This thesis consists of two parts. In the first chapter, we give some results on modulation spaces. First the relationship between the classical spaces and the modulation spaces is established. It is proved that certain modulation spaces defined on R² lie in the BMO space. Another result is that the Zak transform, a discrete time-frequency transform, maps a modulation space into a higher dimensional modulation space. And by using these results, an uncertainty principle for Gabor frames via modulation spaces is obtained. In the second part, we deal with optimization of an increasing positively homogeneous functions on the unit simplex. The class of increasing positively homogeneous functions is one of the function classes obtained via min-type functions in the context of abstract convexity. The cutting angle method is used for the minimization of this type functions. The most important step of this method is the minimization of a function which is the maximum of a number of min-type functions on the unit simplex. We propose a numerical algorithm for the minimization of such functions on the unit simplex and we mathematically prove that this algorithm finds the exact solution of the minimization problem. Some experiments have been carried out and the results of the experiments have been presented.

In this thesis, we study local Tb theorems for singular integral operators in the setting of spaces of homogeneous type. We give a direct proof of the local Tb theorem with L^2 integrability on the pseudo- accretive system. Our argument relies on the Beylkin-Coifman-Rokhlin algorithm applied in adapted Haar wavelet basis and some stopping time results. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2, with an additional weak boundedness type hypothesis, which incorporates some Hardy type inequalities. We study the possibility of relaxing the support conditions on the pseudo-accretive system to a slight enlargement of the dyadic cubes. We also give a result in the case when, for practical reasons, hypotheses on the pseudo-accretive system are made on balls rather than dyadic cubes. Finally we study the particular case of perfect dyadic operators for which the proof gets much simpler. Our argument gives us the opportunity to study Hardy type inequalities. The latter are well known in the Euclidean setting, but seem to have been overlooked in spaces of homogeneous type. We prove that they hold without restriction in the dyadic setting. In the more general case of a ball B and its corona 2B\B, they can be obtained from some geometric conditions relative to the distribution of points in the homogeneous space. For example, we prove that some relative layer decay property suffices. We also prove that this property is implied by the monotone geodesic property of Tessera. Finally, we give some explicit examples and counterexamples in the complex plane to illustrate the relationship between the geometry of the homogeneous space and the validity of the Hardy type inequalities.

On étudie dans cette thèse les théorèmes Tb locaux pour les opérateurs d’intégrale singulière, dans le cadre des espaces de type homogène. On donne une preuve directe du théorème Tb local avec hypothèses d’intégrabilité L^2 sur le système pseudo-accrétif. Notre argument repose sur l’algorithme Beylkin-Coifman-Rokhlin, appliqué dans des bases d’ondelettes de Haar adaptées, et sur des résultats de temps d’arrêt. Motivés par une question posée par S. Hofmann, on étend notre résultat au cas où les conditions d’intégrabilité sont inférieures à 2, avec une hypothèse supplémentaire de type faible bornitude, qui incorpore des inégalités de type Hardy. On étudie la possibilité d’affaiblir les conditions de support du système pseudo-accrétif en l’autorisant à être défini sur un petit élargissement des cubes dyadiques. On donne également un résultat dans le cas où, pour des raisons pratiques, les hypothèses sur le système pseudo-accrétif sont faites sur les boules au lieu des cubes dyadiques. Enfin, on s’intéresse au cas des opérateurs parfaitement dyadiques pour lesquels la démonstration est grandement simplifiée. Notre argument nous donne l’opportunité de nous intéresser aux inégalités de type Hardy. Ces estimations sont bien connues des spécialistes dans le cadre Euclidien, mais elles ne semblent pas avoir été étudiées dans les espaces de type homogène. On montre qu’elles sont vérifiées sans restriction dans le cadre dyadique. Dans le cas plus général d’une boule B et de sa couronne 2B\B, elles peuvent être déduites de certaines conditions géométriques de distribution des points dans l’espace de type homogène. Par exemple, on prouve qu’une condition de petite couche relative est suffisante. On montre aussi que cette propriété est impliquée par la propriété de monotonie géodésique de Tessera. Enfin, on présente quelques exemples et contre-exemples explicites dans le plan complexe, afin d’illustrer le lien entre la géométrie de l’espace de type homogène et la validité des inégalités de type Hardy. / In this thesis, we study local Tb theorems for singular integral operators in the setting of spaces of homogeneous type. We give a direct proof of the local Tb theorem with L^2 integrability on the pseudo- accretive system. Our argument relies on the Beylkin-Coifman-Rokhlin algorithm applied in adapted Haar wavelet basis and some stopping time results. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2, with an additional weak boundedness type hypothesis, which incorporates some Hardy type inequalities. We study the possibility of relaxing the support conditions on the pseudo-accretive system to a slight enlargement of the dyadic cubes. We also give a result in the case when, for practical reasons, hypotheses on the pseudo-accretive system are made on balls rather than dyadic cubes. Finally we study the particular case of perfect dyadic operators for which the proof gets much simpler. Our argument gives us the opportunity to study Hardy type inequalities. The latter are well known in the Euclidean setting, but seem to have been overlooked in spaces of homogeneous type. We prove that they hold without restriction in the dyadic setting. In the more general case of a ball B and its corona 2B\B, they can be obtained from some geometric conditions relative to the distribution of points in the homogeneous space. For example, we prove that some relative layer decay property suffices. We also prove that this property is implied by the monotone geodesic property of Tessera. Finally, we give some explicit examples and counterexamples in the complex plane to illustrate the relationship between the geometry of the homogeneous space and the validity of the Hardy type inequalities.