L^p theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

We consider the operator L= - div (A∇) , where A is an n× n matrix of real coefficients and satisfies the ellipticity condition, with n≥ 2. We assume that the coefficients of the symmetric part of A are in L^∞(Rⁿ) , and those of the anti-symmetric part of A only belong to the space BMO(Rⁿ). We create a complete narrative of the L^p theory for the square root of L and show that it satisfies the L^p estimates ∥Lf∥Lp≲∥∇f∥Lp for 1 < p< ∞, and ∥∇f∥Lp≲∥Lf∥Lp for 1 < p< 2 + ϵ for some ϵ> 0 depending on the ellipticity constant and the BMO semi-norm of the coefficients. Moreover, we prove the L^p estimates for some vertical square functions associated to e^-tL. In another article of the authors, these results are used to establish the solvability of the Dirichlet problem for elliptic equation div (A(x) ∇ u) = 0 in the upper half-space (x,t)∈R+n+1 with the boundary data in L^p(Rⁿ, dx) for some p∈ (1 , ∞).