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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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.

Article information

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

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

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Zentralblatt MATH identifier

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)

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


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.

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