Weak type inequality for generalized maximal operator in Orlicz space on a space of homogeneous type (Q2709960)

In this paper the author considers a general Hardy-Littlewood type maximal operator \(M_{\Omega,\Upsilon}\) in an Orlicz space defined on a space of homogeneous type. This operator generalizes the maximal operator \({\mathcal M}\), which controls the Poisson integral and it is given by NEWLINE\[NEWLINE{\mathcal M}f(x, t)= \sup_Q {1\over|Q|} \int_Q|f(y)|dy,\quad x\in \mathbb{R}^n,\;t\geq 0,NEWLINE\]NEWLINE where the supremum is taken over the cubes \(Q\) in \(\mathbb{R}^n\), containing \(x\) and having side length at least \(t\).NEWLINENEWLINENEWLINEThe boundedness of \({\mathcal M}\) in spaces of \(L_p\) type has been investigated by several authors ([\textit{L. Carleson}, Ann. Math. (2) 76, 547-559 (1962; Zbl 0112.29702)], [\textit{C. Fefferman} and \textit{E. M. Stein}, Am. J. Math. 93, 107-115 (1971; Zbl 0222.26019)], amongst others). Other authors have carried out similar studies concerning the Poisson integral operator in the setting of spaces of homogeneous type [\textit{J. Sueiro}, Trans. Am. Math. Soc. 298, 653-669 (1986; Zbl 0612.32007)] and of Orlicz spaces [\textit{J. Cheng}, Isr. J. Math. 81, No. 1-2, 193-202 (1993; Zbl 0801.47036)].NEWLINENEWLINENEWLINEThis motivates the introduction of a generalized Hardy-Littlewood maximal operator \({\mathcal M}_{\Omega,\Upsilon}\) on a space of homogeneous type and the investigation of the boundedness of this operator in Orlicz spaces.NEWLINENEWLINENEWLINEThe main result of the paper contains a necessary and sufficient condition for \({\mathcal M}_{\Omega,\Upsilon}\) to be of weak type \((\Phi_1,\Phi_2)\) with respect to weights \((w,s,u,v)\), where \(\Phi_1\), \(\Phi_2\) are \(N\)-functions.