces under type and cotype conditions. Banach J. Math. Anal. 11 (2017), no. 4, 713--743. doi:10.1215/17358787-2017-0011. https://projecteuclid.org/euclid.bjma/1495094417


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