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تسجيل مستخدم جديدMixed Estimates for Degenerate Multilinear Operators Associated to Simplexes
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Robert Kesler
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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^{2pi 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.
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