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An oscillating multiplier on Herz type spaces. (English) Zbl 1240.42031

Summary: The boundedness of an oscillating multiplier \(m_{\gamma,\beta}\) for different \(\beta\) on the Herz type spaces is obtained. This operator was initially studied by S. Wainger [Mem. Am. Math. Soc. 59, 98p. (1965; Zbl 0136.36601)] and C. L. Fefferman and E. M. Stein [Acta Math. 129, 137–193 (1972; Zbl 0257.46078)]. Our results extend one of the main results in a paper by X. Li and S. Lu [Acta Math. Sin., New Ser. 14, No. 1, 67–76 (1998; Zbl 0905.42008)] for the non-weighted case, when \(\beta\) is close to 1 or \(\alpha\) is suitably large. For \(\beta\geq 1\), the results with no weights on the Herz type spaces are also new.

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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.)
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References:

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