Jd5-10, 1982)
.ibstract. In this paper we prore weighted norm estimates for vector valued integral operators with positive kernels. In addition weighted norm inequalities for certain general vector valued singular integral operators are obtained. Applications of these results include a generalized SOBOLEY Tlieorern for LIZORKIK-TRIEBEL spaces and estimates of various LITTLEWOOD-PALEIopera tors.
6 1. Introdtiction and Preliminaries. The fractional maximal function JI,,f, O;=y<?j i h defined for all locally integrable functions f on R" I)? (1.1) where x,:Rn and the supremum is taken over all wires &. centered at r of Lehergue measure Q . If y = 0, 2kl.r,f=Jl/ is the HARliT--~JITTLEWOOD niaximal function. The well-known theorein of HARDY and LIT.~ZEWOOT) [6] on the Lp 1)onndedness of f -..]I/ was generalized I);.. c'. F E W F X W ~X a i d E. 31. STEIK [ 5 ] to Z'-valued fuiirtionr and l)>-JIL-CKENHOX-PT [lo] and ASDERSBT anti JOHX [21 to weighted, respectiyely. weighted Z'-valued functions. KOI(IT,A~VII.I TO1 ohserved that it mas straightforward to proye a weighted L'-valued theorein which included the I)reviom results as special cases. He appLed tiis reiiultto tlie study of inultiplierh on weighted LIZORKIS-TRIEBEL spaces. In this paper we stud!-tlie fractioual inasimal fiiiiction X,,j for Lr-[ alued functionh (for y = 0. treated by KoKrrAg\rrr.l ) I'he ~nam observation following Tlieorem 2.1 is that for any nonnegative kernel a weighted scalar inequalit? implies ; i weighted vec*tor-ralued iiiecluality. ' h s result is applied specifically to results of MLTKESIIOTPT and J'HEEJ:IIE?; 11 r' j tor. the KIESZ potential to o1)tain a vector. valued form of SOBOLEV'S inecplity. Since the RIESZ potential dominates the fractional maximal function X:,. we oh1 ain corresponding estimates for this operator. The peneralized So1)ole.c. t heorein gives emheddings between LIZORKIK-(*ll;,f) ( 3 ) = sup -7 I'fir)'df -'"d *) This research w-as undertaken while this author attended the Special Year in Harmonic Analysis ;it the University of Marvland. College Park. The financial support bF the Universitv of Maryland and SSERC of Canada Grant So. -1 4837 is gratefully iwknowledged.