Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

Let $E({\mathbb{T}}_{\theta }^d),F({\mathbb{T}}_{\theta }^d)$ be two symmetric operator spaces on noncommutative torus ${\mathbb{T}}_{\theta }^d$ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo–Nirenberg interpolation inequality with respect to ${\mathbb{T}}_{\theta }^d$: if $G=E^{1-\frac{l}{k}}{F}^{\frac{l}{k}}$ with $0\le l\le k$ and if the Cesàro operator is bounded on $E$ and $F$, then \begin{align*} |\nabla^lx|{G($\mathbb{T}$^{d}θ)}\leq 2^{3\cdot 2^{k-2}-2}(k+1)^d|C|{E\to E}^{1-\frac{l}{k}}|C|{F\to F}^{\frac{l}{k}}|x|_{E($\mathbb{T}$^{d}θ)}^{1-\frac{l}{k}}|\nabla^kx|{F($\mathbb{T}$^{d}_θ)}^{\frac{l}{k}},; x\in W^{k,1}($\mathbb{T}$^{d}_θ), \end{align*} where $W^{k,1}({\mathbb{T}}_{\theta }^d)$ is the Sobolev space on ${\mathbb{T}}_{\theta }^d$ of order $k\in \mathbb{N}$. Our method is different from the previous settings, which is of interest in its own right.