The lattice bump multiplier problem
We study the lattice bump multiplier problem. Precisely, given a smooth bump supported in a ball centered at the origin, we consider the multiplier formed by adding the translations of this bump centered at $N$ distinct lattice points. We investigate the dependence on $N$ of the $L^p$ norm of the linear and bilinear operators associated with this multiplier. We obtain sharp dependence on $N$ in the linear case and in the bilinear case when $p \gt 1$.