<p>This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. The work in this direction originates in a paper due to Gagliardo (1958), and was continued by Fournier (1988) and by Kolyada (2005).</p><p><p><p>We consider fully anisotropic mixed norm spaces. Our main theorem states an embedding of these spaces into Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic fractional Sobolev spaces and anisotropic Sobolev-Besov spaces. The methods used are based on non-increasing rearrangements and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces $B_p^\alpha$ by Bourgain, Brezis, and Mironescu, and for Besov spaces $B^{\alpha_1,\dots,\alpha_n}_p$ by Kolyada.</p><p>We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces and in the description of Sobolev type embeddings.</p></p></p>

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. The work in this direction originates in a paper due to Gagliardo (1958), and was continued by Fournier (1988) and by Kolyada (2005). We consider fully anisotropic mixed norm spaces. Our main theorem states an embedding of these spaces into Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic fractional Sobolev spaces and anisotropic Sobolev-Besov spaces. The methods used are based on non-increasing rearrangements and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces $B_p^\alpha$ by Bourgain, Brezis, and Mironescu, and for Besov spaces $B^{\alpha_1,\dots,\alpha_n}_p$ by Kolyada. We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces and in the description of Sobolev type embeddings.

Dans cette thèse, nous nous proposons d'étudier et de traiter conjointement plusieurs problèmes phares en traitement d'images incluant le recalage d'images qui vise à apparier deux images via une transformation, la segmentation d'images dont le but est de délimiter les contours des objets présents au sein d'une image, et la décomposition d'images intimement liée au débruitage, partitionnant une image en une version plus régulière de celle-ci et sa partie complémentaire oscillante appelée texture, par des approches variationnelles locales et non locales. Les relations étroites existant entre ces différents problèmes motivent l'introduction de modèles conjoints dans lesquels chaque tâche aide les autres, surmontant ainsi certaines difficultés inhérentes au problème isolé. Le premier modèle proposé aborde la problématique de recalage d'images guidé par des résultats intermédiaires de segmentation préservant la topologie, dans un cadre variationnel. Un second modèle de segmentation et de recalage conjoint est introduit, étudié théoriquement et numériquement puis mis à l'épreuve à travers plusieurs simulations numériques. Le dernier modèle présenté tente de répondre à un besoin précis du CEREMA (Centre d'Études et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement) à savoir la détection automatique de fissures sur des images d'enrobés bitumineux. De part la complexité des images à traiter, une méthode conjointe de décomposition et de segmentation de structures fines est mise en place, puis justifiée théoriquement et numériquement, et enfin validée sur les images fournies. / In this thesis, we study and jointly address several important image processing problems including registration that aims at aligning images through a deformation, image segmentation whose goal consists in finding the edges delineating the objects inside an image, and image decomposition closely related to image denoising, and attempting to partition an image into a smoother version of it named cartoon and its complementary oscillatory part called texture, with both local and nonlocal variational approaches. The first proposed model addresses the topology-preserving segmentation-guided registration problem in a variational framework. A second joint segmentation and registration model is introduced, theoretically and numerically studied, then tested on various numerical simulations. The last model presented in this work tries to answer a more specific need expressed by the CEREMA (Centre of analysis and expertise on risks, environment, mobility and planning), namely automatic crack recovery detection on bituminous surface images. Due to the image complexity, a joint fine structure decomposition and segmentation model is proposed to deal with this problem. It is then theoretically and numerically justified and validated on the provided images.