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Convolution with measures on flat curves in low dimensions. (English) Zbl 1203.42024

Let γ be a curve in d given by

γ(t)=(t,t 2 2,,t d-1 (d-1)!,ϕ(t)),

where ϕC d (a,b), where ϕ (j) (t)>0 for t(a,b) and j=0,1,2,,d, and where ϕ (d) is nondecreasing. Such curves are termed simple. In the paper under review the author proves L p L q convolution estimates for the affine arclength measure λ on γ, given by dλ=ϕ (d) (t) 2/(d 2 +d) dt, when d=2,3,4. For d=2,3, he also establishes certain related Lorentz space estimates.

MSC:
42B20Singular and oscillatory integrals, several variables
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