Oscillatory hyper Hilbert transforms along general curves

Abstract

We consider the oscillatory hyper Hilbert transform {{\displaystyle H}}_{\gamma ,\alpha ,\beta }f(x)={\displaystyle {\int }_{\mathrm{0}}^{\infty }f(x-\Gamma (t))}{e}^{it-\beta }{t}^{-(\mathrm{1}+\alpha )}\mathrm{d}t, where Γ(t) = (t, γ(t)) in {ℝ}^{\mathrm{2}} is a general curve. When γ is convex, we give a simple condition on γ such that H_γ,_α,_β is bounded on L² when \beta \ge \mathrm{3}\alpha ,\beta >\mathrm{0}. As a corollary, under this condition, we obtain the L^p-boundedness of H_γ,_α,_β when . When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H_γ,_α,_β is bounded on L². As an application, we construct a class of strictly convex curves along which H_γ,_α,_β is bounded on L² only if β>2α>0.

Keywords

Hilbert transform / oscillatory integral / oscillatory hyper Hilbert transform

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