Quantitative Weighted Estimates for Some Singular Integrals Related to Critical Functions

Let $$(X, d, \mu )$$ be a space of homogeneous type with a metric d and a doubling measure $$\mu $$ . Assume that $$\rho $$ is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note that the assumptions on the kernels of these singular integrals do not have any regularity conditions. Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators.