A restriction estimate using polynomial partitioning (Q2792314)

This paper presents the latest and interesting progress on the three-dimensional restriction problem. To be precise, suppose that \(S\subset \mathbb{R}^3\) is a compact smooth surface which maybe with boundary and define the extension operator \(E_S\) as NEWLINE\[NEWLINEE_Sf(x):=\int_S e^{iwx}f(w)\,\mathrm{dvol}_S(w)NEWLINE\]NEWLINE for complex functions \(f\) on \(S\). The well-known restriction conjecture of E. M. Stein says that the boundedness of \(E_S\) from \(L^\infty(S)\) to \(L^p(\mathbb{R}^3)\) should hold for all \(p>3\). An important milestone on this problem were the works of \textit{T. Tao} [Duke Math. J. 96, No. 2, 363--375 (1999; Zbl 0980.42006)] and \textit{T. Wolff} [Ann. Math. (2) 153, No. 3, 661--698 (2001; Zbl 1125.42302)], which proved the aforementioned boundedness for \(p>10/3\). Later on in 2011, J. Bourgain and the author of this paper improved the above result to \(p>56/17=3.29\dots\) In this paper, the author further improves the result of 2011 and proves the boundedness of \(E_S\) from \(L^\infty(S)\) to \(L^p(\mathbb{R}^3)\) for \(p>3.25\), via employing the polynomial partitioning approach from incidence geometry. This paper seems to be the first one that uses the polynomial method to estimate oscillatory integrals.