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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.17532 |
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| _version_ | 1866913216665223168 |
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| author | Iosevich, A. Palsson, E. Zhai, Y. Wyman, E. |
| author_facet | Iosevich, A. Palsson, E. Zhai, Y. Wyman, E. |
| contents | Consider the operator $$T_Kf(x)=\int_{{\mathbb R}^d} K(x,y) f(y) dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$ Λ^K_G(f_1, \dots, f_n)=\int \dots \int \prod_{ \{(i,j): 1 \leq i<j \leq n; E(i,j)=1 \} } K(x^i,x^j) \prod_{i=1}^n f_i(x^i) dx^i, $$ where $G$ is a connected graph on $n$ vertices, $E$ is the edge map on $G$, i.e $E(i,j)=1$ if and only if the $i$'th and $j$'th vertices are connected by an edge, $K$ is the aforementioned kernel, and $f_i: {\mathbb R}^d \to {\mathbb R}$, measurable.
This paper establishes multi-linear inequalities of the form $$ Λ^K_G(f_1,f_2, \dots,f_n) \leq C {||f_1||}_{L^{p_1}({\mathbb R}^d)} {||f_2||}_{L^{p_2}({\mathbb R}^d)} \dots {||f_n||}_{L^{p_n}({\mathbb R}^d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-linear forms, structure of graphs and Lebesgue spaces Iosevich, A. Palsson, E. Zhai, Y. Wyman, E. Classical Analysis and ODEs Combinatorics Consider the operator $$T_Kf(x)=\int_{{\mathbb R}^d} K(x,y) f(y) dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$ Λ^K_G(f_1, \dots, f_n)=\int \dots \int \prod_{ \{(i,j): 1 \leq i<j \leq n; E(i,j)=1 \} } K(x^i,x^j) \prod_{i=1}^n f_i(x^i) dx^i, $$ where $G$ is a connected graph on $n$ vertices, $E$ is the edge map on $G$, i.e $E(i,j)=1$ if and only if the $i$'th and $j$'th vertices are connected by an edge, $K$ is the aforementioned kernel, and $f_i: {\mathbb R}^d \to {\mathbb R}$, measurable. This paper establishes multi-linear inequalities of the form $$ Λ^K_G(f_1,f_2, \dots,f_n) \leq C {||f_1||}_{L^{p_1}({\mathbb R}^d)} {||f_2||}_{L^{p_2}({\mathbb R}^d)} \dots {||f_n||}_{L^{p_n}({\mathbb R}^d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$. |
| title | Multi-linear forms, structure of graphs and Lebesgue spaces |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2401.17532 |