An invariant variant of the little Grothendieck theorem for Sobolev spaces

TL;DR

This paper establishes that Hilbert space operators factoring invariantly through Sobolev spaces belong to specific Schatten classes, extending the Little Grothendieck theorem to Sobolev spaces.

Contribution

It introduces an invariant variant of the Little Grothendieck theorem applicable to Sobolev spaces, linking operator factorization to Schatten class membership.

Findings

Operators invariantly factorizing through Sobolev spaces are in Schatten classes.

Extension of Little Grothendieck theorem to Sobolev space context.

Provides new insights into operator theory on Sobolev spaces.

Abstract

We 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.