Improved direct sum theorem in classical communication complexity

For a function f : X ×Y → Z , the m-fold direct sum is the function f : X × Y → Z, defined by f(〈x1, . . . , xm〉, 〈y1, . . . , ym〉) ∆ = 〈f(x1, y1), . . . , f(xm, ym)〉. We show the following direct sum theorem for classical communication protocols, R(f) = Ω( m k R(f)) where R(f) is the the k-round private coins communication complexity of f and R(f) is the k-round public coin complexity of f . In particular for constant round protocols it implies R(f) = Θ(mR(f)).