On the maximal Bochner-Riesz conjecture in the plane for \(p<2\) (Q2781393)

In this paper on the plane \(\mathbb{R}^2\) the author attacks one of the most famous problem in harmonic analysis, the almost everywhere convergence problem of the Bochner-Riesz means \(S^\delta_t f\) that are defined as \(S^\delta_t= (1-|D|^2/t)^\delta_+\), where \(m(D)\) denotes the multiplier operator \(\{m(D)f\}^{\widehat{}}(\xi)= m(\xi) \widehat f(\xi)\). A basic question of \(S^\delta_t\) is to find pairs \((\delta, 1/p)\) in the rectangle \(R= \{(\delta, 1/p):0\leq \delta\leq 1/2\), \(0\leq 1/p< 1\}\) for which \(S^\delta_t f\) converges to \(f\) a.e., for arbitrary \(L^p\) functions \(f\). This question is equivalent to establish a weak \((p,p)\) estimate for the maximal operator \(\sup_{t> 0}|S^\delta_t f(x)|\) for test functions \(f\). After all the positive and negative results, known so far, for the pairs in \(R\), there is still an open problem unsolved in the triangle with vertices \((1/2,1)\), \((0,1/2)\) and \((0,2/3)\). In this paper, the author achieves a very remarkable step to obtain the positive answer in the triangle with vertices \((1/2,1)\), \((0,1/2)\) and \((3/20,2/3)\). More precisely, the author obtains the followingNEWLINENEWLINENEWLINETheorem 1.2. Suppose that \(1< p< 2\) and \(\delta> \max\{{3\over 4p}-{3\over 8}, {7\over 6p}-{2\over 3}\}\).NEWLINENEWLINENEWLINEThen one has the almost everywhere convergence of the Bochner-Riesz means \(S^\delta_t f(x)\) for all \(f\in L^p\). The proof is based on a mixed norm estimate for the oscillatory integral \(Sf(x,t)= \int e^{2\pi i\lambda t|x-y|}a(x,y,t)f(y) dy\), where \(a(x,y,t)\) is any smooth function supported on the region \(t\), \(|x-y|\sim 1\), \(|x|,|y|\lesssim 1\).