Some necessary conditions for functional calculus in Besov-type spaces

This paper is devoted to the study of composition operators, denoted as Tf(g) := f o g, under the scope of Besov-type spaces. Considering a complex variable function f acting on Besov-type space Bs,tp,q(ℝn) via composition, we observe that when 0 < s ≤ n/q – nt or q > 1 and s = n/p – nt, f is globally-Lipschitz. Conversely, for s > n/q – nt or q = 1 and s = n/q – nt, f is locally-Lipschitz.