By M.D. Guzman
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COVERING THEOREMS OF THE VITALI TYPE. The most classical covering theorem in differentiation theory is that of Vitali ~908], which has traditionally been the tool to obtain the Lebesgue diffe r- entiation theorem in R n. In its original form the theorem of Vitali refers to closed cubic intervals and the Lebesgue measure. Later on Lebesgue ~910] and others gave it a less rigid geometric form replacing cubes by other sets "regular" with respect to cubes, keeping always the restriction to the Lebesgue measure.
DIFFERENTIATION BASES AND THE MAXIMAL OPERATOR ASSOCIATED TO THEM. Here we present a generalization of the Hardy-Littlewood maximal operator that will be very useful in the problems on differentiation we shall consider. For each x e R n l e t ~ ( x ) be a collection of bounded measurable sets with positive measure containing x and such that there is at least a sequence { ~ } ~ ( x ) with ~(R u)~ + O. The whole c o l l e c t i o n ~ = __~n~(X) will be called a differentiation xeR basis. For example i f ~ l ( x ) is the collection of all open bounded cubic intervals containing x we obtain a b a s i s ~ I.
Assume that H i satisfies the following weak type inequality for each I > 0 and f i e Lloc(R hi) 9 9 Ifi (xi)l mi({xleR nl : H i fi(x I) > l}) ~ ) [ ~ i ( ~ ) d m i ( x i ) where m i means ni-dlmensional Lebesgue measure and $i is a strictly increasin$ continuous function from [0,~3 to [0,~] with $i(0) = 0. e. [B 1 x B 2 : B 1 e ~ l , B2 e ~ 2 } and let H be its maximal operator. < 4__/_f dm. (3) The theorem of Jessen-Marcinkiewicz-ZT~mund. When one applies the previous remark iteratively one obtains the following result: I__n_nR n one considers the b a s l s ~ s such that,S(x) fo___rx e Rn is the collection of all open bounded intervals containing x such that s of the n side lengths are 51 equal and the other n - s are arbitrary.