Necessity of differentiability and derivative Hölder regularity for local Lipschitz continuity of C_f: bv_1(E)→bv_p(E)

Determine whether, for p≥1 and general Banach spaces E (excluding the known case p=1 with E=ℝ), the conditions that f is Fréchet differentiable and its derivative f′ is Hölder continuous on bounded sets with exponent 1/p are necessary for the composition operator C_f: bv_1(E)→bv_p(E) to be Lipschitz continuous on bounded sets.

Background

Theorem 'thm:LH1_bv_1_bv_p' provides sufficient conditions for C_f: bv_1(E)→bv_p(E) to be Lipschitz on bounded sets: f must be Fréchet differentiable and f′ must be Hölder on bounded sets with exponent 1/p. In the special case E=ℝ, p=1, Theorem 'thm:LH1_bv_1_bv_p_ver2' shows these conditions are also necessary.

Beyond this special case, it is unknown whether these sufficient conditions are also necessary in general Banach spaces or for p>1.