Weighted Inequalities and Lipschitz Spaces

In this thesis I have characterized the trace measures for particular potential spaces of functions defined on R^n, but "mollified" so that the potentials are de facto defined on the upper half-space of R^n. The potential functions are kind Riesz-Bessel. The characterization of trace measures for these spaces is a test condition on elementary sets of the upper half-space. To prove the test condition as sufficient condition for trace measures, I had give an extension to the case of upper half-space of the Muckenhoupt-Wheeden and Wolff inequalities. Finally I characterized the Carleson-trace measures for Besov spaces of discrete martingales. This is a simplified discrete model for harmonic extensions of Lipschitz-Besov spaces.

Magnetic Resonance Imaging-Derived biomarkers of Isocitrate-Dehydrogenase mutation in diffuse gliomas: a conventional MR and Diffusion Weighted Imaging study.

The introduction of molecular criteria into the classification of diffuse gliomas has added interesting practical implications to glioma management. This has created a new clinical need for correlating imaging characteristics with glioma genotypes, also known as radiogenomics or imaging genomics. Whilst many studies have primarily focused on the use of advanced magnetic resonance imaging (MRI) techniques for radiogenomics purposes, conventional MRI sequences still remain the reference point in the study and characterization of brain tumours. Moreover, a different approach may rely on diffusion-weighted imaging (DWI) usage, which is considered a “conventional” sequence in line with recently published directions on glioma imaging. In a non-invasive way, it can provide direct insight into the microscopic physical properties of tissues. Considering that Isocitrate-Dehydrogenase gene mutations may reflect alterations in metabolism, cellularity, and angiogenesis, which may manifest characteristic features on an MRI, the identification of specific MRI biomarkers could be of great interest in managing patients with brain gliomas. My study aimed to evaluate the presence of specific MRI-derived biomarkers of IDH molecular status through conventional MRI and DWI sequences.