Weighted Morrey Spaces

This monograph is a testament to the potency of the method of singular integrals of layer potential type in solving boundary value problems for weakly elliptic systems in the setting of Muckenhoupt-weighted Morrey spaces and their pre-duals.

A functional analytic framework for Muckenhoupt-weighted Morrey spaces in the rough setting of Ahlfors regular sets is built from the ground up and subsequently supports a Calderón-Zygmund theory on this brand of Morrey space in the optimal geometric environment of uniformly rectifiable sets. A thorough duality theory for such Morrey spaces is also developed and ushers in a never-before-seen Calderón-Zygmund theory for Muckenhoupt-weighted Block spaces. Both weighted Morrey and Block spaces are also considered through the lens of (generalized) Banach function spaces, and ultimately, a variety of boundary value problems are formulated and solved with boundary data arbitrarily prescribed from either scale of space.

The fairly self-contained nature of this monograph ensures that graduate students, researchers, and professionals in a variety of fields, e.g., function space theory, harmonic analysis, and PDE, will find this monograph a welcome and valuable addition to the mathematical literature.

• Offers a rich theory for weighted Morrey spaces in inclusive geometric settings
• Provides a blueprint to do PDE in a variety of functional analytic contexts
• Contains sophisticated well-posedness results for a large spectrum of boundary problems.

Dr. Marius Mitrea, formerly Professor of Mathematics at the University of Missouri, became a member of the Department of Mathematics at Baylor University in 2019. He joins his wife, Dorina Mitrea, who became the new Chair of Baylor's Department of Mathematics. Marius Mitrea earned his M.S. degree in mathematics from the University of Bucharest in 1988. Prior to moving to the U.S., he taught at the University of Bucharest and then held a research position at the Institute of Mathematics of the Romanian Academy until 1996.

Marius Mitrea earned his Ph.D. in mathematics in 1994 from the University of South Carolina under the direction of Björn Jawerth. From 1994-1996, he held a three-year post-doctoral position at the University of Minnesota, working with Eugene Fabes. He then joined the faculty at the University of Missouri-Columbia in August 1996. Professor Mitrea is a Fellow of AMS, and has more than 180 journal publications and 17 research monographs.

Marcus Laurel is a young mathematician affiliated with Baylor university working at the confluence of geometry, harmonic analysis, and PDE. His interests lie in layer potential methods to solve boundary value problems for elliptic systems, as well as function space theory in rough geometric settings. He received his B.S. in Mathematics in 2018 and earned the distinction of an outstanding thesis.