By Bor-Luh Lin

This quantity includes the lawsuits from a learn Workshop on Banach house concept held on the college of Iowa in Iowa urban in July 1987. The workshop supplied contributors with a collaborative operating surroundings during which principles will be exchanged informally. a number of papers have been initiated throughout the workshop and are awarded the following of their ultimate shape. additionally incorporated are contributions from numerous specialists who have been not able to wait the workshop. not one of the papers may be released somewhere else. through the workshop, hours on a daily basis have been dedicated to seminars on present difficulties in such components as vulnerable Hilbert areas, zonoids, analytic martingales, and operator concept, and those subject matters are mirrored in many of the papers within the assortment

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**Additional info for Banach Space Theory: Proceedings of a Research Workshop Held July 5-25, 1987 With Support from the National Science Foundation**

**Sample text**

M are easy to estimate: < 1. 18) and hence the proof We note in passing that the calculations above, together with the dominated convergence theorem and T1 = 0, yield the following integral representation: LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE < Trp,'I/J > = p +q = 45 lim p 1 u + q 0'-tO I = AK(x,y) bo(Y)[rp(y)- rp(x})- xl(y)rp(x}} '1/J(x) dp(y)dp(x). Next we prove the following lemma which has a somewhat stronger conclusion than one of the results in [FHJW]. 20) Suppose that T: C~(x) ...

16), we have p(zj-l'zj) 5 -s. C 2 J for ~+1 5 j 5 m+l. - On the other hand, if E 1E2 ... E~_ 1 # 0, then arguing - -s. 47) h). ) h LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE 31 I(RN[l,k](x' ,y) - RN[l,k](x,y)) - [RN[l,k](x' ,y') - RN[l,k)(x,y' )] I for a, fJ > 0 and a + fJ = 1. 43)). 13) (iv). S. T. 17) with f- in place of f'. Our assertion follows from this. We now establish the strong weak boundedness property for the operator TN1. S. T. SAWYER 34 -k Set r=2 °. 53) -{l~l+ ... +llml)af -{lk1+'--k l+...

5), since we may assume '7 ~ 0. 12). m are easy to estimate: < 1. 18) and hence the proof We note in passing that the calculations above, together with the dominated convergence theorem and T1 = 0, yield the following integral representation: LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE < Trp,'I/J > = p +q = 45 lim p 1 u + q 0'-tO I = AK(x,y) bo(Y)[rp(y)- rp(x})- xl(y)rp(x}} '1/J(x) dp(y)dp(x). Next we prove the following lemma which has a somewhat stronger conclusion than one of the results in [FHJW].