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Convolution estimates related to surfaces of half the ambient dimension. (English) Zbl 0739.42011
Let $\varphi: \bbfR\sp n\to \bbfR\sp n$ be continuously differentiable, and define $\beta: \bbfR\sp n\to \bbfR\sp{2n}$ by $\beta(x)=(x,\varphi(x))$. Let $\lambda\sb n$ be the $n$-dimensional Lebesgue measure and $\sigma=\beta(\lambda\sb n)$ the image of $\lambda\sb n$ by $\beta$. The authors study the validity of the convolution inequality $$\Vert \sigma* f\Vert \sb{L\sp 3(\bbfR\sp{2n})}\leq C\Vert f\Vert \sb{L\sp{3/2}(\bbfR\sp{2n})}.$$ Let $J(h):=\inf\sb x\vert \det(\varphi'(x+h)-\varphi'(x))\vert$. The authors assume that $x\to (\varphi(x+h)-\varphi(x))$ is injective unless $J(h)=0$. They then prove that the above convolution inequality is implied by a certain estimate for the exotic Riesz potential $R\sb \alpha\theta(x)=\int J(x-y)\sp{-1+\alpha}\theta(y)d\lambda\sb n(y)$, i.e. $$\Vert R\sb{1/2}\theta\Vert \sb{L\sp 6(\bbfR\sp n)}\leq C\Vert \theta\Vert \sb{L\sp{3/2}(\bbfR\sp n)}.$$ They discuss the validity of this estimate in a variety of situations.
Reviewer: J.Marschall (Neubiberg)

42B20Singular and oscillatory integrals, several variables
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