Multifractal dimensions of product measures

We study the multifractal structure of product measures. For a Borel probability measure mu and q, t is an element of R, let H-mu(q,t) and P-mu(q,t) denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [011] Let mu be a Borel probability merasure on R(k) and let v be a Borel probability measure on R(l). Fix q, s, t is an element of R. We prove that there exists a number c > 0 such that integral H-mu(q,s) (H-y) dH(nu)(q,t) (y) less than or equal to cH(mu x nu)(q, s+t) (H), H-mu x nu(q, s+t) (E x F) less than or equal to cH(mu)(q,s) (E) P-nu(q,t) (F), integral H-mu(q,s) (H-y) dP(nu)(q,t) (y) less than or equal to cP(mu x nu)(q, s+t) (H), P-mu x nu(q, s+t) (E x F) less than or equal to cP(mu)(q, s) (E) P-nu(q, t) (F), for E subset of or equal to R(k), F subset of or equal to R(l) and H subset of or equal to H subset of or equal to R(k+l) provided that mu and nu satisfy the so-called Pederer condition. Using these inequalities we give upper and lower bounds for the multifractal spectrum of mu + nu in terms of the multifractal spectra of mu and nu.