Abstract
We study the local regularity of solutions f to the integro-differential equation
for open sets \(U \subseteq \mathbb {R}^{d}\), where A is the infinitesimal generator 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.
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I am grateful to René Schilling for valuable comments which helped to improve the presentation of this paper.
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Appendix: Proof of Lemma 1
Appendix: Proof of Lemma 1
For the proof of our results we used the following lemma, which was already stated in Section 2.
Lemma 1
Let \(\beta \in (0,\infty )\), and let be open. The following statements hold for any j ≥ k := ⌊β⌋ + 1:
-
i).
There exists a constant c > 0 such that
$$ \sup\limits_{0<|h| \leq r} \frac{|{{\Delta}_{h}^{k}} f(x)|}{|h|^{\beta}} \leq c r^{-\beta} \|f\|_{\infty,U} + c \sup\limits_{0<|h| \leq r/j} \sup\limits_{z \in B(x,r(k+1))} \frac{|{{\Delta}_{h}^{j}} f(z)|}{|h|^{\beta}} $$(31)for all f ∈ Cb(U), r > 0, and x ∈ U with \(B(x,r(k+2)) \subseteq U\).
-
ii).
If β > 1 then there exists a constant c > 0 such that
$$ \max\limits_{i=1,\ldots,d} |\partial_{x_{i}} f(x)| \leq c r^{-\beta} \|f\|_{\infty,U} + c \sup\limits_{0<|h| \leq r/j} \sup\limits_{z \in B(x,r(k+1))} \frac{|{{\Delta}_{h}^{j}} f(z)|}{|h|^{\beta}} $$(32)for all \(f \in {C_{b}^{1}}(U)\), r > 0 and x ∈ U with \(B(x,r(k+2)) \subseteq U\).
Proof
First of all, we note that it suffices to prove both statements for \(f \in {C_{b}^{2}}(U)\); the inequalities can be extended using a standard approximation technique, e.g. by considering fi := f ∗ φi for a sequence of mollifiers (φi)i≥ 1.
Denote by τhf(x) := f(x + h) the shift operator. A straightforward computation shows that
holds for any , and any two functions u,v.
To prove i) we note that the assertion is obvious for j = k, and so it suffices to consider j > k. We will first establish the following auxiliary statement: There exists a constant C > 0 such that
for any twice differentiable bounded function . To this end, pick such that . Clearly,
Using the equivalence of the norms on , cf. Eq. 7, we get
for some constants c1 and \(c_{1}^{\prime }\). As χ = 0 on , we have \({\Delta }_{t}^{2j} g(y)=0\) for all \(|y|>k+\frac {2}{3}\) and \(|t| \leq \frac {1}{6j}\). Consequently,
Since
and
an application of the product formula (33) gives
for all \(|y| \leq k+\frac {2}{3}\) and \(|t| \leq \frac {1}{6j}\). Combining this estimate with Eq. 35 and noting that β ≤ k proves (34). Now if \(f \in {C_{b}^{2}}(U)\), then we apply (34) with g(t) := f(x + rth) for fixed |h| = 1 to get the desired inequality.
It remains to prove ii). First we consider the case β ∈ (1, 2) and j = 2. The auxiliary inequality which we need is
for a uniform constant C > 0 where is differentiable on (− 3, 3). To this end, choose with . By the equivalence of the norms on the Hölder–Zygmund space , cf. Eq. 7, we get
for some finite constant c4 > 0. Following the reasoning in the first part of the proof (with k = 1 and j = 2) yields (36). Applying (36) for g(t) := f(x + rtej), where ej is the j-th unit vector in , gives ii) for β ∈ (1, 2) and j = 2. In combination with i), this yields the desired inequality for every β > 1 and \(j \geq \left \lfloor {\beta } \right \rfloor +1\). □
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Kühn, F. Interior Schauder Estimates for Elliptic Equations Associated with Lévy Operators. Potential Anal 56, 459–481 (2022). https://doi.org/10.1007/s11118-020-09892-y
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DOI: https://doi.org/10.1007/s11118-020-09892-y