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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.09399 |
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| _version_ | 1866909686870048768 |
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| author | Hejna, Agnieszka Nagel, Alexander Ricci, Fulvio |
| author_facet | Hejna, Agnieszka Nagel, Alexander Ricci, Fulvio |
| contents | Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calderón-Zygmund kernels adapted to $n$ different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix $\mathbf E$, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on $\mathbb{R}^d$ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their $L^1$-equivalence, we use these square functions to define a local multi-norm Hardy space $\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$. We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the $2$-dilation case, by other authors. The general $n$-dilation case treated here is considerably harder and requires new ideas and a more systematic approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on $\mathbb{R}^{d}$ Hejna, Agnieszka Nagel, Alexander Ricci, Fulvio Functional Analysis Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calderón-Zygmund kernels adapted to $n$ different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix $\mathbf E$, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on $\mathbb{R}^d$ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their $L^1$-equivalence, we use these square functions to define a local multi-norm Hardy space $\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$. We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the $2$-dilation case, by other authors. The general $n$-dilation case treated here is considerably harder and requires new ideas and a more systematic approach. |
| title | Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on $\mathbb{R}^{d}$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.09399 |