The Square Root of a Parabolic Operator

Let L ( t ) = - div A ( x , t ) ∇ x for t ∈ ( 0 , τ ) be a uniformly elliptic operator with boundary conditions on a domain Ω of R d and ∂ = ∂ ∂ t . Define the parabolic operator L = ∂ + L on L 2 ( 0 , τ , L 2 ( Ω ) ) by ( L u ) ( t ) : = ∂ u ( t ) ∂ t + L ( t ) u ( t ) . We assume a very little of regularity for the boundary of Ω and we assume that the coefficients A ( x , t ) are measurable in x and piecewise C α in t (uniformly in x ∈ Ω ) for some α > 1 2 . We prove the Kato square root property for L and the estimate ‖ L u ‖ L 2 ( 0 , τ , L 2 ( Ω ) ) ≈ ‖ ∇ x u ‖ L 2 ( 0 , τ , L 2 ( Ω ) ) + ‖ u ‖ H 1 2 ( 0 , τ , L 2 ( Ω ) ) + ∫ 0 τ ‖ u ( t ) ‖ L 2 ( Ω ) 2 dt t 1 / 2 . We also prove L p -versions of this result.