  We prove a nonlinear regularity principle in sequence spaces which produces
universal estimates for special series defined therein. Some consequences are
obtained and, in particular, we establish new inclusion theorems for multiple
summing operators. Of independent interest, we settle all Grothendieck's type
$(\ell_{1},\ell_{2})$ theorems for multilinear operators. We further employ the
new regularity principle to solve the classification problem concerning all
pairs of admissible exponents in the anisotropic Hardy--Littlewood inequality.
