Littlewood-Paley functions associated to second order elliptic operators (Q1434207)

Let \(A\) be an \(n\times n\) matrix of complex, \(L^\infty\) coefficients defined on \({\mathbb R}^n\), and satisfying the ellipticity condition. Define a divergence form operator \[ Lf\equiv-\text{div}(A\nabla f). \] The generalized vertical Littlewood-Paley \(G_L\)-function associated with the divergence form operator \(L\) is defined by \[ G_L(f)(x)=\left(\int_0^\infty t| \nabla e^{-t\sqrt{L}}f(x)| ^2\,dt\right)^{1/2}. \] Let \(p_n=\frac{2n}{n+2}\). In this paper, the author proves that \(G_L\) is bounded on \(L^q({\mathbb R}^n)\) for \(p_n<q\leq 2\), and \(G_L\) is of weak-type \((p_n,p_n)\). Note that if \(L\) is the Laplacian \(-\triangle\) on \({\mathbb R}^n\), then \(G_L\) is the classical Littlewood-Paley \(g_x\)-function defined by \[ g_x(f)(x)=\left(\int_0^\infty | \nabla_x u(x,y)| ^2y\,dy\right)^{1/2}, \] where \(u(x,y)=P_y\ast f(x)\) and \(P_y(x)\) is the Poisson kernel.