On sectoriality of degenerate elliptic operators

Let $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$ , where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

Conical square functions for degenerate elliptic operators

The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.