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Convolution estimates related to surfaces of half the ambient dimension. (English) Zbl 0739.42011

Let \(\varphi: \mathbb{R}^ n\to \mathbb{R}^ n\) be continuously differentiable, and define \(\beta: \mathbb{R}^ n\to \mathbb{R}^{2n}\) by \(\beta(x)=(x,\varphi(x))\). Let \(\lambda_ n\) be the \(n\)-dimensional Lebesgue measure and \(\sigma=\beta(\lambda_ n)\) the image of \(\lambda_ n\) by \(\beta\). The authors study the validity of the convolution inequality \[ \| \sigma* f\| _{L^ 3(\mathbb{R}^{2n})}\leq C\| f\| _{L^{3/2}(\mathbb{R}^{2n})}. \] Let \(J(h):=\inf_ x| \det(\varphi'(x+h)-\varphi'(x))|\). 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 \(R_ \alpha\theta(x)=\int J(x-y)^{-1+\alpha}\theta(y)d\lambda_ n(y)\), i.e. \[ \| R_{1/2}\theta\| _{L^ 6(\mathbb{R}^ n)}\leq C\| \theta\| _{L^{3/2}(\mathbb{R}^ n)}. \] They discuss the validity of this estimate in a variety of situations.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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[1] DOI: 10.2307/2046270 · Zbl 0613.43002 · doi:10.2307/2046270
[2] DOI: 10.2307/2000407 · Zbl 0563.42010 · doi:10.2307/2000407
[3] Hunt, Enseign. Math. 12 pp 249– (1966)
[4] Littman, Proc. Sympos. Pure Math. 23 pp 479– (1973) · doi:10.1090/pspum/023/9948
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