Let \(S\subset \mathbb{R}^n\) be a smooth, compact hyper-surface with positive definite second fundamental form. Let \(\sigma\) be its surface measure. Assume the exponent \(p\) satisfies

(i)

\(p>2\frac{4n+3}{4n-3}\) if \(n\equiv 0\pmod{3}\);

(ii)

\(p>\frac{2n+1}{n-1}\) if \(n\equiv 1\pmod{3}\);

(iii)

\(p>\frac{4(n+1)}{2n-1}\) if \(n\equiv 2\pmod{3}\).

Applying the Bennett-Carbery-Tao multilinear restriction estimate, the authors prove that if \(p\) satisfies (i), (ii) or (iii), then the inequality \(\|\widehat{\mu}\|_p\lesssim C_p \|\frac{d\mu}{d\sigma}\|_{\infty}\) holds for all measures \(\mu\ll\sigma\) such that \(\frac{d\mu}{d\sigma}\in L^{\infty}(S,d\sigma)\) and, for \(n=3\) and \(p>3{3 \over 10}\), then \(\|\widehat{\mu}\|_p \leq C_p\|\frac{d\mu}{d\sigma}\|_{\infty}\) if \(\mu\ll\sigma\) and \(\frac{d\mu}{d\sigma}\in L^{\infty}(S,d\sigma)\). Here \(\widehat{\mu}\) denotes the Fourier transform of the measure \(\mu\). These results give improved \(L^p\) estimates in the Stein restriction problem for dimension at least \(5\) and a small improvement in dimension \(3\).
The authors also consider the Hörmander-type oscillatory integral operators
\[ (T_{\lambda})f(x):=\int_{\mathbb{R}^{n-1}} e^{i\lambda\psi(x,y)}f(y)\,dy \quad (\|f\|_{\infty}\leq1) \]
with real analytic phase function
\[ \psi(x,y)=x_1y_1+\cdots+x_{n-1}y_{n-1}+x_n\langle Ay,y\rangle +O(|x||y|^3)+O(|x|^2|y|^2){(\ast_1)} \]
and \(A\) non-degenerate (\(x\in \mathbb{R}^n,\,y\in\mathbb{R}^{n-1}\) are restricted to a neighborhood of \(0\)). Let \(T_{\lambda}\) be as above with \(A\) positive or negative definite in \((\ast_1)\). The authors show that \(\|T_{\lambda}f\|_p \leq C_p\lambda^{-n/p} \|f\|_{\infty}\) holds for \(p\) satisfying {(i)}, {(ii)} or {(iii)}, what is new when \(n\geq 5\). If \(n\) is even and \(T_{\lambda}\) as above, assuming in \((\ast_1)\) that \(A\) is non-degenerate, the authors also prove the inequalities \(\|T_{\lambda}f\|_p\leq C_p\lambda^{-n/p} \|f\|_{\infty}\) for \(p>\frac{2(n+2)}{n}\) which is the best possible range of \(p\).
Recall that the Bochner-Riesz multiplier \(S_{\delta}\) is defined by \((S_{\delta}f)^{\wedge}(\xi):=(1-|\xi|^2)_+\widehat{f}(\xi)\) for all \(\xi\in{\mathbb R}^n\). Equivalently \(S_{\delta}f=f*K_{\delta}\), where \(K_{\delta}\) has the asymptotic \[ K_{\delta}(x)\sim e^{\pm2\pi i|x|}/|x|^{\frac{n+1}2+\delta}.{(\ast_2)} \] The Bochner-Riesz multilinear problem is then to obtain the optimal condition on \(\delta\geq 0\) to satisfy \[ \|S_{\delta}f\|_{L^p(\mathbb{R}^n)}\leq C\|f\|_{L^p(\mathbb{R}^n)}.{(\ast_3)} \]
According to the paper, the condition \[ \delta>\max\left(0,\left|\frac12-\frac1p\right|n-\frac12\right){(\ast_4)} \] is clearly necessary in view of \((\ast_2)\). It is conjectured that \((\ast_4)\) also suffices for \((\ast_3)\) to hold and this was proven for \(n=2\) by L. Carleson and P. Sjölin [Stud. Math. 44, 287–299 (1972; Zbl 0215.18303)]. When \(n\geq3\), the authors verify that the Bochner-Riesz conjecture holds provided \(\max(p,p^{\prime})\) satisfies {(i)}, {(ii)} or {(iii)}.