$L^p$-improving properties of certain singular measures on the Heisenberg group

Rocha, Pablo. "$L^{p}$-improving properties of certain singular measures on the Heisenberg group." Mathematica Bohemica 147.1 (2022): 131-140. <http://eudml.org/doc/297674>.

@article{Rocha2022,
abstract = {Let $\mu _A$ be the singular measure on the Heisenberg group $\mathbb \{H\}^\{n\}$ supported on the graph of the quadratic function $\varphi (y) = y^\{t\}Ay$, where $A$ is a $2n \times 2n$ real symmetric matrix. If $\det (2A \pm J) \ne 0$, 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 $0 \le \gamma < 2n$, where $\eta $ is a cut-off function around the origin on $\mathbb \{R\}^\{2n\}$. Moreover, for $\gamma =0$ we characterize the type set of $\nu _\{0\}$.},
author = {Rocha, Pablo},
journal = {Mathematica Bohemica},
keywords = {Heisenberg group; singular Borel measure; $L^\{p\}$-improving property},
language = {eng},
number = {1},
pages = {131-140},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$L^\{p\}$-improving properties of certain singular measures on the Heisenberg group},
url = {http://eudml.org/doc/297674},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Rocha, Pablo
TI - $L^{p}$-improving properties of certain singular measures on the Heisenberg group
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 1
SP - 131
EP - 140
AB - 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 $\det (2A \pm J) \ne 0$, 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 $0 \le \gamma < 2n$, where $\eta $ is a cut-off function around the origin on $\mathbb {R}^{2n}$. Moreover, for $\gamma =0$ we characterize the type set of $\nu _{0}$.
LA - eng
KW - Heisenberg group; singular Borel measure; $L^{p}$-improving property
UR - http://eudml.org/doc/297674
ER -