Coordinatewise multiple summing operators in Banach spaces

We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust–Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P:ℓ∞→ℓp, as well as for the convergence theory of products of vector valued Dirichlet series.