Convolution estimates for some measures on flat curves

AbstractA measure μ is said to be Lp-improving if μ ∗ Lp ⊂ Lq for some q > p. It is known that certain singular measures supported on curves in R2 are Lp-improving. If μ is a smooth measure supported on a flat curve Γ (the curvature of Γ vanishes to infinite order at some point), μ need not be Lp-improving. Under certain hypotheses, it is proved that in this situation, although μ is not LP-improving, it does satisfy an analogous property with respect to Orlicz spaces: μ ∗ LΦ ⊂ L2 for some Orlicz function Φ with limt → ∝ Φ(t)t2 = 0. Estimates on the distribution function of the Fourier transform of μ are obtained