| Title:
|
$L^{p}$-improving properties of certain singular measures on the Heisenberg group (English) |
| Author:
|
Rocha, Pablo |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
131-140 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\nu _{0}$. (English) |
| Keyword:
|
Heisenberg group |
| Keyword:
|
singular Borel measure |
| Keyword:
|
$L^{p}$-improving property |
| MSC:
|
42A38 |
| MSC:
|
42B10 |
| MSC:
|
43A80 |
| idZBL:
|
Zbl 07547245 |
| idMR:
|
MR4387472 |
| DOI:
|
10.21136/MB.2021.0014-20 |
| . |
| Date available:
|
2022-04-17T18:11:13Z |
| Last updated:
|
2022-09-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149590 |
| . |
| Reference:
|
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