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^{2πi 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 $2<p_1\le \infty ,1<p_2,p_3<\infty ,\frac{1}{p_1}+\frac{1}{p_2}<1$, 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.

References in corpus (1)

• Singular integrals meet modulation invariance

Cited by in corpus (3)

• Multiple Vector Valued Inequalities via the Helicoidal Method
• Variational estimates for the bilinear iterated Fourier integral
• Generic Multilinear Multipliers Associated to Degenerate Simplexes