Sparse bounds for the bilinear spherical maximal function

We derive sparse bounds for the bilinear spherical maximal function in any dimension d ⩾ 1 $d\geqslant 1$ . When 2 2$ , this immediately recovers sharp L p × q → r $L^p\times L^q\rightarrow L^r$ bound of operator and implies quantitative weighted norm inequalities with respect to Muckenhoupt weights, which seems be first their kind operator. The key innovation is a group newly developed continuity $L^p$ improving estimates single-scale averaging