Sharp Boundedness of Quasiconformal Composition Operators on Triebel-Lizorkin Type Spaces

For a given quasiconformal mapping f : R-n -> R-n, the authors show that the boundedness or the unboundedness of the composition operator C-f on Triebel-Lizorkin type spaces F-p,q(alpha,1/p-alpha/n) (R-n) or, more generally, Hajlasz-Triebel-Lizorkin type spaces M-p,q(alpha,1/p-alpha/n) (R-n) depends on the indexes alpha, p and the degenerate sets of the Jacobian J(f), but not on the index q. Actually, the following dual relation is proved to be sharp to obtain the boundedness of C-f on F-p,q(alpha,1/p-alpha/n) (R-n) and M-p,q(alpha,1/p-alpha/n) (R-n): [GRAPHIC] where alpha is the regularity index, p the integration index and dim(L)E (resp., dim(G)E) the local (resp., global) self-similar Minkowski dimension of the degenerate set E of J(f). This is completely different from the results for Sobolev, BMO, Besov and Triebel-Lizorkin spaces, and extends the recent result for Q-spaces. Finally, the authors show that, if n - 1 < alpha p < n, then a homeomorphism for which the composition operator is bounded on M-p,q(alpha,1/p-alpha/n) (R-n) must be quasiconformal.