Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

Let E(Tᵈθ),F(Tᵈθ) be two symmetric operator spaces on noncommutative torus Tᵈθ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo--Nirenberg interpolation inequality with respect to Tᵈθ: if G=E1-l/kFl/k with 0≤ l≤ k and if the Ces\`{a}ro operator is bounded on E and F, then {align*} \|∇^lx\|_{G(Tᵈθ)}≤ 2^{3· 2ᵏ⁻²-2}(k+1)^d\|C\|E→ E1-l/k\|C\|F→ Fl/k\|x\|_{E(Tᵈθ)}1-l/k\|∇^kx\|_{F(Tᵈθ)}l/k,\; x∈ Wk,1(Tᵈθ), {align*} where Wk,1(Tᵈθ) is the Sobolev space on Tᵈθ of order k. Our method is different from the previous settings, which is of interest in its own right.