Unions of Lebesgue spaces and \(A_1\) majorants (Q906130)

In this paper, the authors show that the union of \(L^p_w (\mathbb R^ n)\) spaces with \(w\in A_p\) is equal to the union of all Banach function spaces for which the Hardy-Littlewood maximal function is bounded on the space itself and its associate space. The main results are: Theorem. Suppose \(1<p<\infty\), then \[ \bigcup_{w\in A_p} L^p_w(\mathbb R^n)=\mathcal M_{A_1}(\mathbb R^n)\cap\Big(\bigcup_{w\in A_1} L^1_w(\mathbb R^n)\Big), \] where \(\mathcal M_{A_1}\) is the set of all measurable functions possessing an \(A_1\) majorant, i.e., \(f \in \mathcal{M}_{A_1}\) if exists \(w \in A_1\) such that \[ |f|\leq w. \] Theorem. Suppose \(1<p<\infty\) then \[ \bigcup_{w\in A_p}L^p_w(\mathbb R^n)=\bigcup\{X:M\in \mathcal B(X) \cap \mathcal B(X')\}, \] where the second union is over all Banach function spaces such that the Hardy-Littlewood maximal operator is bounded on \(X\) and \(X'\). The paper ends with the following open problems: {\parindent=0.7cm \begin{itemize}\item[--] Let \(A_p^*=\bigcap_{q>p}A_q\). Is there a characterization of the union \[ \bigcup_{w\in A_p^*}L^p_w? \] In the local case we have \[ \bigcup_{w\in A_p^*}L^p_w(Q)\subset \bigcap_{s<p}\bigcup_{r>s}L^r(Q)=\limsup_{r\to p^-} L^r(Q). \] Are these two sets equal? \item[--] It is well known that \[ L^1\cap L^\infty \subset \bigcap_{1<p<\infty} L^p\subset \bigcup_{1<p<\infty} L^p \subset L^1+L^\infty. \] When can we write a function as the sum of a function in \(\mathcal M_{A_1}\) and \(\bigcup_{w\in A_1}L^1_w\)? That is, what conditions on a function guarantee it belongs to \(\mathcal M_{A_1}+\bigcup_{w\in A_1}L^1_w\)? \item[--] What can one say about \[ \bigcup_{w\in A_p} L^{p,\infty}_w? \] If \(w\in A_1\) and \(p>1\) then \(M\in \mathcal B(L^{p,\infty}_w)\), so for \(p>1\) \[ \bigcup_{w\in A_1} L^{p,\infty}_w\subset \mathcal M_{A_1}. \] \item[--] Do these results transfer to more general domains? It is possible to consider a general open set \(\Omega\) as our domain of interest. \end{itemize}}