preharmonicity

Preharmonicity is a concept in mathematics, specifically in the field of harmonic analysis, which is a branch of Fourier analysis. It refers to the study of functions that are not necessarily harmonic but share certain properties with harmonic functions. Harmonic functions are solutions to Laplace's equation and have smooth, regular behavior. Preharmonic functions, on the other hand, may exhibit more irregular or singular behavior but still possess some of the fundamental properties of harmonic functions, such as mean-value properties or subharmonicity.

The concept of preharmonicity was introduced to extend the theory of harmonic functions to more general classes of functions. This extension is useful in various areas of mathematics, including complex analysis, potential theory, and partial differential equations. Preharmonic functions often arise in the study of boundary value problems, where they can provide insights into the behavior of solutions near the boundary of a domain.

One of the key properties of preharmonic functions is that they satisfy a certain maximum principle, which states that the maximum value of a preharmonic function on a domain is attained on the boundary of the domain. This property is analogous to the maximum principle for harmonic functions and is crucial in the study of preharmonic functions.

In recent years, there has been significant interest in the study of preharmonic functions, particularly in the context of non-smooth or irregular domains. This has led to the development of new techniques and tools for analyzing preharmonic functions, as well as applications in areas such as image processing and machine learning. Overall, the study of preharmonicity continues to be an active and vibrant area of research in harmonic analysis.