On the boundedness of operators in Lp (lq) and Triebel-Lizorkin Spaces

Given a bounded linear operator T : L-po (R-n) -> L-p1 (R-n), for 1 <= p(0),p(1) <= infinity, we state conditions under which T defines a bounded operator between corresponding pairs of L-p (R-n; l(q)) spaces and Triebel-Lizorkin spaces F-p,q(s) (R-n). Applications are given to linear parabolic equations and to Schriodinger sernigroups.