Abstract. We construct the fundamental solution of ∂t − ∆y − q(t, y) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition.The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces.We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator. Main results and overview. Let d be a natural number. The Gausand we let g(s, x, t, y) = 0 if s ≥ t. Here x, y ∈ R d are arbitrary. It is well-known that g is a time-homogeneous transition density with respect to the Lebesgue measure, dz, on R d . In particular, for x, y ∈ R d , R d g(s, x, u, z)g(u, z, t, y) dz = g(s, x, t, y) if s < u < t.2000 Mathematics Subject Classification: 47A55, 60J35, 60J57.

Let D be a bounded open subset in R d , d 2, and let G denote the Green function for D with respect to (− ) /2 , 0 < 2, < d. If < 2, assume that D satisfies the interior corkscrew condition; if =2, i.e., if G is the classical Green function on D, assume-more restrictively-that D is a uniform domain. Let g = G(·, y 0 ) ∧ 1 for some y 0 ∈ D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that G(z, y)/g(z) C G(x, y)/g(x) when d (z, x) d(z, y). An intermediate step is the approximation G(x, y) ≈ |x − y| −d g(x)g(y)/g(A) 2 , where A is an arbitrary point in a certain set B(x, y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D ×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.

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Given a Green function G (e. g. with respect to (− ) α/2 , 0 < α ≤ 2) on a region X where G has a generalized triangle property and given a (G-bounded) signed measure µ on X, necessary and sufficient conditions are given for the existence of a perturbed Green function which is comparable with G. This is done in the general setting of measurable spaces. Applications to C 1,1 -regions in R d and, for the classical case α = 2, to finitely connected regions in R 2 are given.

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