Some remarks on the dyadic Rademacher maximal function (Q2839282)

For locally integrable Banach-valued functions on \({\mathbb R}^n\), we set NEWLINE\[NEWLINE Mf (x) := \sup \left\{ \left( {\mathbb E} \Big\| \sum_{Q \ni x} \varepsilon_Q \lambda_Q \langle f \rangle_Q \Big\|^2 \right)^{1/2} : \left( \sum_{Q} | \lambda_Q |^2 \right)^{1/2} \leq 1 \right\} , NEWLINE\]NEWLINE where \({\mathbb E}\) denotes the expectation for independent random variable \(\varepsilon_Q\) attaining values \(+1\) and \(-1\), each with probability \(1/2\), and \(\langle f \rangle_Q\) is the average of \(f\) over a dyadic cube \(Q\). We say that a Banach space \(X\) has the RMF property if NEWLINE\[NEWLINE \int_{\mathbb{R}^n} Mf(x)^p \, dx \leq C \int_{\mathbb{R}^n} \| f(x) \|^p \, dx NEWLINE\]NEWLINE where \(1<p<\infty\). \textit{T. Hytönen, A. McIntosh} and \textit{P. Portal} [J. Funct. Anal. 254, No. 3, 675--726 (2008; Zbl 1143.47013)] showed that this property is independent of \(p \in (0,\infty)\).NEWLINENEWLINEThe author considers extended characterizations of the RMF property.NEWLINENEWLINEA set \(S \subset X\) is said to be R-bounded if NEWLINE\[NEWLINE \left( {\mathbb E} \Big\| \sum_k \varepsilon_k \lambda_k \xi_k \Big\|^2 \right)^{1/2} \leq C \left( \sum_k | \lambda_k |^2 \right)^{1/2} NEWLINE\]NEWLINE for all finite collections of vectors \(\{ \xi_k \} \subset S\) and scalars \(\{ \lambda_k \}\). The smallest such \(C\) is the R-bound \(R(S)\). Recall that \(X\) is said to have type \(2\) if NEWLINE\[NEWLINE \left( {\mathbb E} \Big\| \sum_k \varepsilon_k \xi_k \Big\|^2 \right)^{1/2} \leq C \left( \sum_k | \xi_k |^2 \right)^{1/2}. NEWLINE\]NEWLINE If \(X\) has type \(2\), then \(Mf(x) \) is controlled pointwise by the standard dyadic maximal function \( \text{M}f (x) := \sup_{Q \ni x} \| \langle f \rangle_Q \|. \)NEWLINENEWLINEThe author proves the following: the next conditions are equivalent for any Banach space \(X\):NEWLINENEWLINE(1)\quad \(M: L^p(w;X) \to L^p(w)\) for all \(p \in (0,\infty)\) and any \(w \in A_p\),NEWLINENEWLINE(2)\quad \(M: L^p(X) \to L^p\) for some \(p \in (0,\infty)\),NEWLINENEWLINE(3)\quad \(M: L^1(X) \to L^{1, \infty}\),NEWLINENEWLINE(4)\quad \(M: H^1(X) \to L^{1, \infty}\).