A restriction estimate for a class of oscillatory integral operators along paraboloid

Abstract

Abstract In this paper we establish a restriction estimate for a class of oscillatory integral operators along a paraboloid, Pd−1:={(x1,…,xd):xd=x12+⋯+xd−12}. $$ {\mathbb{P}^{d-1}:=\bigl\{ (x_{1},\ldots ,x_{d}):x_{d}=x_{1}^{2}+ \cdots +x_{d-1}^{2}\bigr\} .} $$ Specifically, we consider the oscillatory integral operators defined by 1 Tm,nf(x)=∫Rdei(x1mξ1n+⋯+xdmξdn)f(ξ)dξ, $$ T_{m,n}f(x)= \int _{\mathbb{R}^{d}}e^{i(x_{1}^{m} \xi _{1}^{n}+\cdots +x _{d}^{m}\xi _{d}^{n})}f(\xi )\,d\xi , $$ where n, m are integers satisfying 2≤d<n≤2md $2\leq d< n\leq 2md$, then ∥Tm,nf∥L2(dσ,Pd−1∩Bd(0,1))≤Cm,n,d∥f∥Lp(Rd) $$ \Vert T_{m,n}f \Vert _{L^{2} (d\sigma , \mathbb{P}^{d-1}\cap B^{d}(0,1) )} \leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})} $$ holds for 1<p≤4md4md−n $1< p\leq \frac{4md}{4md-n}$. A necessary condition is also given to ensure this boundedness.

Keywords

• Restriction estimate
• Oscillatory integral operator
• Multidimensional Van der Corput
• Knapp’s counterexample