Inequalities associated to Riesz potentials and non-doubling measures with applications

• Main Author: Bhandari, Mukta Bahadur
• Language: en_US
• Published: Kansas State University 2010
• Subjects: Riesz Potentials; Non-doubling Measures; Good lambda inequality; Hedberg Inequality; Maximal Functions; Weight Functions; Mathematics (0405)
• Online Access: http://hdl.handle.net/2097/4375

Doctor of Philosophy === Department of Mathematics === Charles N. Moore === The main focus of this work is to study the classical Calder\'n-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.