Schauder Estimates for Equations Associated with Lévy Generators

Abstract

We study the local regularity of solutions f to the integro-differential equation Af = g in U associated with the infinitesimal generator A of a Lévy process (Xt) t≥0 . Under the assumption that the transition density of (Xt) t≥0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f . Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.1991 Mathematics Subject Classification. Primary 60G51; Secondary 45K05, 60J35.

A Liouville theorem for Lévy generators

“…Choquet and Deny [4] characterized the bounded solutions u to convolution equations of the form u = u * μ; these equations play a central role in the study of the "Laplace" equation Au = 0, see Lemma 1. Since the Liouville theorem is an assertion on the smoothness of harmonic functions, there is a close connection between the Liouville theorem and Schauder estimates; see [13,19] and the references therein for recent results. We would like to mention that there are also Liouville theorems in the half-space, see e.g.…”

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

“…This follows from Minkowski's integral inequality (8) where we use Lebesgue measure µ = λ and set v k (y) := f k (x − y)g(y) keeping x ∈ R n fixed. From this together with (11) we thus get…”

“…If p 1 = ∞, then, by ( 9), 1 + 1/p = 1/p 2 , which implies p = ∞ and p 2 = 1, i.e. we need to show that (11) and monotone convergence, we get…”

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

“…Intuitively, (X t ) t≥0 behaves locally like an isotropic stable Lévy process but its index of stability depends on the current position of the process. In view of the results in [25,27], it is an educated guess that any function f ∈ D(A) is "almost" locally Hölder continuous with Hölder exponent α(⋅), in the sense that…”

“…[2] studied operators with functional order of differentiability (ν(x, dy) = c(x, y) ( y d ϕ(y) dy) for "nice" ϕ). The recent article [25] establishes Schauder estimates for a large class of Lévy generators using gradient estimate for the transition density p t of the associated Lévy process. Moreover, we would like to mention the article [27] which studies a complementary question -namely, what are sufficient conditions for the existence of the limit (1) in the space C ∞ (R d ) of continuous functions vanishing at infinity -and which shows that certain Hölder space of variable order are contained in the domain of the (strong) infinitesimal generator.…”