Some inequalities related to Riesz transform on exterior Lipschitz domains

Let n≥2 and L=-div(A∇·) be an elliptic operator on Rⁿ. Given an exterior Lipschitz domain Ω, let LD and LN be the elliptic operators L on Ω subject to the Dirichlet and the Neumann boundary {conditions}, respectively. For the Neumann operator, we show that the reverse inequality \|LN1/2f\|Lᵖ(Ω) ≤ C\|∇ f\|Lᵖ(Ω) holds true for any p∈(1,∞). For the Dirichlet operator, it was known that the Riesz operator ∇ LD-1/2 is not bounded for $p>2$ and p≥ n, even if L=-Δ being the Laplace operator. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and ∂Ω is C¹, we prove that for $p>2$ and p∈ [n,∞), it holds infφᵖ₀(Ω)\|∇ f-∇φ\|Lᵖ(Ω)≤ C\|L1/2D f\|Lᵖ(Ω) for f∈ Ẇ1,p₀(Ω). Here Aᵖ₀(Ω)=∈ Ẇ1,p₀(Ω):\,LDf=0\ is a non-trivial subspace generated by harmonic function in Ω with zero boundary value.