Abstract :
[en] The study of function regularity has been extensively explored, with numerous tools developed to describe this property, such as continuity, differentiability, and H\"older conditions. One way to characterize regularity is by examining the functional spaces to which a function belongs, particularly those constructed through interpolation. A key objective of this work is to refine interpolation methods to define regularity spaces more precisely. A central idea is the use of Boyd functions, which replace the usual power functions in interpolation spaces. These functions, governed by specific continuity and growth conditions, provide a more precise means of capturing fine details in regularity, such as logarithmic effects, which appear, for instance, in Brownian motion. We divide this work into two main parts: the first one is about generalized interpolation spaces and the second one about pointwise regularity through continued fractions.
In the first Chapter, Boyd functions are studied to establish a solid foundation for defining new spaces. They are decomposed into two germs, leading to a representation Theorem that clarifies their structure. This also provides an opportunity to deepen the understanding of the connections between these functions and the concept of admissible sequences, which are likewise employed in the generalization of spaces. In particular, we demonstrate how to construct an adapted Boyd function from a given admissible sequence. It is noteworthy that Boyd functions are well suited for uniform spaces, whereas admissible sequences are more appropriate for pointwise spaces. Furthermore, an improved version of Merucci's Theorem is presented, facilitating work with generalized functional spaces involving parameter functions of higher regularity.
Next, in the second Chapter, generalized real interpolation methods are examined from a functorial perspective. Classical methods, such as the $K$-method, are extended by incorporating function parameters defined via Boyd functions. It is shown that the $K$-method and the $J$-method remain equivalent in this extended framework, ensuring consistency. A reiteration Theorem is also established, reinforcing the structural robustness of these interpolation spaces.
In the third Chapter, we investigate continuous interpolation spaces defined by function parameters, a construction central to trace theory and the analysis of boundary value problems for partial differential equations. Their functorial interpretation is explored, along with density results in specific cases. These spaces, characterized by asymptotic regularity properties, play a fundamental role in the analysis of operators in weighted functional spaces and in solving PDEs with precisely prescribed boundary behavior. We apply these results to several examples, including H\"older, Lebesgue or Besov spaces. The extreme cases ($\theta = 0$ and $\theta = 1$) are specifically examined, providing an additional reason to use Boyd functions.
In the last Chapter of the first part, the scope is extended from interpolation between two spaces to the interpolation of multiple spaces. We attempt to generalize the results of the second Chapter to this context, emphasizing the usefulness of transitioning to multiple spaces.
The second part focuses on pointwise regularity, examining functions that are not necessarily locally bounded via Calder\'on-Zygmund spaces. A key example is the Brjuno function, which arises in dynamical systems and Diophantine approximation. Generalized versions of this function, linked to $\alpha$-continued fractions, are studied. In the fifth Chapter, we explore the metric properties of these continued fractions, such as the notion of cells in this context, with the aim of providing a solid foundation for studying the pointwise regularity of generalized versions of the Brjuno function. We observe that certain values of $\alpha$ lead to cell structures that are easier to describe.
In the final Chapter, we study the pointwise regularity of the Brjuno-Yoccoz function, which corresponds to $\alpha = 1/2$, and we note that its regularity is identical to that of the standard Brjuno function: the behavior at each point is inversely proportional to the irrationality exponent at that point. Thomae's function is also investigated, an emblematic example of function that is discontinuous at rational points yet continuous elsewhere. This analysis offers a refined perspective on its irregularity, leveraging classical analytical tools to elucidate its fractal nature.
