Abstract
We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on \(\mathbb{R}^{2}\) with the generalized homogeneity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Calderón, A. P., Inequalities for the maximal function relative to a metric, Studia Math. 57 (1976), 297–306.
Calderón, A. P. and Torchinsky, A., Parabolic maximal functions associated with a distribution, Adv. Math. 16 (1975), 1–64.
Calderón, A. P. and Zygmund, A., On singular integrals, Amer. J. Math. 78 (1956), 289–309.
Carbery, A., Hernández, E. and Soria, F., Estimates for the Kakeya maximal operator on radial functions in \(\mathbb{R}^{n}\), in Harmonic Analysis, ICM-90 Satellite Conference Proceedings, pp. 41–50, Springer, Tokyo, 1991.
Christ, M., Hilbert transforms along curves I. Nilpotent groups, Ann. Math. 122 (1985), 575–596.
Christ, M., Weak type (1,1) bounds for rough operators, Ann. Math. 128 (1988), 19–42.
Christ, M. and Rubio de Francia, J. L., Weak type (1,1) bounds for rough operators, II, Invent. Math. 93 (1988), 225–237.
Coifman, R. R. and Weiss, G., Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Math. 242, Springer, Berlin and New York, 1971.
Duoandikoetxea, J., Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869–880.
Duoandikoetxea, J. and Rubio de Francia, J. L., Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561.
Fan, D. and Sato, S., Weak type (1,1) estimates for Marcinkiewicz integrals with rough kernels, Tohoku Math. J. 53 (2001), 265–284.
Fan, D. and Sato, S., Weighted weak type (1,1) estimates for singular integrals and Littlewood-Paley functions, Studia Math. 163 (2004), 119–136.
Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, NJ, 1982.
Garcia-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
Hofmann, S., Weak (1,1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), 260–264.
Hofmann, S., Weighted weak-type (1,1) inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989), 423–435.
Hofmann, S., Weighted norm inequalities and vector-valued inequalities for certain rough operators, Indiana Univ. Math. J. 42 (1993), 1–14.
Koranyi, A. and Vagi, S., Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Sc. Norm. Super. Pisa 25 (1971), 575–648.
Muckenhoupt, B. and Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249–258.
Nagel, A. and Stein, E. M., Lectures on Pseudo-Differential Operators, Mathematical Notes 24, Princeton University Press, Princeton, NJ, 1979.
Rivière, N., Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278.
Sato, S., Estimates for singular integrals along surfaces of revolution, J. Aust. Math. Soc. 86 (2009), 413–430.
Sato, S., A note on L p estimates for singular integrals, Sci. Math. Jpn. 71 (2010), 343–348.
Sato, S., Weak type (1,1) estimates for parabolic singular integrals, Proc. Edinb. Math. Soc. 54 (2011), 221–247.
Sato, S., Estimates for singular integrals on homogeneous groups, J. Math. Anal. Appl. 400 (2013), 311–330.
Seeger, A., Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95–105.
Soria, F. and Weiss, G., A remark on singular integrals and power weights, Indiana Univ. Math. J. 43 (1994), 187–204.
Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. (N.S.) 84 (1978), 1239–1295.
Tao, T., The weak-type (1,1) of LlogL homogeneous convolution operator, Indiana Univ. Math. J. 48 (1999), 1547–1584.
Vargas, A., Weighted weak type (1,1) bounds for rough operators, J. Lond. Math. Soc. (2) 54 (1996), 297–310.
Watson, D., Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389–399.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by Grant-in-Aid for Scientific Research (C) No. 25400130, Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Sato, S. Weighted weak type (1,1) estimates for singular integrals with non-isotropic homogeneity. Ark Mat 54, 157–180 (2016). https://doi.org/10.1007/s11512-015-0215-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-015-0215-1