§1. THE ATOMIC DECOMPOSITION OF THE TRIEBEL—LIZORKIN SPACES In this chapter, we collect some basic facts about Triebel—Lizorkin spaces. In particular, we describe the atomic decomposition of these spaces developed by M. Frazier and B. Jawerth in their work about the (^-transform in [Fr—Jl], [Fr—J2] and [Fr—J3]. We will closely follow their notation and we refer the reader to these papers for further details. 1.1. The homogeneous Triebel—Lizorkin spaces Let if G & satisfy supp Ip c { £ G Kn: 1/2 | f | 2 } and | ^(£) | C 0 if 3/5 \i\ 5/3. Let ^ ( x ) = 2imp(2ux), v G Z. For v G I and k G Zn, let Q^k be the dyadic cube Q^k = {(xi,x2,...,xn) G (Rn : ki 2^xi ki + 1, i = l,...,n}. The "lower left corner" 2~~uk of Q = Q , will be denoted by x and the sidelength 2~v by £(Q). Set also ^ 0 (x) = | Q | ' ip (x-x ) if Q = Q , , so that the L norm of tpQ is independent of Q. For aGlR, 0 p o o , 0 q o o , the homogeneous Triebel—Lizorkin space F^' q is the collection of all f G & ' I & (tempered distributions modulo polynomials) such that "WllI^X"!)')1' L P 0 0 ' 11
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