Regularity principle in sequence spaces and applications

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 Grothendiecks 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.