On the factorization theorem for the tensor product of integrable distributions

Abstract

In this paper, we applied the Factorization Theorem of W. Rudin and H. Cohen for obtaining a factorization of the tensor product of the space $\overline{ HK ( {\mathbb {R}} ) }$ with itself, where $\overline{ HK ( {\mathbb {R}} ) }$ is the completion of the space of the Henstock-Kurzweil integrable functions. This generalizes the results over the factorizations of $L^1 ({\mathbb {R}})$ and $\overline{ HK ( {\mathbb {R}} ) }$. In particular, we prove that for the Banach algebra $\overline{ \big ( HK({\mathbb {R}})\cap BV({\mathbb {R}}) \big ) \otimes _\gamma \big ( HK({\mathbb {R}})\cap BV({\mathbb {R}}) \big ) }$ contained in $\overline{ HK({\mathbb {R}}) \otimes _\gamma HK({\mathbb {R}}) }$, the Factorization Theorem does not hold. Similar results are therefore valid for the space $\overline{ {\mathcal {A}}_C ({\mathbb {R}}) \otimes _\gamma {\mathcal {A}}_C ({\mathbb {R}}) }$. Moreover, we build a new integral which is applied to extend some properties of the Fourier Transform on the classic space $L^{2} ({\mathbb {R}}^2 )$.