The behavior of the bounds of matrix-valued maximal inequality in \(\mathbb{R}^{n}\) for large \(n\) (Q470865)

Let \((X,d,\mu)\) be a metric measure space and \(\mathcal B(\ell^2)\) the matrix algebra of bounded operators on \(\ell^2\). For a locally integrable \(\mathcal B(\ell^2)\)-valued function \(f\), define \(f_r(x) =\mu(B(x, r))^{-1}\int_{B(x,r)}f(y)d\mu(y)\), where \(B(x, r) = \{y\in X : d(x, y)\leq r\}\).NEWLINENEWLINEIn this paper, the author studies the behavior of the bounds of matrix-valued maximal functions in \(\mathbb R^n\) for large \(n\). The main result is that the \(L^p\)-bounds \((p > 1)\) can be taken to be independent of \(n\), which is a generalization of Stein and Strömberg's result in the scalar-valued case. It is also shown that the weak type \((1, 1)\) bound has similar behavior as Stein and Strömberg's one.NEWLINENEWLINEHe points out that Stein and Strömberg's result in the scalar-valued case seems not to be available for matrix-valued functions since one cannot compare any two matrices or operators, which is a source of difficulties in the noncommutative analysis. But the author states several histories how the obstacle has been successfully overcome by the interaction with operator space theory, to go from the scalar-valued case to the matrix-valued or more generally to the operator-valued case.