Geodesic
$L^{p}$-improving properties of certain singular measures on the Heisenberg group
Mathematica Bohemica, Tome 147 (2022) no. 1, pp. 131-140
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $\mu _A$ be the singular measure on the Heisenberg group $\mathbb {H}^{n}$ supported on the graph of the quadratic function $\varphi (y) = y^{t}Ay$, where $A$ is a $2n \times 2n$ real symmetric matrix. If $\det (2A \pm J) \neq 0$, we prove that the operator of convolution by $\mu _A$ on the right is bounded from $L^{\frac {(2n+2)}{(2n+1)}}(\mathbb {H}^{n})$ 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 $\nu _{0}$.
Let $\mu _A$ be the singular measure on the Heisenberg group $\mathbb {H}^{n}$ supported on the graph of the quadratic function $\varphi (y) = y^{t}Ay$, where $A$ is a $2n \times 2n$ real symmetric matrix. If $\det (2A \pm J) \neq 0$, we prove that the operator of convolution by $\mu _A$ on the right is bounded from $L^{\frac {(2n+2)}{(2n+1)}}(\mathbb {H}^{n})$ 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 $\nu _{0}$.
DOI : 10.21136/MB.2021.0014-20
Classification : 42A38, 42B10, 43A80
Keywords: Heisenberg group; singular Borel measure; $L^{p}$-improving property 
@article{10_21136_MB_2021_0014_20,
     author = {Rocha, Pablo},
     title = {$L^{p}$-improving properties of certain singular measures on the {Heisenberg} group},
     journal = {Mathematica Bohemica},
     pages = {131--140},
     year = {2022},
     volume = {147},
     number = {1},
     doi = {10.21136/MB.2021.0014-20},
     mrnumber = {4387472},
     zbl = {07547245},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0014-20/}
}
TY  - JOUR
AU  - Rocha, Pablo
TI  - $L^{p}$-improving properties of certain singular measures on the Heisenberg group
JO  - Mathematica Bohemica
PY  - 2022
SP  - 131
EP  - 140
VL  - 147
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0014-20/
DO  - 10.21136/MB.2021.0014-20
LA  - en
ID  - 10_21136_MB_2021_0014_20
ER  - 
%0 Journal Article
%A Rocha, Pablo
%T $L^{p}$-improving properties of certain singular measures on the Heisenberg group
%J Mathematica Bohemica
%D 2022
%P 131-140
%V 147
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2021.0014-20/
%R 10.21136/MB.2021.0014-20
%G en
%F 10_21136_MB_2021_0014_20
Rocha, Pablo. $L^{p}$-improving properties of certain singular measures on the Heisenberg group. Mathematica Bohemica, Tome 147 (2022) no. 1, pp. 131-140. doi: 10.21136/MB.2021.0014-20

[1] Carmo, M. P. do: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992). | DOI | MR | JFM

[2] Gel'fand, I. M., Shilov, G. E.: Generalized Functions. Vol. I. Properties and Operations. Academic Press, New York (1964). | DOI | MR | JFM

[3] Godoy, T., Rocha, P.: $L^p-L^q$ estimates for some convolution operators with singular measures on the Heisenberg group. Colloq. Math. 132 (2013), 101-111. | DOI | MR | JFM

[4] Godoy, T., Rocha, P.: Convolution operators with singular measures of fractional type on the Heisenberg group. Stud. Math. 245 (2019), 213-228. | DOI | MR | JFM

[5] Littman, W.: $L^p-L^q$ 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 | MR | JFM

[6] Oberlin, D. M.: Convolution estimates for some measures on curves. Proc. Am. Math. Soc. 99 (1987), 56-60. | DOI | MR | JFM

[7] Pan, Y.: A remark on convolution with measures supported on curves. Can. Math. Bull. 36 (1993), 245-250. | DOI | MR | JFM

[8] Ricci, F.: $L^p-L^q$ boundedness of convolution operators defined by singular measures in $\mathbb R^n$. Boll. Unione Mat. Ital., VII. Ser., A 11 (1997), 237-252 Italian. | MR | JFM

[9] Ricci, F., Stein, E. M.: Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds. J. Funct. Anal. 86 (1989), 360-389. | DOI | MR | JFM

[10] Secco, S.: $L^p$-improving properties of measures supported on curves on the Heisenberg group. Stud. Math. 132 (1999), 179-201. | DOI | MR | JFM

[11] Secco, S.: $L^p$-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 | JFM

[12] Stein, E. M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis 2. Princeton University Press, Princeton (2003). | MR | JFM

[13] Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series. Princeton University Press, Princeton (1971). | DOI | MR | JFM

Cité par Sources :