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Two estimates for curves in the plane. (English) Zbl 1061.42011

The author proves a uniform Fourier decay and convolution estimate for curves in the plane. Specifically, if \(\phi: [a,b) \to \mathbb R\) is such that \(\phi(a) = \phi'(a) = \phi''(a) = 0\) and \(\phi''(x) > 0\), \(\phi'''(x) \geq 0\) on \((a,b)\), and \(\omega(x) := \phi'(x)^2 / \phi(x)\), and \(\nu\) is the measure on the graph \(\{ (x,\phi(x)): x \in (a,b) \}\) given by \(d\nu := \omega^{1/3}(x)dx\), then the author establishes the convolution estimate \[ \| \nu* f \| _{L^3(\mathbb R^2)} \leq C \| f \| _{L^{3/2}(\mathbb R^2)} \] for an absolute constant \(C\) (independent of \(\phi\)), a well as the Fourier decay estimate \[ \biggl| \int_c^d e^{i \zeta x + \eta \phi(x)} \omega(x)^{1/2}\,dx\biggr| \leq C/| \eta| ^{1/2} \] for all \(\zeta, \eta \in \mathbb R\) and \([c,d] \subset [a,b)\). The weight \(\omega\) is not completely optimal; the author conjectures it should be replaced by the affine curvature weight \(\phi''(x)\). The methods are mostly elementary.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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