Sharpness results for \(L^2\)-smoothing of oscillatory integrals (Q2758116)

Let \(a>0\). For \(x\in\mathbb{R}^n\) and \(t\in\mathbb{R}\), let NEWLINE\[NEWLINE(S^af)[x](t)={1\over{(2\pi)^n}}\int_{\mathbb{R}^n} e^{i(x\xi+t|\xi|^a)}\widehat f(\xi) d\xi,NEWLINE\]NEWLINE where \(\widehat f\) is the Fourier transform of \(f\). Denote by \(B^{n+1}\) the open ball in \(\mathbb{R}^{n+1}\). The author proves that if there is a constant \(C\), independent of \(f\), such that NEWLINE\[NEWLINE\|S^a f\|_{L^2(B^{n+1})}\leq C\|f\|_{H^s(\mathbb{R}^n)},NEWLINE\]NEWLINE then \(s\geq (1-a)/2\land 0\), by using integration by parts to control remainder terms and by using the oscillation at infinity of Bessel functions to reduce the higher-dimensional problem to the one-dimensional one. The case \(a=2\) of this result was obtained by \textit{P. Sjögren} and \textit{P. Sjölin} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 16, No. 1, 3-12 (1991; Zbl 0713.35016)].