BOUNDEDNESS CRITERIA FOR LINEAR AND MULTILINEAR FRACTIONAL INTEGRAL OPERATORS IN LORENTZ SPACES

In this note we give necessary and sufficient condition on a measure mu guaranteeing the boundedness of the multilinear fractional integral operator T gamma m ,mu defined with respect to mu from the product of Lorentz spaces Pi(m)(k =1) L (R) k(,s)k(mu, X)(R) k(,s)k (mu, X ) to the Lorentz space L-p,L-q(mu, X-p,X- q (mu, X ). The result is new even for linear fractional integrals (gamma ,mu) (i.e., when m = 1). From the general results we have a criterion for the validity of Sobolev-type inequality in Lorentz spaces defined for non-doubling measures.