*(English)*Zbl 0972.42009

The author considers the ${L}^{p}$ improving properties of convolution operators $f\mapsto f*d\sigma $ where $d\sigma $ is a compactly supported measure on a ${C}^{2}$ hypersurface $S$. For surfaces of non-zero curvature the sharp estimate is ${L}^{n+1/n}\to {L}^{n}$. In this paper the author considers the slightly weaker restricted estimate ${L}^{n+1/n,1}\to {L}^{n}$.

Under very mild conditions on $S$ (namely that the Gauss map generically has bounded multiplicity, plus another technical condition of a similar flavor) the author shows that one can obtain the above restricted estimate if and only if $\mu $ obeys the estimate $\mu \left(R\right)\lesssim {\left|R\right|}^{(n-1)/(n+1)}$ for all rectangles $R$. This is in particular achieved if $\mu $ is equal to surface measure times ${\kappa}^{1/(n+1)}$, where $\kappa $ is the Gaussian curvature.

The heart of the argument is a certain ${L}^{n}$ estimate which, after multiplying everything out and changing variables, hinges on the estimation of various Jacobians and on certain multilinear estimates with these Jacobians as kernels.

##### MSC:

42B15 | Multipliers, several variables |

44A12 | Radon transform |

42A20 | Convergence and absolute convergence of Fourier and trigonometric series |