Weighted Estimates for Generalised Conical Square Functions and Applications

Let {A(t)}(t>0) be a family of bounded linear operator on L-2(X) where (X, d, mu) is a metric space with metric d and doubling measure mu. Assume that the family {A(t)}(t>0) satisfies suitable off-diagonal estimates from Lp0 to L-2 for some p(0) < 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.