Weak type estimates for functions of Marcinkiewicz type with fractional integrals of mixed homogeneity

  • Shuichi Sato

Abstract

We prove the endpoint weak type estimate for square functions of Marcinkiewicz type with fractional integrals associated with non-isotropic dilations. This generalizes a result of C. Fefferman on functions of Marcinkiewicz type by considering fractional integrals of mixed homogeneity in place of the Riesz potentials of Euclidean structure.

References

Alabern, R., Mateu, J., and Verdera, J., A new characterization of Sobolev spaces on $\mathbb R^n$, Math. Ann. 354 (2012), no. 2, 589–626. https://doi.org/10.1007/s00208-011-0738-0

Calderón, A.-P. and Torchinsky, A., Parabolic maximal functions associated with a distribution, Advances in Math. 16 (1975), 1–64. https://doi.org/10.1016/0001-8708(75)90099-7

Calderón, A.-P. and Torchinsky, A., Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), no. 2, 101–171. https://doi.org/10.1016/S0001-8708(77)80016-9

Capri, O. N., On an inequality in the theory of parabolic $H^p$ spaces, Rev. Un. Mat. Argentina 32 (1985), no. 1, 17–28.

Chanillo, S. and Wheeden, R. L., Inequalities for Peano maximal functions and Marcinkiewicz integrals, Duke Math. J. 50 (1983), no. 3, 573–603. https://doi.org/10.1215/S0012-7094-83-05027-5

Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, vol. 242, Springer-Verlag, Berlin-New York, 1971.

Coifman, R. R. and Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. https://doi.org/10.1090/S0002-9904-1977-14325-5

Duoandikoetxea, J. and Rubio de Francia, J. L., Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. https://doi.org/10.1007/BF01388746

Fabes, E. B. and Rivière, N. M., Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19–38. https://doi.org/10.4064/sm-27-1-19-38

Fefferman, C., Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. https://doi.org/10.1007/BF02394567

Hajłasz, P. and Liu, Z., A Marcinkiewicz integral type characterization of the Sobolev space, Publ. Mat. 61 (2017), no. 1, 83–104. https://doi.org/10.5565/PUBLMAT_61117_03

Marcinkiewicz, J., Sur quelques intégrales du type de Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50.

Nagel, A. and Stein, E. M., Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems, Mathematical Notes, vol. 24, Princeton University Press, Princeton, N.J., 1979.

Rivière, N. M., Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. https://doi.org/10.1007/BF02383650

Sato, S., Littlewood-Paley operators and Sobolev spaces, Illinois J. Math. 58 (2014), no. 4, 1025–1039.

Sato, S., Square functions related to integral of Marcinkiewicz and Sobolev spaces, Linear Nonlinear Anal. 2 (2016), no. 2, 237–252.

Sato, S., Littlewood-Paley equivalence and homogeneous Fourier multipliers, Integral Equations Operator Theory 87 (2017), no. 1, 15–44. https://doi.org/10.1007/s00020-016-2333-y

Sato, S., Spherical square functions of Marcinkiewicz type with Riesz potentials, Arch. Math. (Basel) 108 (2017), no. 4, 415–426. https://doi.org/10.1007/s00013-017-1027-2

Sato, S., Wang, F., Yang, D., and Yuan, W., Generalized Littlewood-Paley characterizations of fractional Sobolev spaces, Commun. Contemp. Math. 20 (2018), no. 7, 1750077, 48 pp. https://doi.org/10.1142/S0219199717500778

Segovia, C. and Wheeden, R. L., On the function $g_\lambda ^\ast $ and the heat equation, Studia Math. 37 (1970), 57–93. https://doi.org/10.4064/sm-37-1-57-93

Stein, E. M., The characterization of functions arising as potentials, Bull. Amer. Math. Soc. 67 (1961), 102–104. https://doi.org/10.1090/S0002-9904-1961-10517-X

Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, no. 30, Princeton University Press, Princeton, N.J., 1970.

Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, no. 32, Princeton University Press, Princeton, N.J., 1971.
Published
2019-08-29
How to Cite
Sato, S. (2019). Weak type estimates for functions of Marcinkiewicz type with fractional integrals of mixed homogeneity. MATHEMATICA SCANDINAVICA, 125(1), 135-162. https://doi.org/10.7146/math.scand.a-114725
Section
Articles