Let \(S\) be a hypersurface in \(\mathbb R^n\), and \(T_Sf(x)=\int_S f(x-y) d\sigma(y)\), where \(d\sigma\) is a smooth compactly supported measure on \(S\). It is known that if \(S\) has non-vanishing Gaussian curvature, then the estimate \[ \|T_Sf\|_{L^q(\mathbb R^n)}\leq C_{p,q}\|f\|_{L^p(\mathbb R^n)}, \quad f\in \mathcal S(\mathbb R^n), \tag{\(*\)} \] holds if and only if \((1/p,1/q)\) belongs to the triangle with vertices \((0,0)\), \((1,1)\), and \((\frac n{n+1}, \frac 1{n+1})\). If the Gaussian curvature on \(S\) is allowed to vanish, sharp \((L^p,L^q)\) bounds are in general difficult to obtain. The purpose of the authors is to determine the optimal range of exponents \((p,q)\) such that \((*)\) holds in the case when \(S\) is a graph of a homogeneous function of degree \(m\geq 2\). Denote by \(\mathcal Q(N,\rho)\) the trapezoid with vertices \((0,0)\), \((1,1)\), \(\left(\frac{N\rho}{(N-1)(\rho+1)}, \frac{\rho}{(N-1)(\rho+1)}\right)\), and \(\left(1-\frac{N\rho}{(N-1)(\rho+1)}, 1-\frac{\rho}{(N-1)(\rho+1)}\right)\). Let \[ Tf(x) = \int_{\mathbb R^{n-1}}f(x'-y', x_n-\Phi(y'))\psi(y') dy', \] where \(\psi\) is a smooth cutoff function and \(\Phi\in C^\infty(\mathbb R^{n-1}\setminus \{0\})\) is homogeneous of degree \(m\geq 2\). Then one of their main result is: Suppose \(\Phi(\omega)^{-1}\in L^\rho(S^{n-2})\) with \(0<\rho<\min\{\frac{n-1}{m}, \frac 12\}\). Then the norm inequality \((\ast)\) holds if \((1/p,1/q)\) lies in the trapezoid \(\mathcal Q(2,\rho)\). Conversely, for each \(\rho<\frac{n-1}{m}\), there is a \(\Phi\) satisfying the above hypotheses such that \((*)\) fails for \((1/p,1/q)\) outside the trapezoid \(\mathcal Q(2,\rho)\).
They investigate further such mapping properties to be obtained from a given decay on the Fourier transform of a surface carried measure weighted by powers of \(|\Phi|\). An application to a class of dispersive equations is also given.