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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)
##### Keywords:
multiplier operator; bounded operator
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