Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

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