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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/183656
Idempotent Fourier multipliers acting contractively on $H^{P}$ spaces
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Abstract
We describe the idempotent Fourier multipliers that act contractively on $H^{p}$ spaces of the $d$-dimensional torus $\mathbb{T}^{d}$ for $d \geq 1$ and $1 \leq p \leq \infty .$ When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^{p}$ spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n, d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^{p}\left(\mathbb{T}^{\infty}\right)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4, \ldots, 2(n+1)$.
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BREVIG, Ole Fredrik, ORTEGA CERDÀ, Joaquim and SEIP, Kristian. Idempotent Fourier multipliers acting contractively on $H^{P}$ spaces. Geometric and Functional Analysis. 2021. Vol. 31, num. 6, pags. 1377-1413. ISSN 1016-443X. [consulted: 10 of August of 2026]. Available at: https://hdl.handle.net/2445/183656