On families between the Hardy-Littlewood and spherical maximal functions (Q824956)

\textit{E. M. Stein} [Proc. Natl. Acad. Sci. USA 73, 2174--2175 (1976; Zbl 0332.42018)] showed that the spherical maximal function \[ S(f)(x) = \sup_{t>0} \frac1{\omega_{n-1}} \int_{\mathbb S^{n-1}} |f(x-t\theta)|d\sigma_{n-1}(\theta) \] is bounded from \(L^p(\mathbb R^n)\) to \(L^p(\mathbb R^n)\) when \(p>\frac{n}{n-1}\) and \(n\ge 3\), and is unbounded when \(p\le \frac{n}{n-1}\) and \(n\ge 2\). When \(n=2\) the positive direction of this result was proved by \textit{J. Bourgain} [J. Anal. Math. 47, 69--85 (1986; Zbl 0626.42012)]. The boundedness of the operator \(S\) was obtained via the auxiliary family of operators \(\{S_\alpha\},\, 0\le \alpha <1\), defined by \[ S_\alpha(f)(x) = \sup_{t>0} \frac2{\omega_{n-1}B(\frac{n}2, 1-\alpha)} \int_{\mathbb B^n} |f(x-ty)|(1-|y|^2)^{-\alpha} dy, \] where \(B(x,y)\) is the Beta function. Another classical operator is the Hardy-Littlewood maximal function given by \[ M(f)(x) = \sup_{t>0} \frac1{v_n} \int_{\mathbb B^n} |f(x-ty)| dy, \] where \(v_n\) is the volume of \(\mathbb B^n\). In the paper the authors study multilinear versions of \(S, \, S_\alpha\) and of \(M\). Specifically, \[ M^m(f_1, \dots, f_m)(x) = \sup_{t>0} \frac1{v_{mn}}\int_{\mathbb B^{mn}} \prod_{i=1}^m|f_i(x-ty_i)|dy_1 \dots dy_m, \] with analogous definitions for \(S^m_\alpha(f_1, \dots, f_m)(x)\) and \(S^m(f_1,\dots,f_m)(x)\). The main result of the paper is the following. Theorem 2. Let \(0<\alpha<1\). Given \(f_i\in L^1_{loc}(\mathbb R^n)\) and \(x\in\mathbb R^n\) we have \begin{align*} &M^m(f_1, \dots, f_m)(x) \le S_\alpha^m(f_1,\dots,f_m)(x) \le S^m(f_1,\dots,f_m)(x),\\ &\lim_{\alpha \to 1^-} S_\alpha^m(f_1,\dots, f_m)(x) = S^m(f_1,\dots,f_m)(x),\\ &\lim_{\alpha\to 1^+ }S_\alpha^m(f_1,\dots,f_m)(x) = M^m(f_1, \dots, f_m)(x). \end{align*} These statements are valid even when some of the preceding expressions equal \(\infty\). Boundedness of the operators \(S_\alpha^m\) is considered in Theorem 3 where the authors prove the following. Let \(n\ge 2,\, 0\le\alpha <1\), and \(1< p_i\le \infty\). Define \(p\) by \(\sum_{i=1}^m \frac1{p_i} = \frac1{p}\). Then there is a constant \(C=C(m,\alpha, p_1,\dots,p_m)\) such that \[ \|S_\alpha^m(f_1,\dots,f_m)\|_{L^p(\mathbb R^n)} \le C\prod_{i=1}^m \|f_i\|_{L^{p_i}(\mathbb R^n)} \] for all \(f_i\in L^{p_i}(\mathbb R^n)\) if and only if \(\frac{n}{mn-\alpha}<p\le \infty\).