Weighted estimates for generalised conical square functions and applications

Let {A_t}_t>0 be a family of bounded linear operator on L²(X) where (X, d, µ) is a metric space with metric d and doubling measure µ. Assume that the family {A_t}_t>0 satisfies suitable off-diagonal estimates from L^p0 to L² for some p0 < 2. This paper aims to prove weighted bound estimates for conical square functions and g-functions associated to the family {A_t}_t>0. Some applications such as weighted bounds for bilinear estimates associated to certain differential operators are also obtained.