On the halo conjecture and maximal convolution operators for locally compact groups

The halo conjecture in the theory of differentiation of integrals states that if the maximal operator MB corresponding to a differentiation basis B is of restricted weak type φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}, then the basis B differentiates the integrals of functions from the class φ(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi(L)$$\end{document}. Using the technique proposed by Antonov and, Sjölin and Soria, which is based on the approximation by simple functions with respect to the integral metrics generated by the truncated maximal convolution operators, for translation invariant bases in locally compact groups of a rather general type, the result has been established that gives an approximation to the conjecture for functions φ(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi(u)$$\end{document} close to u, while for the case of φ(u)=u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi(u)=u$$\end{document}, this implies the validity of the conjecture. The proved theorems generalize the corresponding results obtained by Sjölin and Soria, Moriyón, and Moon.