By Mario Gonzalez

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Example text

I C X i and kgkL1 . i i ˛i / i . Proof. 2/ is lower semi-continuous. x/j Ä 2DC1 . i of (a) is a standard known fact. 6), together with this observation, we further obtain (b). Finally, from (a), we deduce that f supp X 1 !! RD n wi : i Observe that P i wi . 1. Then we have f 1 X i ! : wi L1 . / On the other hand, if (b) holds true, then we see that k holds true. 2. P i ˛i k . 5 Notes • The original theorem of Besicovitch deals with Euclidean balls in RD by Besicovitch [5] and with more abstract sets by Morse [98].

If Qy; k D fyg D Qy; k 1 or Qy; k D R , then 1 2 QO y; k D Qy; k trivially. Assume that Qy; k D fyg ¤ Qy; k 1 now. 2. t u For a fixed k, cubes of the k-th generation may have very different sizes for different y. Nevertheless, we still have some kind of regularity as follows. 3. Qy ; Ry / then Qx 10 1 ; Ry . As a consequence, for k 2 Z, 1 1 1 (a) if Qx; k \ Qy; k ¤ ;, then Qx; k 2 2 2 (b) if Qx; k \ Qy; k ¤ ;, then Qx; k (c) if Qx; k \ Qy; k ¤ ;, then Qx; k 1 O1 QO y; k and, in particular, x 2 Qy; k I 2 O2 QO y; k and, in particular, x 2 Qy; k I Qy; k 1 .

2. P i ˛i k . 5 Notes • The original theorem of Besicovitch deals with Euclidean balls in RD by Besicovitch [5] and with more abstract sets by Morse [98]. 1 was given by M. de Guzm´an [23, pp. 2–5]. • The maximal functions, M. / and M. / , were introduced by Tolsa [131]. Tolsa also showed that M. / and M. / are both bounded on Lp . 1; 1/ and from L1 . / to L1; 1 . /. When D 1, Journ´e [75, p. 1/ is not bounded from L1 . / to L1; 1 . /. 1; 1/ plays a key role here. 1; 1/, is bounded from L1 . / to L1; 1 .