  We consider a class of constant-coefficient partial differential operators on
a finite-dimensional real vector space which exhibit a natural dilation
invariance. Typically, these operators are anisotropic, allowing for different
degrees in different directions. The heat kernels associated to these so-called
positive-homogeneous operators are seen to arise naturally as the limits of
convolution powers of complex-valued measures, just as the classical heat
kernel appears in the central limit theorem. Building on the
functional-analytic approach developed by E. B. Davies for higher-order
uniformly elliptic operators with measurable coefficients, we formulate a
general theory for (anisotropic) self-adjoint variable-coefficient operators,
each comparable to a positive-homogeneous operator, and study their associated
heat kernels. Specifically, under three abstract hypotheses, we show that the
heat kernels satisfy off-diagonal (Gaussian type) estimates involving the
Legendre-Fenchel transform of the operator's principle symbol. Our results
extend those of E. B. Davies and G. Barbatis and partially extend results of A.
F. M. ter Elst and D. Robinson.
