The Liouville theorem under second order differentiability assumption

Iwaniec and Martin proved that in even dimensions n≥3, Wloc1,n/2 conformal mappings are Möbius transformations and they conjectured that it should also be true in odd dimensions. We prove this theorem for a conformal map f∈Wloc1,1 in dimension n≥3 under one additional assumption that the norm of the first order derivative |Df| satisfies |Df|p∈Wloc1,2 for p≥(n−2)/4. This is optimal in the sense that if |Df|p∈Wloc1,2 for p<(n−2)/4, it may not be a Möbius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec-Martin conjecture. Meanwhile, we show that the Iwaniec-Martin conjecture can be reduced to a conjecture about the Caccioppoli type estimate.