By G. N. Watson

An Unabridged, Digitally Enlarged Printing, to incorporate: The Tabulation Of Bessel features - Bibliography - Index Of Symbols - record Of Authors Quoted, And A finished Index

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**Example text**

3I{supu > A}I Yfi Yo for each positive function u defined on D and all A > O. In particular, an approach family is subordinate to 90. if and only if it is subordinate to 9{3. Proof. If (3 :S a the conclusion follows from (3) in the definition of an approach system, so we assume that a < (3. \Ve may assume that I{suPYO u > A} I is finite, for otherwise there is nothing to prove. Then let f E Ll(W) be the characteristic function of the set {SUPYO u > A}. ,a}. Yfi Assume then that SUPYfi u( wo) > A.

165]. M. Stein in 1961 [Ste61]. He proved that a weak type estimate for the maximal function associated to a sequence of operators is a necessary condition for the existence of pointwise limits for this sequence of operators. The version given below is not the more general one proved in lSte61], but it captures the main ingredients of the original statement. 19 Let {Tn}n be a sequence of bounded linear operators of V (bD) commuting with the rotations. 4. MAXIMAL OPERATORS AND CONVERGENCE 25 exists for almost every w E bD.

Therefore there is a constant e', depending only on Ao, k' and e", such that k' . lffi(wg(z) , Tg(Z)) C e' . lffi(w n , Tn). g (e' ·lffi( Wn , Tn)). 15 is thus completed. It is clear that (2) implies (3). 7 that (3) implies (4). Finally, we show that (4) implies (1). For a fixed ball B, consider the characteristic function XB of B. d. In the construction of approach families, it is useful to consider the following notion. 8) for W E W. Then £ is g-complete if £ = £g. An equivalent way to define the g-completion of £ is obtained by first observing that any approach family 9 for (D, W) extends as follows to a map where, for ( E D, we let Then £g(W) == U gt(().