Abstract
For the frame i in n, let B2(i)(x) (x 2 n) be a family of all n-dimensional rectangles containing x and having edges parallel to the straight lines of i, and let MB2(i) be a maximal operator corresponding to B2(i). The main result of the paper is the following Theorem. For any function f 2 L(1 + ln + L)( n ) (n i 2) there exists a measure preserving and invertible mapping ! : n ! n such that 1: fx : !(x) 6 xg o suppf; 2: sup i2i( n) R fMB2(i)(fe!)>1g MB2(i)(f e !) < 1. This theorem gives a general solution of M.de Guzman's problem that was previously studied by various authors.
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