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In his celebrated article, Aronson established Gaussian bounds for thefundamental solution to the Cauchy problem governed by a second orderdivergence form operator with uniformly elliptic coefficients. We extendAronson's proof of upper heat kernel estimates to nonlocal operators whosejumping kernel satisfies a pointwise upper bound and whose energy form iscoercive. A detailed proof is given in the Euclidean space and extensions todoubling metric measure spaces are discussed.

Upper heat kernel estimates for nonlocal operators via Aronson's method