Weighted HARDY Spaces
By BUI Huu QUI of Hiroshima (Eingegangen am 5.8.1980)
The main purpose of this paper is to study the weighted version hz of the local HARDY space hp considered recently by GOLDBERC [ 113, where the weight w is assumed to satisfy the condition (A,) of MUCKENHOUPT. After giving preliminary results concerning weight functions and harmonic functions on the strip domain in Section 1, we investigate the general properties of hP, by adopting the arguments of GUNDY-WHEEDEN [ 1 2 ] and WHEEDEN [22] in Sections 2 and 3. It is also our intention to obtain results as close as possible to the non-weighted case. To achieve this aim, in the later part of the paper we restrict ourselves to weights in the class A,. IKth this restriction on the weight w , satisfactory results are obtained in the case p = 1 including the characterization of hf, in terms of the n RIESZ transforms I ' ~. . . . , r,, the FEFFERMAN-STEIN characterization ([8]) and the atomic decomposition ([4], [6], [13]) for both Hi and h:. These results are given in Sections 3,4 and 5 , respectively. As a consequence, the connection between H: and h:, and the representation of the dual of h i as the space bmo, are derived.
The author acknowledges helpful correspondences with Professor H. TRIEBEL m t l Professor R. L. WHEEDEN during the preparation of this paper.
1. Preliminaries
\Ye use S to denote the strip domain R n X ] O , 1[. An element of X is denoted by ( x . f). where x=(xl, . . . , x,)β¬ Rn and O -= t -= l . The POISSON kernel for S, denoted 1)y P , is defined as follows. Let Po(., t ) (O-=t-=2) be the function i n S(R77=S, the SCHWARTZ class of rapidly decreasing functions, given by bO(5, t ) = sinh { (1 -t ) 2 n
If:]/sinh (232 /El), where f is the FOURIER transform of fCLI(R") and is given by f ( 5 ) =Je-')nipi/(x) dx, 5~ Rn .
(Here, as usual, the integral is extended over all of R" unless otherwise indicated and xt = xiti + . . . +x&,.) Define P(z, t ) =PO(%, t ) + Pi(s, t ) for (z, t ) β¬8, where f"(z, t)=Po(x, 1 -t ) . Then P(*, t ) is in S and f'(E, t)=cosh ( ( 1 -2 t ) nl5l)lcosh {z ,El}. Furthermore, we can easily obtain the following integral representation
* f ( , + P ' ( . ,
- ; / L 0.r 8. P r o o f . Since (u(., t ) } O < t < l is bounded in LP,, there exist a sequence (tk} tending t o 0 and β¬unctions f o , f , in 1,; such that u ( * , tk) + f o and u ( -, 1 -tk) + f l in the weak*topology of LP, as tk--O, which means that and as t,-O for any go, gl in Lz'. Let (z, b ) c S , O<S-=1/2, be Eixed. Define g,(y)= = Pt ( x -y , 6) w ( y ) -l for i = 0, 1 and yc R". Then i t follows from (P.4) and Lemma 1.2 that qiEL:', i = O , 1. Therefore, we obtain 8 ' J U(Y, t k ) .4dY) 4 Y ) '/y-*J fu(Y) So(?/) 4 Y ) J'U ( Y 7 1 -td gl(Y) U*(Y) dY-Jl,(Y) Bl(Y) wl) dY J u ( y , t k ) P O (z-y, 6) dy $-J u (y, 1 -t k ) PI (x -y, 6) dy -J Po ( X -% 6) M Y ) dY+J PI (Z-Y, 6 ) f * ( Y ) d?.l.