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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 ϕ:[a,b) is such that ϕ(a)=ϕ ' (a)=ϕ '' (a)=0 and ϕ '' (x)>0, ϕ ''' (x)0 on (a,b), and ω(x):=ϕ ' (x) 2 /ϕ(x), and ν is the measure on the graph {(x,ϕ(x)):x(a,b)} given by dν:=ω 1/3 (x)dx, then the author establishes the convolution estimate

ν*f L 3 ( 2 ) Cf L 3/2 ( 2 )

for an absolute constant C (independent of ϕ), a well as the Fourier decay estimate

c d e iζx+ηϕ(x) ω (x) 1/2 d xC/|η| 1/2

for all ζ,η and [c,d][a,b). The weight ω is not completely optimal; the author conjectures it should be replaced by the affine curvature weight ϕ '' (x). The methods are mostly elementary.

42B20Singular and oscillatory integrals, several variables