Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Multi-linear forms, structure of graphs and Lebesgue spaces (2401.17532v1)

Published 31 Jan 2024 in math.CA and math.CO

Abstract: 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 $$ \LambdaK_G(f_1, \dots, f_n)=\int \dots \int \prod_{ {(i,j): 1 \leq i<j \leq n; E(i,j)=1 } } K(xi,xj) \prod_{i=1}n f_i(xi) dxi, $$ 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 $$ \LambdaK_G(f_1,f_2, \dots,f_n) \leq C {||f_1||}{L{p_1}({\mathbb R}d)} {||f_2||}{L{p_2}({\mathbb R}d)} \dots {||f_n||}_{L{p_n}({\mathbb R}d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$.

Summary

We haven't generated a summary for this paper yet.