By Y. S. Han, Eric T. Sawyer

During this paintings, Han and Sawyer expand Littlewood-Paley conception, Besov areas, and Triebel-Lizorkin areas to the final environment of an area of homogeneous sort. For this objective, they determine an appropriate analogue of the Calderón reproducing formulation and use it to increase classical effects on atomic decomposition, interpolation, and T1 and Tb theorems. a few new ends up in the classical surroundings also are got: atomic decompositions with vanishing b-moment, and Littlewood-Paley characterizations of Besov and Triebel-Lizorkin areas with basically part the standard smoothness and cancellation stipulations at the approximate id.

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**Additional info for Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces (Memoirs of the American Mathematical Society)**

**Sample text**

Now let {yj}jel e X be a maximal collection of points satisfying for all k. 17) By the maximality of {yj} jel' we have that for each x E X there exists a point Yj such that p(x,yj) S o-. >, = 0[~J and "ijj(y) = [ ~ fli(y) ]-1 flj(y). To see that "ijj is well defined, it suffices to show that for any y e X, there are only finite many '7j with '7j(y) # 0. This follows from the following fact: "'j(y) # 0 if and only if p(y,yj) S 2o- and hence this implies that B(yj,o) ~ B(y, 4Ao). 17) shows B(yj, B(yk, ~ h> = +.

54) which proves R; e SWBP with constant em 2-N 6m. For N so large that C 2-N 6 < 1, we obtain TN1 E SWBP. §3. ldgrOn - tyPe repmdudnr tormula on spaa:s of homnaeneoUI type To establish a Calderon reproducing formula on spaces of homogeneous type, we need to introduce a suitable class of distributions on these spaces. 1) Fix two exponents 0 < {J ~ 1 and 7 > 0. J(x) = 0. This definition was first introduced in [M] for the case X = Rn. A function f is said to be a strong smooth molecule of type ({J,7) centred at Xo e X with width d > 0 iff satisfies the above conditions but with (ii) replaced by] (ii)' lf(x) -f(x')l ~ C (~~~(~'l, 0)]p(d + pt'Y x,~ ~ ~ (d+p(x~)).

S. T. 5) implies where M is the Hardy-Littlewood maximal function. Thus, by Minkowski's or Young's inequality again. 3). 3) follows in the same way. 36). ~replaced - - by Dk , where Dk is LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE 59 We now introduce the following norms which will define the Besov and TriebelLizorkin spaces on spaces of homogeneous type. 14). (((P,7)), with define II~IFa,q = ll{k~l (2kal Dk(f) Dq} 1/qllp for 1 < p, q < Ill. 14). 10) max{O,-a} < 7 < where E, f is as in (4.