The Square Root of a Parabolic Operator

Let $$L(t) = - \mathrm{div} \left( A(x,t) \nabla _x \right) $$ for $$t \in (0, \tau )$$ be a uniformly elliptic operator with boundary conditions on domain $$\Omega of $$\mathbb {R}^d$$ and $$\partial \frac{\partial }{\partial t}$$ . Define the parabolic $${{\mathcal {L}}}= \partial + L$$ $$L^2(0, , L^2(\Omega ))$$ by $$({{\mathcal {L}}}u)(t) := u(t)}{\partial t} L(t)u(t)$$ We assume very little regularity we that coefficients A(x, t) are measurable in x piecewise $$C^\alpha t (uniformly $$x \Omega ) some $$\alpha > \frac{1}{2}$$ prove Kato square root property $$\sqrt{{{\mathcal {L}}}}$$ estimate $$\begin{aligned}&\Vert \sqrt{{{\mathcal {L}}}}\, u \Vert _{L^2(0,\tau ))} \approx _{H^{\frac{1}{2}}(0,\tau ))}\\&\qquad \int _0^\tau u(t) _{L^2(\Omega )}^2\, \frac{dt}{t} ^{1/2}. \end{aligned}$$ also $$L^p$$ -versions this result.