A convolution estimate for two-dimensional hypersurfaces. (English) Zbl 1203.42033

Bejenaru, Ioan; Herr, Sebastian; Tataru, Daniel

Rev. Mat. Iberoam. 26, No. 2, 707-728 (2010).

Let \(\Sigma_1=yz\)-plane, \(\Sigma_2=xz\)-plane, \(\Sigma_3=xy\)-plane be coordinate hyperplanes in \(\mathbb{R}^3\). By the classical Loomis-Whitney inequality, \[ || f * g ||_{L^2 (\Sigma_3) }\leq C\, || f ||_{L^2 (\Sigma_1) } ||g ||_{L^2 (\Sigma_2) }. \] J. Bennet, A. Carbery and J. Wright [Math. Res. Lett. 12, No. 4, 443–457 (2005; Zbl 1106.26020)] extended this result to the case when \(\Sigma_1\), \(\Sigma_2\), and \(\Sigma_3\) are bounded subsets of transversal sufficiently smooth surfaces in \(\mathbb{R}^3\). In the present paper, the result by J. Bennet, A. Carbery, and J. Wright is generalized to a class of \(C^{1,\beta}\) hypersurfaces in \(\mathbb{R}^3\) under scaleable assumptions. Applications to non-linear dispersive equations are indicated.

Reviewer: Boris Rubin (Baton Rouge)

MSC:

42B35	Function spaces arising in harmonic analysis
47B38	Linear operators on function spaces (general)

Keywords:

transversality; hypersurface; convolution; \(L^2\) estimate; induction on scales

Citations:

Zbl 1106.26020

References:

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