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Convolution with measures on flat curves in low dimensions. (English) Zbl 1203.42024

Let $\gamma$ be a curve in ${ℝ}^{d}$ given by

$\gamma \left(t\right)=\left(t,\frac{{t}^{2}}{2},\cdots ,\frac{{t}^{d-1}}{\left(d-1\right)!},\varphi \left(t\right)\right),$

where $\varphi \in {C}^{d}\left(a,b\right)$, where ${\varphi }^{\left(j\right)}\left(t\right)>0$ for $t\in \left(a,b\right)$ and $j=0,1,2,\cdots ,d$, and where ${\varphi }^{\left(d\right)}$ is nondecreasing. Such curves are termed simple. In the paper under review the author proves ${L}^{p}\to {L}^{q}$ convolution estimates for the affine arclength measure $\lambda$ on $\gamma$, given by $d\lambda ={\varphi }^{\left(d\right)}{\left(t\right)}^{2/\left({d}^{2}+d\right)}dt$, when $d=2,3,4$. For $d=2,3$, he also establishes certain related Lorentz space estimates.

MSC:
 42B20 Singular and oscillatory integrals, several variables
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