### Nonnegative solutions of parabolic operators with low-order terms.

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We prove a new 3G-Theorem for the Laplace Green function G on an arbitrary Jordan domain D in ℝ². This theorem extends the recent one proved on a Dini-smooth Jordan domain.

We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a ${C}^{1,1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

We consider the general Schrödinger operator $L=div\left(A\left(x\right){\nabla}_{x}\right)-\mu $ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function ${G}_{\Delta}$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K{\u2099}^{\infty}$ considered by Zhao and Pinchover. As an application we study the cone ${}_{L}\left(\mathbb{R}\u207f\u208a\right)$ of all positive L-solutions continuously vanishing...

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