[vc_row][vc_column][vc_tta_accordion shape=»square» c_icon=»chevron» c_position=»right» active_section=»» no_fill=»true» collapsible_all=»true»][vc_tta_section title=»Resumen» tab_id=»resumen»][vc_column_text]
The proposed research project falls within the field of Harmonic Analysis. This area of study is connected to other fields of work, including partial differential equations, function spaces, and signal theory. In Spain, harmonic analysis is a major focus of research, and several universities have internationally renowned research groups in this branch of analysis. In recent years, significant results have been obtained on linear and multilinear commutators associated with fractional and singular integrals in different spaces. We propose to study the boundedness and compactness of commutators for Riesz transforms, fractional integrals, and Littlewood-Paley operators in Lebesgue and Morrey spaces within the context of the Bessel operator. Within the framework of weighted Lebesgue spaces, we aim to find the optimal space in relation to the initial value problem associated with the heat equation, which contains fractional powers of differential operators in divergent form, including the fractional Laplacian as a special case. Harmonic analysis in the context of orthogonal systems (Hermite, Jacobi, Laguerre) has been an active area of research in the last decade. We propose to extend the study of Riesz transforms and the spectral multipliers associated with these systems to spaces of differential forms. Shell potentials are a useful tool for analyzing the existence and uniqueness of solutions to different boundary value problems (Dirichlet, Neumann, and regularity) for partial differential equations. We intend to use this procedure to analyze the solvability of this class of problems associated with equations in divergent form, with complex matrices of variable coefficients, and perturbed by a potential. The development of harmonic analysis in Gaussian contexts requires specific methods. We pose some problems within this framework. In particular, we propose to study Hardy spaces associated with Laguerre polynomial expansions and Littlewood-Paley operators for non-symmetric Ornstein-Uhlenbeck semigroups. By connecting wavelets with vector harmonic analysis, we aim to characterize different Banach-valued function spaces (BMO, Hardy) defined on homogeneous spaces using their coefficients with respect to wavelet bases. Related to the Laplacian operator p(.), where p is a function, the field of variable-exponent Lebesgue, Besov, Sobolev spaces has been developed. We intend to study variable-exponent Hardy and Hardy-Lorentz spaces associated with anisotropies and also in the context of martingales.
[/vc_column_text][/vc_tta_section][vc_tta_section title=»Abstract» tab_id=»abstract»][vc_column_text]
The proposed research project fits in the Harmonic Analysis. This area of study is connected with other fields. Partial differential equations, function spaces and the theory of signals are some of them. In Spain harmonic analysis employs many researchers, and several universities have working groups with international prestige in this branch of analysis. In recent years important results on linear and multilinear commutators associated with fractional and singular integrals in different spaces have been achieved. We intend to study the boundedness and compactness of commutators for Riesz transforms, fractional integrals and Littlewood-Paley operators in Lebesgue and Morrey spaces in the context of Bessel operators. In the framework of Lebesgue spaces with weights, we plan to find optimal spaces in relation to the initial value problem associated with the heat equation containing fractional powers of differential operators with divergent form, including as a special case the fractional Laplacian. Harmonic analysis in the context of orthogonal systems (Hermite, Jacobi, Laguerre, …) has been an active area of work in the last decade. We intend to extend the study of Riesz transforms and spectral multipliers associated with these systems to spaces of differential forms. Layer potentials are a useful tool for analyzing the existence and uniqueness of solutions for different problems of boundary value (Dirichlet, Neumann and regularity) for partial differential equations. Our purpose is to use this procedure to analyze the solubility of this class of problems associated with divergent form partial differential equations involving complex matrices with variable coefficients and perturbed with potentials. The development of harmonic analysis in Gaussian contexts requires specific methods. We want to study some problems in this framework. In particular, we propose to investigate Hardy spaces associated with Laguerre polynomials expansions and Littlewood-Paley operators for non-symmetrical Ornstein-Uhlenbeck semigroups. Connecting wavelets with harmonic analysis in a vector valued context, we aim to characterize different spaces of functions with values in a Banach space (BMO, Hardy, …) defined on spaces of homogeneous type, with respect to their wavelet coefficients bases. Related to the operator p(.)- Laplacian where p is a function, the field of variable exponent spaces (Lebesgue, Besov, Sobolev, …) has been developed. We intend to study Hardy and Hardy Lorentz variable exponent spaces associated with anisotropy and also in the context of martingales.
[/vc_column_text][/vc_tta_section][/vc_tta_accordion][/vc_column][/vc_row]