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Generic trilinear multipliers associated to degenerate simplexes

Research paper by Robert Kesler

Indexed on: 10 Jun '18Published on: 04 May '18Published in: Collectanea Mathematica



Abstract

For each \(1 \le p \le \infty \) , let \(W_{p}(\mathbb {R}) = \left\{ f \in L^p(\mathbb {R}): \hat{f} \in L^{p^\prime }(\mathbb {R}) \right\} \) be equipped with the norm \( f _{W_{p}(\mathbb {R})} = \left \hat{f}\right _{L^{p^\prime }(\mathbb {R})}\) . Moreover, let \(a_1,a_2 : \mathbb {R}^2 \rightarrow \mathbb {C}\) satisfy the condition that for all \(\mathbf {\alpha } \in \mathbb {Z}_{ \ge 0}^2\) there is \(C_{a_1, a_2,\mathbf {\alpha }}>0\) such that for all \(\mathbf {\xi } \in \mathbb {R}^2\) and \(j \in \{1,2\}\) , \(\left \partial ^{\mathbf {\alpha }} a_j (\mathbf {\xi })\right \le \frac{C_{a_1, a_2,\mathbf {\alpha }}}{dist(\mathbf {\xi }, \varGamma )^{ \mathbf {\alpha } }}\) . Our main result is that the trilinear multiplier given on \(\mathcal {S}(\mathbb {R})^3\) by $$\begin{aligned} B[a_1, a_2] : (f_1, f_2, f_3) \mapsto \int _{\mathbb {R}^3} a_1(\xi _1, \xi _2) a_2(\xi _2, \xi _3) \left[ \prod _{j=1}^3 \hat{f_j} (\xi _j) e^{2 \pi ix \xi _j} \right] d\xi _1 d\xi _2 d\xi _3 \end{aligned}$$ extends to a bounded map from \(L^{p_1}(\mathbb {R}) \times W_{p_2}(\mathbb {R}) \times L^{p_3}(\mathbb {R})\) into \(L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb {R})\) provided $$\begin{aligned} 1< p_1, p_3< \infty , \frac{1}{p_1} + \frac{1}{p_2}<1, \frac{1}{p_2} + \frac{1}{p_3}<1, 2< p_2 < \infty . \end{aligned}$$