Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces

Abstract

Moreover, for α < N p , we provide an explicit description of the domain. Spectral properties of the operator L are also obtained.

Optimal kernel estimates for a Schrödinger type operator

“…Let k µ its associated heat kernel. We define Lyapunov function in the following way (see also [13], [14])…”

“…Recently elliptic operators with unbounded coefficients have been studied in several paper (see for example [12], [13], [14], [15], [10], [7], [4], [9], [8], [5], [6]). In [10] and [4] it is proved that A endowed with domain…”

“…This semigroup is also consistent, irreducible and ultracontractive. As regards the case β = 0 we refer to [7] and [12]. Due to the regularity of the coefficients of the operator A, the semigroup T (t) can be represented in the following integral form through a heat kernel k(t, x, y)…”

Regularity of solutions to Kolmogorov equations with perturbed drifts

“…Sufficient conditions for the boundedness of ̺ can be found in [20], [7], [12] along with some other bounds (see [9] for a survey). Other related questions have been studied in [5], [19], [22], [23], [24], [26], [27], for the infinitedimensional case see [17], [18], and [30].…”

Compactness of semigroups of explosive symmetric Markov processes

“…Let γ 1 , γ 2 > 0 such that γ 1 < d andγ 1 + γ 2 > d. Setting J γ1,γ2 (x) = R d dy |x − y| γ1 (1 + |y| γ2 ) x ∈ R d ,J γ1,γ2 is bounded on R d and there exist positive constants c 1 , c 2 , c 3 such that (4.9)J γ1,γ2 (x) |x| d−(γ1+γ2) , if γ 2 < d, c 2 (1 + |x|) −γ1 log |x| if γ 2 = d, c 3 (1 + |x|) −γ1 if γ 2 > dfor any x ∈ R d . See[10, Lemma 6.1] for the bounds (4.9). We denote by G(x, y) the Green function of X.…”