Factoring the Sobolev embedding operator

The paper studies the factorization and summing properties of the Sobolev embedding operator. We propose two different approaches. One shows that the Sobolev embedding operator $S:W^{1,1}({\mathbb{T}}^2)\hookrightarrow L_2({\mathbb{T}}^2)$ factorises through the identical embedding ${\mathrm{\ell }}_{\mathrm{\Phi }}\hookrightarrow {\mathrm{\ell }}_2$ for some Young function with Matuszewska-Orlicz index 1. Proof of this fact is based on two results of independent interest. First, a necessary and sufficient conditions on a Young function $\mathrm{\Phi }$ and weight $\mathrm{\Psi }$ for boundedness of the embedding of the Sobolev space $W^{1,1}({\mathbb{T}}^2)$ into Besov-Orlicz space $B_{\mathrm{\Phi },1}^{\mathrm{\Psi }}({\mathbb{T}}^2)$. Second, a generalization of the Marcinkiewicz sampling theorem to the context of Orlicz spaces. Another approach is based on the extrapolation of $(p,1)$-summing norm.