Gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes

By Krzysztof Bogdan, Tadeusz Kulczycki, Adam Nowak. 2002

130 citations recorded by the source corpus.

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Citations in the corpus

Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “1 is also known for harmonic functions with respect to Schrödinger operators based on the Laplacian and the fractional Laplacian ([12], [5], [20]).”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “on 4 we prove that the perturbation series indeed converge and yield (1) for small sets D with bounded distortion. In the proofs we use estimates for GD ([38], [20], [30]) and for the gradient of GD ([14]), the boundary Harnack inequality for ∆α/2 ([13], [5]) and the Kato condition (30) for the drif”
Entire $s$-harmonic functions are affine (2014, background): “unction on RN is constant. In the case of the fractional Laplacian −(−∆)s, for s ∈ (0,1), (see Section 2), the strong form of the Liouville theorem holds as well and was proved by K. Bogdan et al. in [4]. Applications of Liouville theorems for nonlocal operators in the study of nonlocal elliptic sys”
Some Liouville theorems for the fractional Laplacian (2014, methodology): “for solutions in PDE analysis. ∗Partially supported by the Simons Foundation Collaboration Grant for Mathematicians245486. 1 This Liouville Theorem has been generalized to the fractional Laplacian in [BKN] and [ZCCY]: Proposition1 Let 0 <α<2 and n≥ 2. Assume that uis a solution of ˆ (−△)α/2u(x”
Monotonicity and nonexistence results for some fractional elliptic problems in the half-space (2013, unknown): “gt;0 such that (6.1) kukC0,α(B r) ≤ Cs,N,r,α kukL∞(B r) +kfkL∞(B 1) . 19 Proof. Let η∈ C∞ c (RN) be such that η= 1 on Br, η= 0 on RN \ B1 and 0 ≤ η≤ 1 on RN. Consider the Riesz potential, see [5] and [7], Φ(x,y) = CN,s|x−y|2s−N for N>2s, 1 π log 1 |x| for 1 = N= 2s, −C1,s|x−y|2s−N for N= 1 &”
Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, background): “We note in passing that (45) agrees with (35) because r → MD(x) is increasing [15].”