Schauder Estimates for Nonlocal Equations with Singular Lévy Measures

In this paper, we establish Schauder’s estimates for the following non-local equations in R d : ∂ t u = L κ , σ ( α ) u + b · ∇ u + f , u ( 0 ) = 0 , where α ∈ ( 1 / 2 , 2 ) and b : R + × R d → R is an unbounded local β -order Hölder function in x uniformly in t , and L κ , σ ( α ) is a non-local α -stable-like operator with form: L κ , σ ( α ) u ( t , x ) : = ∫ R d ( u ( t , x + σ ( t , x ) z ) - u ( t , x ) - σ ( t , x ) z ( α ) · ∇ u ( t , x ) ) κ ( t , x , z ) ν ( α ) ( d z ) , where z ( α ) = z 1 α ∈ ( 1 , 2 ) + z 1 | z | ≤ 1 1 α = 1 , κ : R + × R 2 d → R + is bounded from above and below, σ : R + × R d → R d ⊗ R d is a γ -order Hölder continuous function in x uniformly in t , and ν α ) is a singular non-degenerate α -stable Lévy measure.