The invariant little Grothendieck conjecture for Sobolev spaces

Let $W^{1}_{1}$ denote the Sobolev space on the relevant torus, and let $T$ be an operator between Hilbert spaces that factorizes through $W^{1}_{1}$. Invariant little Grothendieck conjecture. Any operator between Hilbert spaces which factorizes through the Sobolev space $W^{1}_{1}$ belongs to some non-trivial Schatten class. This conjecture is motivated by the failure of the corresponding Hilbert–Schmidt conclusion for the Sobolev embedding from $W^{1}_{1}(\mathbb{T}^2)$ to $L_2(\mathbb{T}^2)$, while related summability results are known for every exponent greater than $1$; the paper proves the asserted conclusion in an invariant setting.