An inequality on weighted Orlicz spaces for a vector-valued extension of the Hardy-Littlewood maximal operator on \(S^n\) and \(P^n(\mathbb{R})\) (Q2770416)

Let \(X\) be a Banach space or a Banach lattice with norm \(\|\cdot\|\) satisfying the UMD property. Let \(W\) be a positive integrable function on the unit sphere \(S^n\) and \(L^p_X(W)\) be the space of \(X\)-valued functions \(f\) on \(S^n\) such that NEWLINE\[NEWLINE \|f\|_{L^p_X(W)}=\left(\int_{S^n} \|f(x)\|^p W(x) d\sigma(x)\right)^{1/p}<\infty.NEWLINE\]NEWLINE Let \(M\) be the Hardy-Littlewood maximal operator on \(S^n\) and let \(A_p(S^n)\) be the Muckenhoupt class of weights. Then, the authors prove that if \(W\in A_\infty(S^n)\), \(X\) is a UMD Banach space with a normalized unconditional basis \((e_j)_{j\geq 1}\) and \(\Phi\) is a convex function, then there exists a constant \(K\), depending only on \(X\), \(\Phi\) and \(W\) such that NEWLINE\[NEWLINE\int_{S^n}\Phi\left(\sup_{k\geq 1} \left\|\sum_{j=1}^k M(f_j)e_j \right\|\right)W d\sigma\leq K \int_{S^n}\Phi(M(\|f\|))W d\sigmaNEWLINE\]NEWLINE for all \(f=\sum_{j=1}^\infty f_j e_j\in L^1_X\). Moreover, if \(1<p<\infty\), \(W\in A_{p}(S^n)\) and \(f\in L^p_X(W)\), then \(\sum_{j=1}^\infty M(f_j) e_j\) converges in \(L^p_X(W)\) to a function \(\tilde M(f)\) and the operator \(\tilde M\) is bounded on \(L^p_X(W)\). To prove this result, the authors first construct some partitions of \(S^n\) which induce partitions of the real projective space \(P^n(\mathbb R)\), similar to the dyadic partitions of \({\mathbb R}^n\), and prove several properties of these partitions.