Let M be a compact m-dimensional Riemannian manifold. Given a n × n symmetric

matrix A = (ai,j) with ai,j ≥ 0, ∀i, j, we give optimal conditions on the vector M =

(M1, . . . , Mn) ∈ R

n

+ which ensure boundedness from below of the functional

Ψ(ρ) =

Xn

i=1

Z

M

ρi

ln ρi +

Xn

i,j=1

ai,j Z

M

Z

M

ρi(x)ln d(x, y)ρj(y) dx dy

over

ΓM =



(ρ1, . . . , ρn) ∈ (LlnL(M, R+))n

,

Z

M

ρi = Mi

, ∀i

.

This result generalizes the logarithmic Hardy-Littlewood-Sobolev inequality of Beckner to

the systems case. In some cases we also address the question of existence of minimizers.

This is a joint work with Gershon Wolansky.