This paper discusses the structure of Morrey spaces, weak Morrey spaces, generalized Morrey spaces, and generalized weak Morrey spaces. Some necessary and sufficient conditions for the inclusion property of these spaces are obtained through a norm estimate for the characteristic functions of balls.

We give the super Catalan numbers S (m, n) and their associated family of rational numbers, called the circular super Catalan numbers Ω (m, n) defined byan algebraic interpretation in terms of values of an algebraic integral of polynomials over the unit circle over a finite field of odd characteristic. We consider three metrical geometries, the Euclidean geometry and two Einstein-Minkowski geometries and show that the problem of polynomial summation over the unit circle in each geometry is intricately connected, which is a phenomenon of chromogeometry.This investigation opens up not only the possibility to do prime characteristic harmonic analysis and a connection to characteristic zero harmonic analysis, but also the study of a wide variety of additional (rational) numbers that complement the super Catalan numbers and circular super Catalan numbers.

In 2003, Deutsch and Elizalde defined bijective maps between Dyck paths which are beneficial in investigating some statistics distributions of Dyck paths and pattern-avoiding permutations. In this paper, we give a generalization of the maps so that they are generated by permutations in S2n. The construction induces several novel ways to partition S2n which give a new interpretation of an existing combinatorial identity involving double factorials and a new integer sequence. Although the generalization does not in general retain bijectivity, we are able to characterize a class of permutations that generates bijections and furthermore imposes an algebraic structure to a certain class of bijections. As a result, we introduce statistics of a Dyck path involving the number of unpaired steps in some subpath whose distribution is identical to other well-known height statistics.

We find an algebraic interpretation of the super Catalan numbers through polynomial summation formulae over unit circles over finite fields of odd characteristic. While traditional Fourier analysis involves Riemann integration over the unit circle in the real number plane, we develop a purely algebraic integration theory without recourse to infinite procedures, and develop an algorithm for explicitly computing such Fourier sums for general monomials.We consider three unit circles that arise from Euclidean geometry and two relativistic geometries, and demonstrate the strong relationship between the integration theory in each geometry. The algebraic integrals in the three geometries are called the Fourier summation functionals and take values in the same finite field.The key results in this thesis are the existence and uniqueness of the Fourier summation functionals, as well as the explicit formulae for them in terms of the super Catalan numbers and their rational variants which we call the circular super Catalan numbers.This investigation not only opens up new avenues in developing finite field harmonic analysis from a completely algebraic point of view, but also highlights many similarities to the traditional integration theory over the unit circle.