Mathematical Sciences: Weighted norm inequalities and applications

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Work will continue on weighted norm inequalities for various classical operators and operators related to orthogonal polynomial expansions. Applications of these inequalities to differential equations will also be examined. The norm inequalities to be considered concern fractional integrals, Sobolev interpolation inequalities, Fourier transforms, weighted Hardy and Lebesgue product spaces, n-dimensional Hardy's inequality, Littlewood-Paley functions and Hilbert transforms. The basic problem in all these situations is that of understanding the character of the weight functions for which weighted norm inequalities hold. The operators related to orthogonal polynomial expansions are Cesaro sum operators and transplantation operators for Hermite and Laguerre polynomials. Weak and strong type norm inequalities will be proved by obtaining accurate kernel estimates. Applications to differential equations concern the regularity of solutions of degenerate elliptic equations. In case the degeneracy is governed by two weight functions, the problems include estimates of the Green function and norm inequalities for the potentials associated with it. With equal weights, the problem is how the weight affects the smoothness of the solutions when the coefficients are assumed to have some smoothness. A local version of Harnack's inequality will also be investigated for equations whose coefficients are not smooth and degenerate differently in different directions.