Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

Abstract

We prove heat kernel bounds for the operator (1 + |x| α )∆ in R N , through Nash inequalities and weighted Hardy inequalities.Mathematics subject classification (2000): 47D07, 35B50, 35J25, 35J70.

Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponents

“…Observe that we have taken ϕ(x) = (1 + |x| α ) γ α with γ = α(2−N ) 4 and since 2 < α < α 0 we have γ > γ 1 and then ϕ is a Lypaunov function. In any case we have obtained (13) from which follows that…”

“…In [7] the operator (1 + |x| α ) has been considered for α < 2. In [13] kernel estimates for the same operator have been proved for an arbitrary α. In [15] has been proved that the more general elliptic operator (1 + |x| 2 ) α 2 N i, j=1 a i j (x)D i j for α < 2 generates an analytic semigroup in L p (R N ) when the diffusion coefficients a i j admit a limit at infinity.…”

“…The arguments used are in [13,14]. For the convenience of the reader we report the fundamental statement.…”

“…In the next Theorem we need to use the following weighted Sobolev inequality. For the proof we refer to [13,Proposition 3.5] and the reference therein.…”

“…In particular taking β ≥ 0 we have α ≥ 2. The case α = 2 has been discussed in [13] so it remains to consider only the case α > 2.…”