Gagliardo–Nirenberg interpolation inequality for symmetric spaces on noncommutative torus

Abstract:

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 $0\leq l\leq k$ and if the Cesàro operator is bounded on $E$ and $F$, then \begin{multline*} \|\nabla ^lx\|_{G(\mathbb {T}_{\theta }^{d})} \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}_{\theta }^{d})}^{1-\frac {l}{k}}\|\nabla ^kx\|_{F(\mathbb {T}_{\theta }^{d})}^{\frac {l}{k}},\;\\ x\in W^{k,1}(\mathbb {T}_{\theta }^{d}), \end{multline*} 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.