Heating of the Beurling operator: Sufficient conditions for the two-weight case
We establish sucient conditions on the two weights w and v so that the Beurling-Ahlfors transform acts continuously from L 2 (w 1 ) to L 2 (v). Our conditions are simple estimates involving heat extensions and Green's potentials of the weights. Notation. C a constant not depending on any functions, other than the weights. C will not necessarily denote the same constant at each occurrence. x := (x1;x2), @1f(x) := @f=@x1. (x;t) = t(x) := (4t ) 1 e k xk 2 =4t