Weak and strong type estimates for square functions associated with operators

Let L be a linear operator on L²(Rⁿ) which generates a semigroup e^−tL whose kernels p_t(x,y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function S_α,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding to Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p,p) for S_α,L and the endpoint estimate for commutators of S_α,L. Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.