: The paper considers heat kernels of second--order parabolic equations in R N , with constant uniform diffusion and advective coefficients bounded in the maximum norm. Two critical cases, corresponding to upper and lower solutions, are identified, and explicit solutions are constructed for them in terms of the error function. They are shown to bound above and below all other heat kernels satisfying the same constraints on their advective coefficients, using a method of proof which relates two heat kernels together in a way which resembles the classical parametrix construction. Sharp bounds on the corresponding parabolic solution operators in L 1 (R N ) are obtained as a consequence. Keywords and phrases: Heat kernels, non--selfadjoint operators. AMS(MOS) Subject classification: 35B45, 35K05 To appear: SIAM J. Math. Anal. 1 Introduction Upper and lower pointwise bounds on the heat kernels of second--order parabolic operators were first derived, for the general case of variab..