\(L^p\to L^q\) estimates for the circular maximal function (Q2709027)

In this thesis the author proves the sharp \((L^p,L^q)\) mapping properties for the circular maximal function NEWLINE\[NEWLINE Mf(x) = \sup_{1 \leq r \leq 2} \int f(x + r e^{i\theta})\;d\thetaNEWLINE\]NEWLINE in the plane, except for some logarithmic losses at endpoints. Specifically, the author obtains a restricted weak-type \(L^{5/2} \to L^5\) estimate with a logarithmic loss in derivatives, and then interpolates that with Bourgain's circular maximal theorem to obtain a triangle of strong-type estimates. The techniques are combinatorial, in the same spirit as that of Wolff's circular maximal theorem, with no use of the Fourier transform or other oscillatory integral techniques. However, in a later paper of the author and Sogge, essentially the same theorem was proven using oscillatory integral methods, and specifically local smoothing estimates for the wave equation. The thesis also contains a short exposition of the connection of circular maximal theorems to wave equations and also to certain questions in geometric measure theory concerning circle packings.NEWLINENEWLINEFor the entire collection see [Zbl 0955.43001].