Transplanting maximal inequalities between Laguerre and Hankel multipliers (Q1924475)

Suppose that \(m\) is a function on \(R^+\). For a function \(f\) on \(R^+\), with a Laguerre expansion \(\sum^\infty_{n=0}c_n\phi^a_n\), define \(L^*_mf(x)= \sup_{\varepsilon>0}|\sum^\infty_{n=0}c_nm (\varepsilon n^{1/2})\phi^a_n(x)|\). Similarly, define the maximal function \(H^*_nf(x)= \sup_{\varepsilon>0}|[m(\varepsilon\cdot) \widehat{f}(\infty) ]^\vee|\), where the Hankel transform and its inverse are denoted by \({}^\wedge\) and \({}^\vee\), respectively. The paper under review modifies techniques of \textit{Y. Kanjin} [Proc. Am. Math. Soc. 103, No. 4, 1063-1069 (1988; Zbl 0671.42016)] to show that, if \(L^*_m\) is bounded on \(L^p(R^+,x^{2a+1}dx)\), then so is \(H^*_m\). Some extensions and consequences of this result are described.