A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

We prove some weighted $L^p{\mathrm{\ell }}^p$-decoupling estimates when $p=2n\mathrm{/}(n-1)$. As an application, we give a result beyond the real interpolation exponents for the maximal Bochner-Riesz operator in ${\mathbb{R}}^3$. We also make an improvement in the planar case.