Boundedness of the dyadic maximal function on graded Lie groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916089614565376 |
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| author | Cardona, Duván Delgado, Julio Ruzhansky, Michael |
| author_facet | Cardona, Duván Delgado, Julio Ruzhansky, Michael |
| contents | Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calderón, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*}
\mathcal{M}^{dσ}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)dσ(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $dσ(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $dσ$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{dσ}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{dσ}$ of $dσ$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of $\mathcal{M}_D^{dσ}$ on $L^p(G)$ for all $1<p\leq \infty.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08964 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundedness of the dyadic maximal function on graded Lie groups Cardona, Duván Delgado, Julio Ruzhansky, Michael Functional Analysis Analysis of PDEs Representation Theory Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calderón, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} \mathcal{M}^{dσ}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)dσ(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $dσ(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $dσ$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{dσ}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{dσ}$ of $dσ$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of $\mathcal{M}_D^{dσ}$ on $L^p(G)$ for all $1<p\leq \infty.$ |
| title | Boundedness of the dyadic maximal function on graded Lie groups |
| topic | Functional Analysis Analysis of PDEs Representation Theory |
| url | https://arxiv.org/abs/2301.08964 |