An invariant version of the little Grothendieck theorem for Sobolev spaces (Q2054295)

An important theorem of Grothendieck states that any bounded linear operator \(T: L_1\to L_2\) is absolutely summing. A weaker (and simpler) version of Grothendieck's theorem claims that such operators are 2-absolutely summing. This weaker statement is equivalent to the property that every operator between Hilbert spaces which factorizes through \(L_1\) is a Hilbert-Schmidt operator. As observed by Kislyakov, the space \(L_1\) cannot be replaced by the Sobolev space \(W^1_1\). Indeed, the classical embedding operator of \(W^1_ 1 (\mathbb{T}^2)\) to \(L_2(\mathbb{T}^2)\) is not 2-abolutely summing. However, it is \((p, 1)\)-summing for every \(p > 1\). This suggests the following conjecture: \textit{Any operator between Hilbert spaces which factorizes through the Sobolev space \(W^1_1\) belongs to some non-trivial Schatten class}. In the paper under review, assuming some additional structure, the authors prove such a conjecture. Precisely, the authors prove that every Hilbert space operator which factorizes invariantly through Sobolev space \(W^1_1 (\mathbb{T}^d)\) belongs to some non-trivial Schatten class.