A restriction theorem for oscillatory integral operator with certain polynomial phase

Abstract

We consider the oscillatory integral operator {{\displaystyle T}}_{\alpha ,m}f(x)={\int }_{{ℝ}^n}{\mathrm{e}}^{\mathrm{i}({{\displaystyle x}}_{\mathrm{1}}^{{\alpha }_1}{{\displaystyle y}}_{\mathrm{1}}^m+\cdots +{{\displaystyle x}}_n^{{\alpha }_n}{{\displaystyle y}}_n^m)}f(y)\mathrm{d}y, where the function f is a Schwartz function. In this paper, the restriction theorem on S^{n-\mathrm{1}} for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.

Keywords

Restriction theorem / oscillatory integral operator / L² boundedness / optimal estimate / necessary condition

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