Affinity kernels on measure spaces and maximal operators

Authors

  • Hugo Aimar Instituto de Matemática Aplicada del Litoral, UNL, CONICET; CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac. 168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina
  • Ivana Gómez Instituto de Matemática Aplicada del Litoral, UNL, CONICET; CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac. 168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina
  • Luis Nowak Instituto de Investigación en Tecnologías y Ciencias de la Ingeniería, CONICET, UNComa; Departamento de Matemática, FaEA, Neuquén, Argentina

DOI:

https://doi.org/10.33044/revuma.2518

Abstract

In this note we consider maximal operators defined in terms of families of kernels and families of their level sets. We prove a general  estimate that extends some classical Euclidean cases and, under some mild transitivity property, we show their basic boundedness  properties on Lebesgue spaces. The motivation of these problems has its roots in the analysis associated to affinity kernels on large  data sets.

Downloads

Download data is not yet available.

References

H. Aimar, Elliptic and parabolic BMO and Harnack's inequality, Trans. Amer. Math. Soc. 306 (1988), no. 1, 265–276. MR 0927690.

H. Aimar and I. Gómez, Affinity and distance. On the Newtonian structure of some data kernels, Anal. Geom. Metr. Spaces 6 (2018), no. 1, 89–95. MR 3816950.

H. Aimar and R. A. Macías, Weighted norm inequalities for the Hardy–Littlewood maximal operator on spaces of homogeneous type, Proc. Amer. Math. Soc. 91 (1984), no. 2, 213–216. MR 0740173.

R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 0358205.

M. de Guzmán, Real Variable Methods in Fourier Analysis, Notas de Matemática, 75, North-Holland Mathematics Studies, 46, North-Holland, Amsterdam-New York, 1981. MR 0596037.

B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), no. 1, 217–234. https://doi.org/10.4064/fm-25-1-217-234.

R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 0546295.

O. Nikodym, Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fund. Math. 10 (1927), no. 1, 116–168. https://doi.org/10.4064/fm-10-1-116-168.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970. MR 0290095.

Downloads

Published

2022-12-20

Issue

Section

Article