We prove higher Hölder regularity for solutions of equations involving the fractional p−Laplacian of order s, when p ≥ 2 and 0 < s < 1. In particular, we provide an explicit Hölder exponent for solutions of the non-homogeneous equation

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma.We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to Hélein's for harmonic maps into spheres.Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivière's celebrated argument in the local case.Lastly, the fractional div-curl quantities provide also a new, simpler, proof for Hölder continuity of W s,n/s -harmonic maps into spheres and we extend this to an argument for W s,n/s -harmonic maps into homogeneous targets. This is an analogue of Strzelecki's and Toro-Wang's proof for n-harmonic maps into spheres and homogeneous target manifolds, respectively.2010 Mathematics Subject Classification. 42B37, 42B30, 35R11, 58E20, 35B65.

We prove that if a curve γ ∈ H 3/2 (R/Z, R n ) parametrized by arc length is a stationary point of the Möbius energy introduced by Jun O'Hara in [O'H91], then γ is smooth. Our methods only rely on purely analytical arguments, entirely without using Möbius invariance. Furthermore, they are not fundamentally restricted to one-dimensional domains, but are generalizable to arbitrary dimensions. 2The price we pay is that, instead of the very appealing geometric argument in [FHW94], we have to adapt some sophisticated techniques originally developed by Tristan Rivière and Francesca Da Lio [DLR11a, DLR11b, DL11] and the third author [Sch12, Sch11] to deal with n 2 -harmonic maps into manifolds. The first task in order to prove this result, is to derive the Euler-Lagrange equation for such stationary points. In [FHW94], it was shown that for simple closed curves γ ∈ C 1,1 (R/Z, R n ) and h ∈ C 1,1 (R/Z, R n ) we have δE (2) (γ; h) := lim τց0

No abstract

Abstract. We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional p-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weak fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded n s -harmonic maps converge strongly outside at most finitely many points.

We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations.The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-McIntosh-Meyer, Kato-Ponce-Vega type commutators, the fractional Leibniz rule, and the Da Lio-Rivière three-term commutators. We also give a limiting L 1 -estimate for a double commutator of Coifman-Rochberg-Weiss-type, and several intermediate estimates. Some of the estimates obtained seem to be new or known only to some experts.The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general paraproduct arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the "cancellation effect" responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.

In this note we give a shorter proof of recent regularity results in [Riv07], [RS08]. We differ from the mentioned articles only in using the direct method of Hélein's moving frame to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using besides the duality of Hardy-and BMO-space only elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandt's conjecture one can avoid the Nash-Moser imbedding theorem. There are no new results presented here, nor are there any techniques we could claim originality for.

For s ∈ (0, 1) we introduce (integro-differential) harmonic maps v : Ω ⊂ R n → R N , which are defined as critical points of the Besov-Slobodeckij energywith the side-condition that v(Ω) ⊂ S N −1 , for the (N − 1)-spherethis are the classical fractional harmonic maps first considered by Da Lio and Rivière. For p s = 2 this is a new energy which has degenerate, non-local Euler-Lagrange equations. They are different from the n/s-harmonic maps introduced by Francesca Da Lio and the author, and have to be treated with new arguments, which might be of independent interest for further applications on geometric energies. For the critical case p s = n s we show Hölder continuity of these maps.