Small Divisor Problems and A_p weights with an Application

We show the Muckenhoupt theory of A_p weights can be used to understand certain small divisor problems. We shall use this link to obtain a quantitative version of the Ehrenpreis-Malgrange local solvability theorem for constant coefficient partial differential operators, where the loss of derivatives in the scale of Sobolev spaces is related to the A_p class of the symbol of the differential operator.