Mixed Estimates for Degenerate Multilinear Operators Associated to Simplexes

Abstract

We prove that the degenerate trilinear operator $C_3^{-1,1,1}$ given by the formula \begin{eqnarray*} C_3^{-1,1,1}(f_1, f_2, f_3)(x)=\int_{x_1 < x_2 < x_3} \hat{f_1}(x_1) \hat{f_2}(x_2) \hat{f_3}(x_3) e^{2\pi i x (-x_1 + x_2 + x_3)} dx_1dx_2 dx_3 \end{eqnarray*} satisfies the new estimates \begin{eqnarray*} ||C_3^{-1,1,1}(f_1, f_2, f_3)||_{\frac{1}{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}} \lesssim_{p_1, p_2, p_3} ||\hat{f}_1||_{p^\prime_1} ||f_2||_{p_2}||f_3||_{p_3} \end{eqnarray*} for all $f_1 \in L^{p_1}(\mathbb{R}): \hat{f}_1 \in L^{p_1^\prime}(\mathbb{R}) , f_2 \in L^{p_2}(\mathbb{R})$, and $f_3 \in L^{p_3}(\mathbb{R})$ such that $2 <p_1 \leq \infty, 1 < p_2, p_3 < \infty, \frac{1}{p_1}+\frac{1}{p_2} <1$, and $\frac{1}{p_2}+\frac{1}{p_3} <3/2$. Mixed estimates for some generalizations of $C_3^{-1,1,1}$ are also shown.