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Convolution with measures on flat curves in low dimensions. (English) Zbl 1203.42024
Let $\gamma$ be a curve in $\mathbb R^d$ given by $$ \gamma (t)=(t, \frac {t^2}{2}, \dots, \frac{t^{d-1}}{(d-1)!}, \phi (t)),$$ where $\phi \in C^d(a,b)$, where $\phi^{(j)}(t)>0$ for $t \in (a,b)$ and $j=0, 1, 2, \dots, d$, and where $\phi^{(d)}$ is nondecreasing. Such curves are termed {\it 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 = \phi^{(d)}(t)^{2/(d^2+d)}dt$, when $d=2, 3, 4$. For $d=2, 3$, he also establishes certain related Lorentz space estimates.

42B20Singular and oscillatory integrals, several variables
Full Text: DOI arXiv
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