The Moser's Iterative Method for a Class of Ultraparabolic Equations

Abstract

We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions.

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

“…Thus, by using some potential estimate for the fundamental solution of K , we prove the needed bound for the L p loc norm of u. The proof of the Caccioppoli type inequality plainly extends to non-homogeneous groups, whereas the Sobolev inequalities used in [28] heavily rely on the homogeneity of the fundamental solution. The main results of this paper are some L p potential estimates for the convolution with the non-homogeneous fundamental solution Γ of K and with the derivatives ∂ x 1 Γ, .…”

“…In [28] we proved some pointwise estimate for the weak solutions to (1) by adapting a classical iterative method introduced by Moser [26,27] to the non-Euclidean framework of the homogeneous Lie groups. The main goal in the papers by Moser is a Harnack inequality.…”

“…The Moser's method is based on a combination of a Caccioppoli type estimate with the classical embedding Sobolev inequality. Due to the strong degeneracy of the KFP operators, we encountered in [28] a new difficulty: the natural extension of the Caccioppoli estimates gives an L 2 loc bound only of the first order derivatives ∂ x 1 u, . .…”

“…, ∂ x m 0 u of the solution u of (1), but it does not give any information on the other spatial directions. The main idea used in [28] is to prove a Sobolev type inequality only for the solutions to (1), by using a representation formula for the solution u in terms of the fundamental solution of the principal part K of L. More specifically, let u be a solution to (1), then…”

“…in the first integral and note that detE λ (−s) = e sλ 2 trB , by (26) and (28). On the other hand, in the second integral we use the change of variable Ψ (y, s) = (y+E λ (s)ξ, τ+s), and note that detJ Ψ (w) = 1.…”