Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

Let E (T^d_), F (T^d_) be two symmetric operator spaces on noncommutative torus T^d_ corresponding to symmetric function spaces E, F on (0, 1). We obtain the Gagliardo--Nirenberg interpolation inequality with respect to T^d_: if G=E^1-l{k}F^l{k} with 0 l k and if the Ces\`aro operator is bounded on E and F, then align* \|ˡx\|₆ (ₓ^₃_{) } 2^3 2^{k-2-2} (k+1) ᵈ\|C\|₄ ₄^1-l{k}\|C\|₅ ₅^l{k}\|x\|₄ (ₓ^₃_{) }^1-l{k}\|ᵏx\|₅ (ₓ^₃_{) }^l{k}, \; x W^k, 1 (T^d_), align* where W^k, 1 (T^d_) is the Sobolev space on T^d_ of order k. Our method is different from the previous settings, which is of interest in its own right.