Some inequalities related to Riesz transform on exterior Lipschitz domains

Let n2 and L=-div (A) be an elliptic operator on Rⁿ. Given an exterior Lipschitz domain, let LD and LN be the elliptic operators L on subject to the Dirichlet and the Neumann boundary conditions, respectively. For the Neumann operator, we show that the reverse inequality \|LN^1/2f\|₋㵵 () C\| f\|₋㵵 () 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=- being the Laplace operator. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and is C¹, we prove that for p>2 and p [n, ), it holds 㵵䃐 () \| f-\|₋㵵 () C\|L^1/2D f\|₋㵵 () for f Ẇ^1, p₀ (). Here Aᵖ₀ () =\f Ẇ^{1, p₀ (): \, LDf=0\} is a non-trivial subspace generated by harmonic function in with zero boundary value.