Fourier multipliers in Banach function spaces with UMD concavifications. (English) Zbl 1425.42014

Amenta, Alex; Lorist, Emiel; Veraar, Mark

Trans. Am. Math. Soc. 371, No. 7, 4837-4868 (2019).

Let \(\Delta\) be a family of non-overlapping intervals in \(\mathbb{R}\). For \(I\in \Delta\) let \(S_I\) be the partial sum operator defined by using the Fourier transform as \((S_I f)\hat{\ }=\chi_I\hat{f}\), where \(\chi_I\) denotes the characteristic function for \(I\). J. L. Rubio de Francia [Rev. Mat. Iberoam. 1, No. 2, 1–14 (1985; Zbl 0611.42005)] proved that \[ \left\|\left(\sum_{I\in \Delta} |S_If|^2\right)^{1/2} \right\|_p \leq C_p\|f||_p, \] for \(p\in [2,\infty)\), where \(\|\cdot\|_p\) denotes the \(L^p(\mathbb{R})\) norm and the constant \(C_p\) is independent of \(\Delta\). This is a generalization of the Littlewood-Paley inequality considered for the case when \(\Delta\) is the family of dyadic intervals. Higher dimensional version of the inequality on \(\mathbb{R}^n\), \(n\geq 2\), was obtained by J.-L. Journé [Rev. Mat. Iberoam. 1, No. 3, 55–91 (1985; Zbl 0634.42015)]. As the Littlewood-Paley inequality for the dyadic decomposition is used to get the Marcinkiewicz multiplier theorem, one would expect that the Littlewood-Paley inequality of Rubio de Francia for arbitrary non-overlapping intervals would be available to prove some new Fourier multiplier theorem which would improve the multiplier theorem of Marcinkiewicz. Indeed, such a theorem of Fourier multiplier was found by the work of R. Coifman et al. [C. R. Acad. Sci., Paris, Sér. I 306, No. 8, 351–354 (1988; Zbl 0643.42010)].
In the article under review the authors extend the result of R. Coifman et al. [loc. cit.] to the case of operator valued multipliers on Banach function spaces. The abstract of the paper written by the authors is as follows.
Abstract. “We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call \(\ell^r(\ell^s)\)-boundedness, which implies \(\mathcal{R}\)-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.”

Reviewer: Shuichi Sato (Kanazawa)

MSC:

42B15	Multipliers for harmonic analysis in several variables
42B25	Maximal functions, Littlewood-Paley theory
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47A56	Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)

Keywords:

Fourier multipliers; UMD Banach function spaces; bounded \(s\)-variation; Littlewood-Paley-Rubio de Francia inequalities; Muckenhoupt weights; complex interpolation

Citations:

Zbl 0611.42005; Zbl 0643.42010; Zbl 0634.42015

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