Regularity of harmonic functions for anisotropic fractional Laplacians

Sztonyk, Paweł

doi:10.1002/mana.200711116

Cited by 34 publications

(38 citation statements)

References 36 publications

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“…For example, when n = 2 we have µ = δ (1,0) + δ (0,1) + δ (−1,0) + δ (0,−1) (up to a multiplicative constant). The regularity of solutions to Lu = f (or Lu = 0) for operators L like (1.3), (1.1), or related ones, has been widely investigated; see the works by Bass, Kassmann, Schwab, Silvestre, Sztonyk, and Bogdan, among others [1,23,22,3,39,21,2,37,5,6,7,9,24]. A typical assumption in some of these results is that 0 < c ≤ a(θ) ≤ C in S n−1 .…”

Section: Introduction and Resultsmentioning

confidence: 99%

Regularity theory for general stable operators

Ros‐Oton

Serra

2016

Journal of Differential Equations

158

206

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Abstract. We establish sharp regularity estimates for solutions to Lu = f in Ω ⊂ R n , being L the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on S n−1 , called the spectral measure.First, we study the interior regularity of solutions to Lu = f in B 1 . We prove that if f is C α then u belong to C α+2s whenever α + 2s is not an integer. In case f ∈ L ∞ , we show that the solution u is C 2s when s = 1/2, and C 2s−ǫ for all ǫ > 0 when s = 1/2.Then, we study the boundary regularity of solutions to Lu = f in Ω, u = 0 in R n \ Ω, in C 1,1 domains Ω. We show that solutions u satisfy u/d s ∈ C s−ǫ (Ω) for all ǫ > 0, where d is the distance to ∂Ω.Finally, we show that our results are sharp by constructing two counterexamples.

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Section: Introduction and Resultsmentioning

confidence: 99%

Regularity theory for general stable operators

Ros‐Oton

Serra

2016

Journal of Differential Equations

158

206

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“…Estimates of kernels for processes which are solutions of SDEs driven by Lévy processes were obtained in [32,20,35,36,21]. For estimates of derivatives of Lévy densities we refer the reader to [41,5,40,26,31,27]. In [23] the authors gave a very interesting geometric interpretation of the transition densities for symmetric Lévy processes.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Small-time sharp bounds for kernels of convolution semigroups

Kaleta

Sztonyk

2017

JAMA

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We study small time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in R d , d ≥ 1, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric nor absolutely continuous with respect to the Lebesgue measure, we find the optimal, both in time and space, upper bound for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous, with densities g such that g(x) ≍ f (|x|) for nonincreasing profile functions f , we also prove the full characterization of the sharp two-sided transition densities bounds of the formThis is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to an interesting and surprising dichotomy of the decay properties at infinity for transition kernels of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

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“…It is proved in [29] that the potential kernel K associated to L α (i.e., the fundamental solution of the operator) satisfies c 1 |y| n−α ≤ K(y) ≤ c 2 |y| n−α , for suitable positive constants c 1 ≤ c 2 , and K(y) = |y| α−n K y |y| .…”

Section: 1mentioning

confidence: 99%