Abstract
This paper is devoted to studying the mapping properties for the spherical maximal operator \({\mathbf {S}}_G\) defined on finite connected graphs G. Some operator norms of \({\mathbf {S}}_G\) on the \(\ell ^p(G)\), \(\ell ^{p,\infty }(G)\) and the spaces of bounded p-variation functions defined on G are investigated. Particularly, as some special examples of finite connected graphs, the complete graph \(K_n\) and star graph \(S_n\) are discussed.
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Aldaz, J.M., Lázaro, J.P.: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Am. Math. Soc. 359(5), 2443–2461 (2007)
Badr, N., Martell, J.M.: Weighted norm inequalities on graphs. J. Geom. Anal. 22, 1173–1210 (2012)
Bourgain, J.: Averages over convex curves and maximal operators. J. Anal. Math. 47, 69–85 (1986)
Carneiro, E., Moreira, D.: On the regularity of maximal operators. Proc. Am. Math. Soc. 136(12), 4395–4404 (2008)
Cowling, M., Mauceri, G.: On maximal functions. Rend. Sem. Mat. Fis. Milano 49, 79–87 (1979)
Cowling, M., Meda, S., Setti, A.G.: Estimates for functions of the Laplace operator on homogeneous trees. Trans. Am. Math. Soc. 352(9), 4271–4293 (2000)
Gonzalez–Riquelme, C., Madrid, J.: Sharp inequalities for maximal operators on finite graphs.J. Geom. Anal. 31, 9708–9744 (2021)
Hajłasz, P., Liu, Z.: Maximal potentials, maximal singular integrals, and the spherical maximal function. Proc. Am. Math. Soc. 142(11), 3965–3974 (2013)
Hajłasz, P., Malý, J.: On approximate differentiability of the maximal function. Proc. Am. Math. Soc. 138(1), 165–174 (2010)
Hajłasz, P., Onninen, J.: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29, 167–176 (2004)
Ionescu, A.D.: An endpoint estimate for the discrete spherical maximal function. Proc. Am. Math. Soc. 132(5), 1411–1417 (2003)
Kinnunen, J.: The Hardy–Littlewood maximal function of a Sobolev function. Isr. J. Math. 100, 117–124 (1997)
Kinnunen, J., Lindqvist, P.: The derivative of the maximal function. J. Reine Angew. Math. 503, 161–167 (1998)
Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. Lond. Math. Soc. 35(4), 529–535 (2003)
Kurka, O.: On the variation of the Hardy–Littlewood maximal function. Ann. Acad. Sci. Fenn. Math. 40, 109–133 (2015)
Liu, F., Xue, Q.: On the variation of the Hardy–Littlewood maximal functions on finite graphs. Collect. Math. 72(2), 333–349 (2021)
Luiro, H.: The variation of the maximal function of a radial function. Ark. Mat. 56(1), 147–161 (2018)
Magyar, A., Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis: Spherical averages. Ann. Math. 155, 189–208 (2002)
Mockenhaupt, G., Seeger, A., Sogge, C.: Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. Math. 136, 207–218 (1992)
Rubio de Francia, J.L.: Maximal functions and Fourier transforms. Duke Math J. 53, 395–404 (1986)
Stein, E.M.: Maximal functions I: Spherical means. Proc. Nat. Acad. Sci. 73, 2174–2175 (1976)
Stein, E.M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc. 84, 1239–1295 (1978)
Tanaka, H.: A remark on the derivative of the one-dimensional Hardy–Littlewood maximal function. Bull. Aust. Math. Soc. 65(2), 253–258 (2002)
Korányi, A., Picardello, M.A.: Boundary behaviour of eigenfunctions of the Laplace operator on trees. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13(3), 389–399 (1986)
Schlag, W.: A geometric proof of the circular maximal theorem. Duke Math. J. 93, 505–533 (1998)
Soria, J., Tradacete, P.: Best constants for the Hardy–Littlewood maximal operator on finite graphs. J. Math. Anal. Appl. 436(2), 661–682 (2016)
Soria, J., Tradacete, P.: Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator. J. Anal. Math. 137(2), 913–937 (2019)
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The authors would like to express their sincerely thanks to the referee for his or her valuable remarks and suggestions, which help us to improve partial results of this paper and made this paper more readable.
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The work was supported partly by the NNSF of China (Grant No. 11701333).
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Zhang, X., Liu, F. On the spherical maximal function on finite graphs. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 186 (2021). https://doi.org/10.1007/s13398-021-01127-y
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DOI: https://doi.org/10.1007/s13398-021-01127-y