Quadratic difference operators in Lp-spaces

Let (G, +, ∑,μ) be an abelian complete measurable group with μ(G) < +∞ and let E be a Banach space. For any f: G → E we define the quadratic difference operator Qf by Qf(x,y):= 2f(x) + 2f(y) - f(x + y) - f(x - y), x, y ∈ G. We will prove that if Qf ∈ L μ×μp(G × G, E) for a certain 1 ≤ p ≤ +∞, then there exists exactly one quadratic function K: G → E, i.e. K satisfying the functional equation 2f(x) + 2f(y) = f(x + y) + f(x - y), x,y ∈ G and a constant G such that ||f - K||p≤C||Qf|| p. Moreover, the operator Qf: LP(G, E) → L p(G × G, E) is linear, continuous and invertible.