Conical square functions for degenerate elliptic operators

Abstract

The aim of this paper is to study the boundedness of di erent conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let Lw = w1 div(w A r) where w 2 A2 and A is an n n bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and C. Rios solved the L2(w)-Kato square root problem obtaining that p Lw is equivalent to the gradient on L2(w). The same authors in collaboration with the second named author of this paper studied the Lp(w)-boundedness of operators that are naturally associated with Lw, such as the functional calculus, Riesz transforms, or vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp(vdw) for v 2 A1(w)), and in particular a class of “degeneracy” weights w was found in such a way that the classical L2-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 Lp(w) and on Lp(vdw), with v 2 A1(w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw. As a consequence of our methods, we find a class of degeneracy weights w for which L2-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 Lw.