Schauder estimates for nonlocal equations with singular Lévy measures (Q6564504)

The authors ``establish Schauder's estimates for the following non-local equations in \(\mathbb{R}^d\):\N\[\N\partial_t u=\mathcal{L}_{k,\sigma}^{(\alpha)}u +b \cdot \nabla u +f , u(0)=0\N\]\Nwhere \(\alpha\in(1/2,2)\) and \(b:\mathbb{R}_+ \times \mathbb{R}^d\) is an unbounded local \(\beta\)-order Hölder function in \(x\) uniformly in \(t\)', and \(\mathcal{L}_{k,\sigma}^{(\alpha)}\) is a certain non-local \(\alpha\)-stable-like operator.'' ``It is well-known that Schauder's estimates play a basic role in constructing the classical solution for quasilinear partial differential equations (...), and also give an approach to show the well-posedness of stochastic differential equations (...).''