Four lectures on real Hp spaces by Shanzhen Lu
By Shanzhen Lu
This ebook introduces the genuine variable idea of HP areas in short and concentrates on its functions to varied facets of research fields. It involves 4 chapters. bankruptcy 1 introduces the fundamental idea of Fefferman-Stein on genuine HP areas. bankruptcy 2 describes the atomic decomposition conception and the molecular decomposition conception of genuine HP areas. additionally, the twin areas of actual HP areas, the interpolation of operators in HP areas, and the interpolation of HP areas also are mentioned in bankruptcy 2. The houses of numerous uncomplicated operators in HP areas are mentioned in bankruptcy three intimately. between them, a few easy effects are contributed by means of chinese language mathematicians, reminiscent of the decomposition conception of vulnerable HP areas and its functions to the learn at the sharpness of singular integrals, a brand new technique to take care of the elliptic Riesz skill in HP areas, and the transference theorem of HP-multipliers and so forth. The final bankruptcy is dedicated to functions of genuine HP areas to approximation concept.
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1)tf+1» Note that £V £f = Afo*, and s u p p ^ can be written as j +1 C tlk+i C fi*. e. /(*) = Obviously, supp/i* C Bk. 6), it is easy to see that the equality k\{x)Q{x)dx --= 0 / I ' h (x)Q(x)dx = Q holds for any Q e Vs. < 2': C. Ul\ It remains to show that there exists a I £ BMO(R) such that Lf = f(x)l(x)dx = lim \^ m JR -°° fr[ / a,k(x)l(x)dx JR for any / € Ha'q,0(M). The proof of the above assertion is divided into the following three steps. (i) Let us first prove (Hl'q'°)' C (£/)', where J is an interval and Lq = {f&Lq(I):Jf(x)dx = 0}. In fact, when / € L\, it is easy to see that a(x)^\I\^l\\f\\llmHx) is a (l,q,0) atom. Thus, L* C Hl'9>°. And therefore, (i) holds. As a direct result of (i), it follows from the Hahn-Banach extension theorem and Riesz representation theorem that there exists a I € Lq (I), 1/q+l/q' = 1, such that Lf= [ f(x)l(x)dx,VfeLqj. 3 (i) in Chapter I, we obtain {$*+(a)(x)ydx < [ ) \am(x)\rdx JB nOO pap-l\{x = / Jo r\B\-1/p € B : a*m(x) > a}\dx \B\pap-rda