The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy–Riemann system in complex analysis. It hinges on global Lp-bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse Holder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.