Mean oscillation and related function spaces

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My research lies in the general area of harmonic analysis (Fourier analysis), which has at its core the idea of decomposing a function (which can represent a sound signal or an image) into basic components which are in some sense simpler.  It is then easier to act on these components with operators, or transformations, such as those arising in the solution of partial differential equations.  In order to reconstruct the whole from its parts, which are usually infinite in number, one needs to have a notion of convergence, as well as boundedness of the operators.   This is where the importance of function spaces comes into the picture, namely choosing the appropriate class of functions in which to take our input, and determining what is the appropriate class for the output.  The finer the function spaces we choose, the better is our understanding of the behavior of these operators.  One of the main challenges is to be able to recognize a function space in different guises, prove inclusion results and distinguish different spaces. Another is to understand how the function spaces relate to the geometric setting of the problem. The proposed research is largely motivated by the following question: what do we know about a function from control of its mean oscillation? Mean oscillation measures how much, on the average, the function deviates from its mean on a given set. While much is already known, we still understand significantly more about the relationship between a function and its derivative, for example, than between the mean oscillation and the function. The space of functions of bounded mean oscillation (BMO) was introduced by John and Nirenberg in 1961, motivated by questions in elasticity theory, and imposes uniform control of the mean oscillation over all subsets. Variants of this space have been widely studied recently and can give more nuanced information about a function in terms of size and smoothness.  Our understanding of mean oscillation can also be applied to probability and statistics, where it has important connections to Brownian motion and stochastic differential equations. A celebrated result of C. Fefferman links BMO with the Hardy space H1.  Hardy spaces have played an essential role in harmonic analysis since the early 20th century, initially in relation to the convergence of Fourier series, and more recently in connection with partial differential equations.  Of particular interest are "local" or non-homogeneous versions of Hardy spaces, and the corresponding BMO spaces, which are well suited to certain types of partial differential equations, as well as allowing more flexibility in the underlying geometry.  In many applications, one only considers the problem in a bounded setting, for example in the case of the lake equations of fluid dynamics.  The shape of the domain and its boundary play a crucial role.  In other situations, results from Euclidean space need to be extended to a different setting, for example graphs.