An \(L^p-L^q\) estimate for Radon transforms associated to polynomials
Let \(S(x,y)\) be a real polynomial of degree \(n\geq 2\) defined by \[ S(x,y)=\sum_{d=0}^{n}\sum_{j+k=d} a_{jk}x^jy^k. \] The Radon transforms \(R\), and \(T\) of \(f\) are defined by \[ Rf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y)\psi(t,x,y) dy, \] and \[ Tf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y) dy, \] respectively, where \(\psi\in C_c^\infty(\mathbb R^3)\) is a cutoff function. Let \(\Delta\) be the closed convex hull of the points \(O=(0,0)\), \(A=(2/(n+1)\), \(1/(n+1))\), \(A'=(n/(n+1)\), \((n-1)/(n+1))\), and \(O'=(1,1)\). When \(S(x,y)\) is a homogeneous polynomial, and \(a_{1, n-1}\neq 0\) and \(a_{n-1, 1}\neq 0\), Phong and Stein proved that \(R\) is bounded from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\), if \((1/p, 1/q)\) is in the set \(\Delta\) minus the half-open segments \((O, A]\) and \([A', O')\). They also proved that for \(R\) to be bounded from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\), it is necessary that \((1/p, 1/q)\) is in \(\Delta\), and in the case \(n=2, 3\) it is known that \(R\) is bounded precisely for \((1/p, 1/q)\in\Delta\). The author gives a positive result for endpoint estimates in the case \(n\geq 4\), and more. His main result is (for not necessarily homogeneous polynomials): if \(a_{1, n-1}\neq 0\), then there is \(C_n>0\) such that \[ \|Tf\|_{L^{n+1}}\leq C_n|a_{1,n-1}|^{-1/(n+1)}\|f\|_{L^{(n+1)/2}}, \] where \(C_n\) is independent of the coefficients \(a_{jk}\). Using this, he gets endpoint estimates, not treated by Phong and Stein, in the homogeneous polynomial case.
scientific article; zbMATH DE number 1442142
scientific article