New weighted inequalities with applications to pseudodifferential operators

Abstract: This thesis is devoted to weighted integral inequalities and their applications to the  study of boundary behavior of solutions to Dirichlet's problem for fractional powers of the Laplacian.We obtain a necessary and sufficient condition on μ for the operator (—Δ)μin R 0 , 0 < μ < n/2 to have the so called weighted positivity property, the weight being the fundamental solution of the operator. This property is also studied for ordinary differential operators and we provide various examples of operators with and without the property.The optimal constants in a two parameter family of Hardy-Rellich type inequalities are found. A number of other weighted inequalities related to fractional derivative are obtained.A sufficient Wiener type condition for regularity of a boundary point with respect to (-,;:.)μ is obtained for the range of μ ensuring the weighted positivity property. For the same μ's, we also study the behavior of the μ-harmonic Dirichlet capacitarypotential near a boundary point which do not satisfy the Wiener condition