Dimension of images of subspaces under mappings in Triebel‐Lizorkin spaces

Let m < α < p and let f : R n → R kbe a s , p ‐quasicontinuous representative of a mapping in the Triebel‐Lizorkin space F p , q s . We find an optimal value of β ( n , m , p , α , s ) such that for H β a.e. y ∈ ( 0 , 1 ) n − mthe Hausdorff dimension of f ( ( 0 , 1 ) m × { y } ) is at most α. We construct examples to show that the value of β is optimal and we show that it does not increase once p goes below the critical value α.