Zhao, Junyan; Fan, Dashan Certain averaging operators on Lebesgue spaces. (English) Zbl 1431.42024 Banach J. Math. Anal. 14, No. 1, 116-139 (2020). Summary: In this paper, we study some multiplier operator \(\mu_{\gamma ,\alpha}\) raised from studying the \(L^p\)-approximation of the spherical mean \(S_t^{\gamma}.\) We obtain the optimal range of exponents \((\alpha ,\gamma ,p)\) such that \(\mu_{\gamma ,\alpha}\) is an \(L^p\) multiplier. As an application, we obtain the convergence rate for \(S_t^{\gamma}\left( f\right)\) in the \(L^p\) spaces. Cited in 3 Documents MSC: 42B15 Multipliers for harmonic analysis in several variables 42B25 Maximal functions, Littlewood-Paley theory 41A35 Approximation by operators (in particular, by integral operators) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B38 Linear operators on function spaces (general) Keywords:spherical mean; Sobolev spaces; \(L^p\) norm convergence; Bessel function; wave operator PDFBibTeX XMLCite \textit{J. Zhao} and \textit{D. Fan}, Banach J. Math. Anal. 14, No. 1, 116--139 (2020; Zbl 1431.42024) Full Text: DOI References: [1] Chen, Jiecheng; Fan, Dashan; Zhao, Fayou, On the Rate of Almost Everywhere Convergence of Combinations and Multivariate Averages, Potential Analysis, 51, 3, 397-423 (2018) · Zbl 1428.41039 · doi:10.1007/s11118-018-9716-4 [2] Dai, F.; Ditzian, Z., Combinations of multivariate averages, J. Approx. Theory, 131, 2, 268-283 (2004) · Zbl 1109.41010 · doi:10.1016/j.jat.2004.10.003 [3] Fan, D.; Lou, Z.; Wang, Z., A note on iterated spherical average on Lebesgue spaces, Nonlinear Anal., 180, 170-183 (2019) · Zbl 1458.41006 · doi:10.1016/j.na.2018.10.004 [4] Fan, D.; Zhao, F., Approximation properties of combination of multivariate averages on Hardy spaces, J. Approx. Theory, 223, 77-95 (2017) · Zbl 1377.41004 · doi:10.1016/j.jat.2017.07.008 [5] Fan, D.; Zhao, F., Block-Sobolev spaces and the rate of almost everywhere convergence of Bochner-Riesz Means, Constr. Approx., 45, 3, 391-405 (2017) · Zbl 1373.42012 · doi:10.1007/s00365-016-9343-5 [6] Gelfand, I.; Moiseevich, S.; Georgi, E., Generalized Functions. Vol. I: Properties and Operations, Translated by Eugene Saletan (1964), New York: Academic, New York · Zbl 0115.33101 [7] Miyachi, A., On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, 2, 267-315 (1981) · Zbl 0469.42003 [8] Juan, C., Peral, \(L^p\) estimates for the wave equation, J. Funct. Anal., 36, 1, 114-145 (1980) · Zbl 0442.35017 · doi:10.1016/0022-1236(80)90110-X [9] Stein, Em, Maximal functions I: Spherical means, Proc. Natl. Acad. Sci. USA, 73, 7, 2174-2175 (1976) · Zbl 0332.42018 · doi:10.1073/pnas.73.7.2174 [10] Stein, Em, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series (1993), Princeton: Princeton University Press, Princeton · Zbl 0821.42001 [11] Stein, Em; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton, N.J: Princeton University Press, Princeton, N.J · Zbl 0232.42007 [12] Strichartz, Rs, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16, 1031-1060 (1967) · Zbl 0145.38301 [13] Strichartz, Robert S., \(H^p\) Sobolev spaces, Colloquium Mathematicum, 60, 1, 129-139 (1990) · Zbl 0782.46034 · doi:10.4064/cm-60-61-1-129-139 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.