Generalized parabolic-type potentials and associated new anisotropic wavelet transforms

Classical parabolic Bessel and parabolic Riesz potentials, which are important technical tools of Harmonic Analysis, are defined as negative fractional powers of differential operators $\left( I-\Delta +\partial /\partial t\right) $ and $\left( -\Delta +\partial /\partial t\right) $ , where $\Delta $ is the Laplacian and $I$ is the unit operator. In this thesis, the operators $\mathcal{H}_{\beta }^{\alpha }f=\left( I+(-\Delta )^{\beta /2}+\partial /\partial t\right) ^{-\alpha /\beta }f$ and $H_{\beta }^{\alpha }f=\left( (-\Delta )^{\beta /2}+\partial /\partial t\right) ^{-\alpha /\beta }f$ are defined and their behavior are examined in Lebesgue spaces. These operators being the generalizations of the classical parabolic Bessel and parabolic Riesz potentials, turn into them in special case $\beta =2$. Similar results have been found for singular parabolic-type potentials created by using the Laplace-Bessel differential operator instead of the Laplace operator. Moreover, new anisotropic wavelet transforms are defined with the help of a special "wavelet measure" and a kernel function that generalizes both the classical Gauss and Poisson kernels, and then the Calderon type inversion formulas have been obtained for these transforms. In addition, one-dimensional integral representations relating parabolic type potentials $\mathcal{H}_{\beta }^{\alpha }f$ and $H_{\beta }^{\alpha }f$ with the anisotropic wavelet transforms, are established.