Pathological Complex Interpolation Involving the Space Wl,∞$W^{l,\infty }$

In 2024 it was shown that a pathological situation occurs in the complex interpolation between Wl,∞$W^{l,\infty }$ (with l≥0$l\ge 0$ integer) and other Sobolev spaces. Specifically, if d≥2$d\ge 2$ is an integer, p∈(1,∞)$p\in (1,\infty)$, and s>1/p$s>1/p$, then *${\ast }$ Hσ,ρ(Rd)¬↪(Hs,p(Rd),Wl,∞(Rd))θ,$$\begin{equation} H^{\sigma,\rho }(\mathbb {R}^{d})\not\hookrightarrow (H^{s,p}(\mathbb {R}^{d}),W^{l,\infty }(\mathbb {R}^{d}))_{\theta }\text{,} \end{equation}$$for any θ∈0,1$\theta \in \left(0,1\right)$, where σ:=1−θs+θl$\sigma:=\left(1-\theta \right) s+\theta l$ and ρ:=p/(1−θ)$\rho:=p/(1-\theta)$. In this paper we extend this result by showing that (*$\ast$) still holds under the weaker condition s>0$s>0$. While the case s>1/p$s>1/p$ can be handled by using traces on hyperplanes, the case s>0$s>0$ requires trace theory on fractal sets.