Fix d > 3 and I 0 e tLy I f are bounded on the weighted Lebesgue space LP (W). The boundedness of the Lv-Riesz potentials Lv'/2 from LT (w) to L' (w'IP) for 0 < c < 2, 1 < p < aand + = i a will also be proved. These weight classes are strictly larger than a class previously introduced by Bongioanni, Harboure and Salinas in H that shares these properties and they contain weights of exponential growth and decay. The classes will also be considered in relation to different generalised forms of Schrodinger operator. In particular, the Schrodinger operator with measure potential A+ it, the uniformly elliptic operator with potential divAV + V and the magnetic Schrodinger operator (V ia)*(7 ia) + V will all be considered. Finally, this article will investigate necessary conditions that a weight w must satisfy in order for the Riesz transforms or the heat maximal operator to be bounded on LT(w). To aid in this task, lower bounds for the heat kernel of the standard Schrodinger operator A+ V will be proved. These estimates provide a lower counterpart to the upper estimates proved in [23] 2021 Elsevier Inc. All rights reserved.
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