Mathematica Bohemica, Vol. 147, No. 1, pp. 131-140, 2022
$L^p$-improving properties of certain singular measures on the Heisenberg group
Pablo Rocha
Received January 20, 2020. Published online April 21, 2021.
Abstract: Let $\mu_A$ be the singular measure on the Heisenberg group $\mathbb{H}^n$ supported on the graph of the quadratic function $\varphi(y) = y^tAy$, where $A$ is a $2n \times2n$ real symmetric matrix. If $\det(2A \pm J) \neq0$, we prove that the operator of convolution by $\mu_A$ on the right is bounded from $L^{\frc{(2n+2)}{(2n+1)}}(\mathbb{H}^n)$ to $L^{2n+2}(\mathbb{H}^n)$. We also study the type set of the measures ${\rm d}\nu_{\gamma}(y,s) = \eta(y) |y|^{-\gamma} {\rm d}\mu_A(y,s)$, for $0 \leq\gamma< 2n$, where $\eta$ is a cut-off function around the origin on $\mathbb{R}^{2n}$. Moreover, for $\gamma=0$ we characterize the type set of $\nu_0$.
References: [1] M. P. do Carmo: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992). DOI 10.1007/978-1-4757-2201-7 | MR 1138207 | Zbl 0752.53001
[2] I. M. Gel'fand, G. E. Shilov: Generalized Functions. Vol. I. Properties and Operations. Academic Press, New York (1964). DOI 10.1090/chel/377 | MR 0166596 | Zbl 0115.33101
[3] T. Godoy, P. Rocha: $L^p-L^q$ estimates for some convolution operators with singular measures on the Heisenberg group. Colloq. Math. 132 (2013), 101-111. DOI 10.4064/cm132-1-8 | MR 3106091 | Zbl 1277.43013
[4] T. Godoy, P. Rocha: Convolution operators with singular measures of fractional type on the Heisenberg group. Stud. Math. 245 (2019), 213-228. DOI 10.4064/sm8781-12-2017 | MR 3865682 | Zbl 1412.43009
[5] W. Littman: $L^p-L^q$ estimates for singular integral operators arising from hyperbolic equations. Partial Differential Equations Proceedings of Symposia in Pure Mathematics 23. American Mathematical Society, Providence (1973), 479-481. DOI 10.1090/pspum/023/9948 | MR 0358443 | Zbl 0263.44006
[6] D. M. Oberlin: Convolution estimates for some measures on curves. Proc. Am. Math. Soc. 99 (1987), 56-60. DOI 10.1090/S0002-9939-1987-0866429-6 | MR 866429 | Zbl 0613.43002
[7] Y. Pan: A remark on convolution with measures supported on curves. Can. Math. Bull. 36 (1993), 245-250. DOI 10.4153/CMB-1993-035-2 | MR 1222541 | Zbl 0820.43002
[8] F. Ricci: $L^p-L^q$ boundedness of convolution operators defined by singular measures in $\mathbb R^n$. Boll. Unione Mat. Ital., VII. Ser., A 11 (1997), 237-252. (In Italian.) MR 1477777 | Zbl 0946.42006
[9] F. Ricci, E. M. Stein: Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds. J. Funct. Anal. 86 (1989), 360-389. DOI 10.1016/0022-1236(89)90057-8 | MR 1021141 | Zbl 0684.22006
[10] S. Secco: $L^p$-improving properties of measures supported on curves on the Heisenberg group. Stud. Math. 132 (1999), 179-201. DOI 10.4064/sm-132-2-179-201 | MR 1669682 | Zbl 0960.43009
[11] S. Secco: $L^p$-improving properties of measures supported on curves on the Heisenberg group. II. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 5 (2002), 527-543. MR 1911204 | Zbl 1113.42012
[12] E. M. Stein, R. Shakarchi: Complex Analysis. Princeton Lectures in Analysis 2. Princeton University Press, Princeton (2003). MR 1976398 | Zbl 1020.30001
[13] E. M. Stein, G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series. Princeton University Press, Princeton (1971). DOI 10.1515/9781400883899 | MR 0304972 | Zbl 0232.42007
Affiliations: Pablo Rocha, Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253-Bahía Blanca 8000, Buenos Aires, Argentina, e-mail: pablo.rocha@uns.edu.ar
