Hardy Spaces for Semigroups with Gaussian Bounds

Let Tt = e−tL be a semigroup of self-adjoint linear operators acting on L(X,μ), where (X, d, μ) is a space of homogeneous type. We assume that Tt has an integral kernel Tt(x, y) which satisfies the upper and lower Gaussian bounds: C1 μ(B(x, √ t)) exp ( −c1d(x, y)/t ) ≤ Tt(x, y) ≤ C2 μ(B(x, √ t)) exp ( −c2d(x, y)/t ) . By definition, f belongs to H L if ‖f‖H1 L = ‖ supt>0 |Ttf(x)|‖L1(X,μ) < ∞. We prove that there is a function ω(x), 0 < c ≤ ω(x) ≤ C, such that H L admits an atomic decomposition with atoms satisfying: supp a ⊂ B, ‖a‖L∞ ≤ μ(B)−1, and the weighted cancellation condition