Gâteaux or Fréchet differentiable norms in duals of complex interpolation spaces

Gâteaux or Fréchet differentiable norms on duals of interpolation spaces

“…Hence, by (iii) we have with $$r = p^{\prime}$$ that, isomorphically, $$$$\begin{equation} (B_\alpha ^*, B_\beta ^*)_{\eta, r} = ((B_\alpha, B_\beta)_{\eta, r^{\prime}})^* = (B_{\theta, p})^* = (B_{\theta, r^{\prime}})^*. \end{equation}$$$$On the other hand, by [3, Theorem 4.4], the spaces $$B_\alpha ^*$$ and $$B_\beta ^*$$ are Gâteaux differentiable when one of $$B_0^*$$ or $$B_1^*$$ is Gâteaux differentiable. Hence, by Lemma 7, the dual $$((B_\alpha, B_\beta)_{\eta, r^{\prime}}, \beta _{(B_\alpha, B_\beta)_{\eta, r^{\prime}}})^*$$ is Gâteaux differentiable.…”

“…Proof of Theorem One has just to follow the proof of Theorem 1, replacing the result of [3, Theorem 4.4] about Gâteaux differentiability by the one about Fréchet differentiability [3, Theorem 5.3], and replacing Lemma 7 by Lemma 11.$$\Box$$…”

“…If we assume only that one of $$B_0^*$$ or $$B_1^*$$ is Gâteaux differentiable (resp., Fréchet differentiable), we are able to prove in Theorem 1 that for every $$p \in (1, +\infty)$$, there exists an equivalent Gâteaux differentiable dual norm on $$((B_0, B_1)_{\theta, p})^*$$ (resp., an equivalent Fréchet differentiable dual norm, see Theorem 2). The proof goes via results obtained in [3] for the spaces constructed by complex interpolation, and via the reiteration theorem [2, Theorem 4.7.2]. This allows us to bring back the problem to the situation of Lemma 7 or 11, but for some equivalent norm on the space $$(B_0, B_1)_{\theta, p}$$.…”

“…Hence, by (iii) we have with 𝑟 = 𝑝 ′ that, isomorphically, 𝜂,𝑟 = ((𝐵 𝛼 , 𝐵 𝛽 ) 𝜂,𝑟 ′ ) * = (𝐵 𝜃,𝑝 ) * = (𝐵 𝜃,𝑟 ′ ) * . (3.3)On the other hand, by[3, Theorem 4.4], the spaces 𝐵 * 𝛼 and 𝐵 * 𝛽 are Gâteaux differentiable when one of 𝐵 * 0 or 𝐵 * 1 is Gâteaux differentiable. Hence, by Lemma 7, the dual ((𝐵 𝛼 , 𝐵 𝛽 ) 𝜂,𝑟 ′ , 𝛽 (𝐵 𝛼 ,𝐵 𝛽 ) 𝜂,𝑟 ′ ) * is Gâteaux differentiable.…”