Abstract
Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu ), where (X,d,mu ) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: C1μ(B(x,t))exp-c1d(x,y)2/t≤Tt(x,y)≤C2μ(B(x,t))exp-c2d(x,y)2/t.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{C_1}{\\mu (B(x,\\sqrt{t}))} \\exp \\left( {-\\,c_1d(x,y)^2/ t} \\right) \\le T_t(x,y)\\le \\frac{C_2}{\\mu (B(x,\\sqrt{t}))} \\exp \\left( {-\\,c_2 d(x,y)^2/ t} \\right) . \\end{aligned}$$\\end{document}By definition, f belongs to H^1(L) if Vert fVert _{{H^1(L)}}=Vert sup _{t>0}|T_tf(x)|Vert _{L^1(X,mu )} <infty . We prove that there is a function omega (x), 0<cle omega (x)le C, such that H^1(L) admits an atomic decomposition with atoms satisfying: {mathrm{supp}}, asubset B, Vert aVert _{L^infty } le mu (B)^{-1}, and the weighted cancelation condition int a(x)omega (x)hbox {d}mu (x)=0.
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