Double Hilbert transforms along polynomial surfaces in \(\mathbb{R}^3\)
Let \(P(s,t)=\sum_{(m,n)\in\Lambda}a_{m,n}s^mt^n\) be a polynomial in \(s\) and \(t\) on \({\mathbb R}^2\), where \((m,n)\in\Lambda\) if and only if \(a_{m,n}\not=0\). For each \((m,n)\in\Lambda\), let \(Q_{m,n}=\{(x,y)\in {\mathbb R}^2; x\geq m\;\text{ and} y\geq n\}\). Set \(Q=\cup_{(m,n)\in\Lambda}Q_{m,n}\). Then the Newton diagram of \(P\) is the smallest (closed) convex set containing \(Q\). The authors consider the \(L^p({\mathbb R}^3)\)-boundedness of the (local) double Hilbert transform along the surface \((s,t, P(s,t))\), defined formally as \[ Hf(x,y,z)=\int_{|s|\leq 1}\int_{|t|\leq 1}f(x-s,y-t,z-P(s,t)) \frac {ds dt}{st}, \] where \(P(s,t)\) is a polynomial in \(s\) and \(t\) with \(P(0,0)=0\) and \(\nabla P(0,0)=0\). The authors prove that for any \(p\in (1,\infty)\), \(\|Hf\|_{L^p}\leq A_p \|f\|_{L^p}\) if and only if for each \((m,n)\) that is a corner point of the Newton diagram corresponding to \(P\), at least one of \(m\) and \(n\) is even. The authors also prove that if \(P(s,t,u)=s^4t^4u^2+s^2t^2u^4+ s^3t^3u^3\), the analogous triple Hilbert transform in \({\mathbb R}^4\) is unbounded, and thus the strict \(4\)-dimensional analogue of the above result is false.
scientific article; zbMATH DE number 1441962