On $L^p$ estimates for square roots of second order elliptic operators on $\mathbb{R}^n$

Abstract

We prove that the square root of a uniformly complex elliptic operator L = -div(A∇) with bounded measurable coefficients in R n satisfies the estimate L 1/2 f p ∇f p for sup(1, 2n n+4ε) < p < 2n n-2 + ε, which is new for n ≥ 5 and p < 2 or for n ≥ 3 and p > 2n n-2 .One feature of our method is a Calderón-Zygmund decomposition for Sobolev functions.We make some further remarks on the topic of the converse L p inequalities (i.e.Riesz transforms bounds), pushing the recent results of [BK2] and [HM] for 2n n+2 < p < 2 when n ≥ 3 to the range sup(1, 2n n+2 -ε) < p < 2+ε .In particular, we obtain that L 1/2 extends to an isomorphism from Ẇ 1,p (R n ) to L p (R n ) for p in this range.We also generalize our method to higher order operators.

Locations

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