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Multiplier transformations on \(H^p\) spaces. (English) Zbl 1050.42010
Summary: The authors obtain some multiplier theorems on \(H^p\) spaces analogous to the classical \(L^p\) multiplier theorems of de Leeuw. The main result is that a multiplier operator \((Tf)\hat{\;} (x)= \lambda(x) \widehat f(x)(\lambda\in C (\mathbb{R}^n))\) is bounded on \(H^p(\mathbb{R}^n)\) if and only if the restriction \(\{\lambda (\varepsilon m)\}_{m\in\Lambda}\) is an \(H^p(\mathbb{T}^n)\) bounded multiplier uniformly for \(\varepsilon>0\), where \(\Lambda\) is the integer lattice in \(\mathbb{R}^n\).

MSC:
42B15 Multipliers for harmonic analysis in several variables
42B30 \(H^p\)-spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
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