“…In this context, Gromov [7] got a vanishing result for L 2 -Dolbeault cohomology, that is, he proved that if (M, J, g, ω) is a complete Kähler manifold of complex dimension n such that ω is d-bounded, i.e., ω = dη, with η bounded, then H p,q dR (M ) = 0 for p + q = n. In his proof, he made use of the L 2 -Hodge decomposition theorem, and showed the vanishing of L 2 de Rham harmonic forms. In [16] and [17] (see also [10] for the almost Kähler setting), the authors of the present note extend Gromov's result for W 1,2 Bott-Chern harmonic forms, giving a characterization of W 1,2 Bott-Chern harmonic forms on Stein d-bounded manifolds, respectively on complete Hermitian manifolds. Along the same line, we prove the following result (see Theorem 3.3), which implies Theorem 1.1.…”