## Final Report Summary - CZOSQP (Noncommutative Calderón-Zygmund theory, operator space geometry and quantum probability)

Our research is focused in developing modern techniques from harmonic analysis in noncommutative scenarios, understanding the interplay with other fields like quantum probability/operator space theory and also finding applications back in classical harmonic analysis.

The analysis of linear operators associated to singular kernels is a central topic in harmonic analysis and partial differential equations. A large subfamily of these maps falls under the scope of Calderón-Zygmund theory, which exploits the relation between metric and measure in the underlying space to give sufficient conditions for Lp boundedness. Namely, the Hörmander smoothness condition for the kernel or the CZ decomposition combine the notions of proximity in terms of the metric with that of smallness in terms of the measure. The doubling and polynomial growth conditions between metric and measure allow to extend Calderón-Zygmund theory to nonEuclidean spaces. The existence of such a nice metric in the underlying space is always given in the literature, an assumption which makes impossible to follow the classical approach to develop a CZ theory for noncommutative von Neumann algebras. The existente of such a theory would allow to face important questions, like Lp boundedness of Fourier multipliers/convergence of Fourier series in discrete group von Neumann algebras. Also, we are interested in Sobolev embedding theorems and logarithmic Sobolev inequalities via hypercontractivity on these algebras and other related topics. On the other hand, our research also leads us naturally to related problems in classical harmonic analysis and martingale theory. So far, our main achievements in the project are:

A criterium for Lp boundedness of Fourier multipliers over locally compact (frequency) groups in terms of smoothness the multiplier (Hörmander--Mihlin) and the geometric structure of the group (cohomology).

Dimension free estimates for noncommutative Riesz tranforms beyond the results of Gundy, Varopoulos, Meyer, Bakry, Pisier and Lust-Piquard. This includes fractional Laplacians in Rn or the standard length over free groups.

An algebraic formulation of Calderón-Zygmund theory under the mild assumption that the space comes equipped with a Markov semigroup, not with a metric. This includes noncommutative von Neumann algebras.

A characterization (via Kakeya sets of directions) of Lp boundedness for u-directional Hilbert transforms (semispace Fourier multipliers) on twisted products in terms of the u-orbit of the crossed product action.

Optimal and nearly optimal time hypercontractivity estimates for the Markov process associated to the word length for free groups, cyclic groups and triangular groups. Logarithmic Sobolev inequalities are also given.

A new notion of lacunarity and a characterization for the Lp boundedness of direccional maximal operators in higher dimensions (n>2) in terms of this geometric behavior, solving a classical problem in harmonic analysis.

A new link between martingale BMO spaces for nonregular filtrations and Tolsa’s RBMO space for nondoubling measures. This includes martingale atomic blocks and dyadic forms of RBMO with deep applications.

Transference results for Fourier multipliers with frequencies in different (locally compact) groups related by restriction (subspaces), compactification (Bohr) and periodization (quotients).

The analysis of linear operators associated to singular kernels is a central topic in harmonic analysis and partial differential equations. A large subfamily of these maps falls under the scope of Calderón-Zygmund theory, which exploits the relation between metric and measure in the underlying space to give sufficient conditions for Lp boundedness. Namely, the Hörmander smoothness condition for the kernel or the CZ decomposition combine the notions of proximity in terms of the metric with that of smallness in terms of the measure. The doubling and polynomial growth conditions between metric and measure allow to extend Calderón-Zygmund theory to nonEuclidean spaces. The existence of such a nice metric in the underlying space is always given in the literature, an assumption which makes impossible to follow the classical approach to develop a CZ theory for noncommutative von Neumann algebras. The existente of such a theory would allow to face important questions, like Lp boundedness of Fourier multipliers/convergence of Fourier series in discrete group von Neumann algebras. Also, we are interested in Sobolev embedding theorems and logarithmic Sobolev inequalities via hypercontractivity on these algebras and other related topics. On the other hand, our research also leads us naturally to related problems in classical harmonic analysis and martingale theory. So far, our main achievements in the project are:

A criterium for Lp boundedness of Fourier multipliers over locally compact (frequency) groups in terms of smoothness the multiplier (Hörmander--Mihlin) and the geometric structure of the group (cohomology).

Dimension free estimates for noncommutative Riesz tranforms beyond the results of Gundy, Varopoulos, Meyer, Bakry, Pisier and Lust-Piquard. This includes fractional Laplacians in Rn or the standard length over free groups.

An algebraic formulation of Calderón-Zygmund theory under the mild assumption that the space comes equipped with a Markov semigroup, not with a metric. This includes noncommutative von Neumann algebras.

A characterization (via Kakeya sets of directions) of Lp boundedness for u-directional Hilbert transforms (semispace Fourier multipliers) on twisted products in terms of the u-orbit of the crossed product action.

Optimal and nearly optimal time hypercontractivity estimates for the Markov process associated to the word length for free groups, cyclic groups and triangular groups. Logarithmic Sobolev inequalities are also given.

A new notion of lacunarity and a characterization for the Lp boundedness of direccional maximal operators in higher dimensions (n>2) in terms of this geometric behavior, solving a classical problem in harmonic analysis.

A new link between martingale BMO spaces for nonregular filtrations and Tolsa’s RBMO space for nondoubling measures. This includes martingale atomic blocks and dyadic forms of RBMO with deep applications.

Transference results for Fourier multipliers with frequencies in different (locally compact) groups related by restriction (subspaces), compactification (Bohr) and periodization (quotients).