Estimates L^r-L^s for solutions of the ∂ equation in strictly pseudo convex domains in ℂⁿ.

2013/12/26 by Eric Amar, Amar, Eric
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV

paper · pdf · doi:10.48550/arxiv.1312.7136

The result in the case Lr-Ls were known, thank to the referee, even in the case of q-convex domains. But here the method is different and we get the BMO and Lipschitz estimates

arxiv created 2014/01/24 · arxiv updated 2014/01/27

Abstract

We prove estimates for solutions of the ∂ u=ω equation in a strictly pseudo convex domain Ω in ℂⁿ. For instance if the (p,q) current ω has its coefficients in L^r(Ω) with 1≤ r<2(n+1) then there is a solution u in L^s(Ω) with (1)/(s)=(1)/(r)-(1)/(2(n+1)). We also have BMO and Lipschitz estimates for r≥ 2(n+1). These results were already done by S. Krantz in the case of (0,1) forms and just for the L^r-L^s part by L. Ma and S. Vassiliadou for general (p,q) forms. To get the complete result we propose another approach, based on Carleson measures of order α and on the subordination lemma.

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