Let \(\{\cal{A}_{t}\}_{t>0}\) 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 \(\{\cal{A}_{t}\}_{t>0}\) satisfies suitable off-diagonal estimates from \(L^{p_{0}}\) 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 \(\{\cal{A}_{t}\}_{t>0}\) . Some applications such as weighted bounds for bilinear estimates associated to certain differential operators are also obtained.
