The existence of unimodular forms on sequence spaces with small norms is crucial in a variety of problems in modern analysis. As a consequence of our results, we prove that the optimal solution f (n 1 , .

We refine a recent technique introduced by Pellegrino, Santos, Serrano and Teixeira and prove a quite general anisotropic regularity principle in sequence spaces. As applications we generalize previous results of several authors regarding Hardy-Littlewood inequalities for multilinear forms. Littlewood inequalities. In this paper we follow this vein, exploring the ideas from the Regularity Principle proven by Pellegrino et al. [21] and providing a couple of applications.The paper is organized as follows. In Section 2, borrowing ideas from [21] we prove an anisotropic inclusion theorem for summing operators that will be useful along the paper. The techniques and arguments explored in Section 2 paves the way to the statement of a kind of anisotropic regularity principle for sequence spaces/series, in Section 3, completing results from [21]. In the final section the bulk of results are combined to prove new Hardy-Littlewood inequalities for multilinear operators.1.1. Summability of multilinear operators. The theory of multiple summing multilinear mappings was introduced in [18, 23]; this class is certainly one of the most useful and fruitful multilinear generalizations of the concept of absolutely summing linear operators, with important connections with the Bohnenblust-Hille and Hardy-Littlewood inequalities and its applications in applied sciences. For recent results on absolutely summing operators and these classical inequalities we refer to [7,11,17] and the references therein. The Hardy-Littlewood inequalities have its starting point in 1930 with Littlewood's 4/3 inequality [16]. In 1934 Hardy and Littlewood extended Littlewood's 4/3 inequality [15] to more general sequence spaces. Both results are for bilinear forms. In 1981 Praciano-Pereira [24] extended these results to the m-linear setting and recently various authors have revisited this subject. In 2016 Dimant and Sevilla-Peris [13] proved the following inequality: for all positive integers m and all m < p ≤ 2m we have (1.1)   ∞ j1,··· ,jm=1|T (e j1 , · · · , e jm )| p p−m   p−m p ≤ √ 2 m−1T for all continuous m-linear forms T : ℓ p × · · · × ℓ p → K. It is also proved that the exponent p p−m cannot be improved. However, this optimality seems to be just apparent, as remarked in some previous works (see [4,11]). Following these lines, the exponent can be potentially improved in the anisotropic viewpoint. In order to do that, the theory of summing operators shall play a fundamental role.Along the years, somewhat puzzling inclusion results for multilinear summing operators were obtained [8,9,23]. In this note we prove an inclusion result for multiple summing operators generalizing recent approaches of 2010 Mathematics Subject Classification. 46G25, 47H60.