Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations

2007/05/07 by Hofmann, S.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.0705.0839

Abstract

We consider divergence form elliptic operators L=-\dv A(x)∇, defined in ℝⁿ⁺¹=\(x,t)∈ℝⁿ×ℝ\, n ≥ 2, where the L^∞ coefficient matrix A is (n+1)×(n+1), uniformly elliptic, complex and t-independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if Lu=0 in ℝⁿ⁺¹₊, then for any vector-valued \bf v ∈ W^1,2_loc, we have the bilinear estimate |\iint_ℝⁿ⁺¹+ ∇ u ⋅ \bf v dx dt |≤ Csup_tgt;0 ‖u(⋅,t)‖_L²(ℝⁿ)(‖|t ∇ \bf v‖| + ‖N_*\bf v‖_L²(ℝⁿ)), where ‖|F‖| ≡ (\iint_ℝⁿ⁺¹+ |F(x,t)|² t^-1 dx dt)^1/2, and where N_* is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients, and generalizes the analogous result of Dahlberg for harmonic functions on Lipschitz graph domains.

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