Abstract. The best known upper estimates for the constants of the Hardy-Littlewood inequality for m-linear forms on ℓp spaces are of the form √ 2 m−1 . We present better estimates which depend on p and m. An interesting consequence is that if p ≥ m 2 then the constants have a subpolynomial growth as m tends to infinity.

The Hardy-Littlewood inequalities for m-linear forms on p spaces are stated for p > m. In this paper, among other results, we investigate similar results for 1 ≤ p ≤ m. Let K be R or C and m ≥ 2 be a positive integer. Our main results are the following sharp inequalities:T for all m-linear forms T : n p × · · · × n p → K and all positive integers n. (ii) If (r, p) ∈ [2, ∞) × (m, 2m], then n j 1 ,...,jm=1 |T (ej 1 , . . . , ej m )| r 1 r ≤ √ 2 m−1 n max p+mr−rp pr ,0T for all m-linear forms T : n p × · · · × n p → K and all positive integers n. Moreover, the exponents max{(2mr + 2mp − mpr − pr)/2pr, 0} in (i) and max{(p + mr − rp)/pr, 0} in (ii) are optimal. The cases (r, p) = (2m/ (m + 1) , ∞) and (r, p) = (2mp/ (mp + p − 2m) , p) for p ≥ 2m and (r, p) = (p/ (p − m) , p) for m < p < 2m recover the classical BohnenblustHille and Hardy-Littlewood inequalities.Mathematics Subject Classification. 32A22, 47H60.

Abstract. It is proved the existence of large algebraic structures -including large vector subspaces or infinitely generated free algebrasinside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

Given an integer m ≥ 2, the Hardy-Littlewood inequality (for real scalars) says that for all 2m ≤ p ≤ ∞, there exists a constant C R m,p ≥ 1 such that, for all continuous m-linear forms A : N p × · · · × N p → R and all positive integers N ,   N j 1 ,...,jm=1|A(e j 1 , ..., e jm )|The limiting case p = ∞ is the well-known Bohnenblust-Hille inequality; the behavior of the constants C R m,p is an open problem. In this note we provide nontrivial lower bounds for these constants.

The main result of the present paper is a new Inclusion Theorem for summing operators, that encompasses several recent similar results as particular cases. As applications, we improve estimates of certain Hardy-Littlewood inequalities for multilinear forms.2010 Mathematics Subject Classification. 46G25, 47H60.