The Square Root of a Parabolic Operator
El Maati Ouhabaz · Journal of Fourier Analysis and Applications · 2021
Let $$L(t) = - \mathrm{div} \left( A(x,t) abla _x \right) $$ for $$t \in (0, \tau )$$ be a uniformly elliptic operator with boundary conditions on a domain $$\Omega $$ of $$\mathbb {R}^d$$ and $$\partial = \frac{\partial }{\partial t}$$ . Define the parabolic operator $${{\mathcal {L}}}= \partial + L$$ on $$L^2(0, \tau , L^2(\Omega ))$$ by $$({{\mathcal {L}}}u)(t) := \frac{\partial u(t)}{\partial t} + L(t)u(t)$$ . We assume a very little of regularity for the boundary of $$\Omega $$ and we assume that the coefficients A(x, t) are measurable in x and piecewise $$C^\alpha $$ in t (uniformly in $$x \in \Omega $$ ) for some $$\alpha > \frac{1}{2}$$ . We prove the Kato square root property for $$\sqrt{{{\mathcal {L}}}}$$ and the estimate $$\begin{aligned}&\Vert \sqrt{{{\mathcal {L}}}}\, u \Vert _{L^2(0,\tau , L^2(\Omega ))} \approx \Vert abla _x u \Vert _{L^2(0,\tau , L^2(\Omega ))} + \Vert u \Vert _{H^{\frac{1}{2}}(0,\tau , L^2(\Omega ))}\\&\qquad + \left( \int _0^\tau \Vert u(t) \Vert _{L^2(\Omega )}^2\, \frac{dt}{t} \right) ^{1/2}. \end{aligned}$$ We also prove $$L^p$$ -versions of this result.