振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·

Boundedness of Oscillatory Hyper-Hilbert Transform on Sobolev Spaces

We study the oscillatory hyper-Hilbert transform H_n,α,βf(x) = ?₀¹f(x - Γ(t))e^it-βt^-1-αdt, along the curve Γ(t) = (t^p₁, t^p₂, …, t^p_n), where 0 < p₁ < p₂ <…< p_n and α > β > 0. For 0 ≤ γ < (nα)/((n+1)(p₁+α)), we prove that H_n,α,β is bounded from L_γ^p (Rⁿ) to L^p(Rⁿ) whenever |1/p-1/2|< (β-(n+1)α-(β+p₁)γ])/(2β), and that H_n,α,β is bounded from L^γ2 (Rⁿ) to L²(Rⁿ) whenever β ≥ (α-γp₁)/(γ+ (1/n+1)).