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First, we establish the theory of fractional powers of first orderdifferential operators with zero order terms, obtaining PDE properties andanalyzing the corresponding fractional Sobolev spaces. In particular, our studyshows that Lebesgue and Sobolev spaces with inverse measures (like the inverseGaussian measure) play a fundamental role in the theory of fractional powers ofthe first order operators. Second, and motivated in part by such a theory, welay out the foundations for the development of the harmonic analysis for\emph{inverse} measures. We discover new families of polynomials related to theinverse Gaussian, Laguerre, and Jacobi measures, and characterize them usinggenerating and Rodrigues formulas, and three-term recurrence relations.Moreover, we prove boundedness of several fundamental singular integraloperators in these inverse measure settings.

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures