论文标题
$ l^p $估计多线卷积运算符，以球形度量定义
$L^p$ estimates for multilinear convolution operators defined with spherical measure
论文作者
论文摘要
令$σ=（σ_{1}，σ_{2}，\ dots，σ_{n}）\ in \ Mathbb {s}^{n-1} $和$dσ$表示符合标准化的lebesgue MATEA，对$ \ m m i \ m athbb {s}}^n-1}，n-1}}，n \ geq，〜n \ geq，〜对于功能，$ f_1，f_2，\ dots，f_n $在$ \ r $上定义$ t（f_ {1}，f_ {2}，\ dots，f_ {n}）（x）= \ int _ {\ mathbb {s}^{n-1}} \ prod^{n} _ {指示符的条件$ p_1，p_2，\ dots，p_n $和$ r $，其$ \ prod_ {j = 1}^n l^{p_j}（\ r）\ rightarrow l^r（\ r），$ 1 \ r^r（\ r），$ 1 \ leq p_j，r \ leq p_j，r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ r \ f iff the $ \ prod_ {j = 1}^n l^{p_j}（\ r）概括在〜\ cite {jbak，oberlin}中获得的结果。
Let $σ=(σ_{1},σ_{2},\dots,σ_{n})\in \mathbb{S}^{n-1}$ and $dσ$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear operator given by $$T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-σ_j)dσ, ~x\in \R.$$ In this paper we obtain necessary and sufficient conditions on exponents $p_1,p_2,\dots,p_n$ and $r$ for which the operator $T$ is bounded from $\prod_{j=1}^n L^{p_j}(\R)\rightarrow L^r(\R),$ where $1\leq p_j,r\leq \infty, j=1,2,\dots,n.$ This generalizes the results obtained in~\cite{jbak,oberlin}.
