HAPDE

Objective:

"This scientific proposal covers a broad scope of problems within the harmonic analysis in Euclidean and non-Euclidean settings, pursuing applications to partial differential equations (PDEs). Here it is an account of the topics that will be treated. -Estimates concerning directional Rubio de Francia square functions. -Regularity of the HardyLittlewood maximal function in various smooth spaces. -Fourier restriction theory, sharp constants in the inequalities as well as what are the extremizing functions. -Fourier interpolation formulas and uncertainty principles in Fourier analysis, connections to number theory and discrete geometry. -Discrete harmonic analysis and connection with ergodic theory. - Non-commutative theory in nilpotent groups. -Harmonic analysis and nonlocal operators in the Heisenberg group. -Fractional discrete Laplacian and unique continuation -Degenerate Elliptic PDE and sharp degenerate Fractional Poincaré and Poincaré-Sobolev inequalities. -Poincaré-Sobolev inequalities in the multiparameter setting. -Theory of weights: the $C_p$ class in connection with various central operators like square functions; optimal bounds for reverse H\""older inequalities; operators on Lorentz spaces.-Multiparameter harmonic analysis, optimal weighted bounds. -Harmonic analysis in the infinite-dimensional torus. -Multilinear operators in harmonic analysis: weighted norm inequalities of operators and vector-valued extensions in the context of general Banach spaces."