Related papers: The square root of a parabolic operator

We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on…

We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in…

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a…

The Kato square root problem for divergence form elliptic operators with potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u…

We establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with bounded measurable coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(\mu \nabla u) +…

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow…

We consider the Kato square root problem for non-divergence second order elliptic operators $L =- a_{ij} D_iD_j$, and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real…

We solve the Kato square root problem for parabolic operators whose coefficients can be written as the sum of a complex part, which is coercive, and a real anti-symmetric part, which is in BMO. In particular, we allow for certain unbounded…

Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla…

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field…

We prove the Kato conjecture for degenerate elliptic operators in R^n. More precisely, we consider the divergence form operator L_w = -1/w div (wA) grad, where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix which is…

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We…

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space…

We study parabolic operators H = $\partial$t -- div $\lambda$,x A(x, t)$\nabla$ $\lambda$,x in the parabolic upper half space R n+2 + = {($\lambda$, x, t) : $\lambda$ > 0}. We assume that the coefficients are real, bounded, measurable,…

Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $$ \Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in…

We prove the Kato square root estimate for second-order divergence form elliptic operators $-div(A\nabla)$ on a bounded, locally uniform domain $D \subseteq \mathbb{R}^n$, for accretive coefficients $A \in L^\infty(D; \mathbb{C}^n)$, under…