A convolution estimate for a measure on a curve in $\mathbb {R}^4$
A convolution estimate for a measure on a curve in $\mathbb {R}^4$
Let $\gamma (t)=(t,t^{2},t^{3},t^{4})$ and fix an interval $I\subset {\mathbb {R}}$. If $T$ is the operator on ${\mathbb {R}}^{4}$ defined by $Tf(x)=\int \nolimits _{I}f(x-\gamma (t)) dt$, then $T$ maps $L^{\frac {5}{3}}({\mathbb {R}}^{4})$ into $L^{2}({\mathbb {R}}^{4})$.