A Complete Characterization of the Generalized Multilinear Sobolev Inequality in Grand Product Lebesgue Spaces Defined on Non-homogeneous Spaces

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator T gamma,mu(f)(x) = integral(m)(X) f1(y(1))...f(y(m)) /(rho(x,y(1))+center dot center dot center dot +rho(x,y(m)))m-gamma d mu((y) over right arrow) defined with respect to a measure mu from grand product space Pi(m)(j=1) L-pj),L-theta (X, mu) to grand Lebesgue space L-q),L-theta q/p (X, mu) is established, where (X,rho, mu) is a quasi-metric measure space, 1/p = Sigma(m)(j=1) 1/pj, p < q and theta > 0. In particular, we show that for that boundedness, the necessary and sufficient condition on mu is that it is upper Ahlfors beta(m)-regular for certain number beta(m). To establish this result, first we derive the boundedness of T-gamma,T-mu between more general grand spaces: T-gamma,T-mu : Pi(m)(j=1) L-pj),L-/psi(center dot)(X, mu) (sic) L-q),L-Phi(center dot)(X, mu). As a corollary, we have a complete characterization of the multilinear Sobolev inequality (i.e., when q is the Sobolev exponent of p) in these spaces. In this case criterion on mu is that it is upper Ahlfors 1- regular. Some weighted inequalities for the bilinear fractional integral operator B gamma(f, g)(x) = integral(n)(R) f(x-t)g(x+t)dt/ |t|n-gamma, 0 < gamma < n, in the classical Lebesgue spaces are also discussed.