Real analysis in non Euclidean contexts such as Lie groups, (sub)-Riemannian manifolds, graphs, fractals or more general metric spaces has evolved quite spectacularly in the recent years. At the same time, a huge amount of work was done on analysis of differential operators with non-smooth coefficients. In these subjects, the properties of the heat semigroup on functions or on differential forms played an essential role. On the other hand, it is crucial to understand the geometrical aspects that are involved when considering problems outside the Euclidean context. There are several challenging problems in which analysis and geometry are intimately linked, and an extremely good knowledge of both aspects is necessary to attack them. This is what we wish to do in the framework of this ANR project. We intend to gather well-known experts and young researchers in analysis and geometry in order to tackle new problems at the interface of these two directions. We develop further connections between real analysis and geometry. We shall investigate the heat kernel on Riemannian manifolds together with bounds on its spacial gradient, the heat kernel and heat semigroup on 1-différential forms (the semigroup of the Hodge-de Rham operator), L^p-boundedness of the Riesz transform, Sobolev and Besov algebras in the setting of sub-Gaussian bounds, the heat kernel and functional inequalities in the setting of sub-Riemannian geometry (such as the Heisenberg group), spectral estimates of Schrödinger operators, bi-parameter analysis, spectral multipliers, parabolic equations/systems, boundary value problems, maximal regularity for non autonomous equations... Most of these problems are at the interface of real analysis and geometry.

Here are some questions that we want to address:

**Question 1 :**

We intend to understand the geometry which encodes Gaussian upper bounds for the heat kernel of 1-differential forms (which hold for manifolds with nonnegative Ricci curvature). In a similar problem, we also intend to generalize to the case of Riemannian manifolds the results obtained by Arnaudon, Bonnefont and Joulin [Interwinnings and generalized Brascamp-Lieb inequalities. *Revista Math. Iberoam. *to appear https://arxiv.org/pdf/1602.03836.pdf] about intertwinings between modified gradients and semigroups and Brascamp-Lieb inequalities in R*n *endowed with a probability measure.

**Question 2**

We conjecture that on the connected sum of two Euclidean space of dimension $n>2$, then the Hardy space $H^p_{1,d}$ of exact 1-forms is an $L^p$ space if and only if $1<p<n$. On such a manifold, it would be also very interesting to characterize $H^p_{1,d}$ for$ p \ge n$.

**Question 3**

One purpose is to prove the inequality $|d*e^{-**tL}* *f *| \le C *e^{**-ctL}*(|d *f *|)$ (where $d$ stands for the “horizontal” gradient and $L$ the associated diffusion operator), under the assumption that the generalized Bakry-Emery curvature criterion (introduced by Baudoin, Bonnefont and Garofalo) holds.We also want to deal with Bochner’s identity in the sub-Riemannian context. In Riemannian geometry, Bochner’s identity and the theory of Jacobi fields lead to a long list of important results for Riemannian manifolds such as Li-Yau estimates, Poincaré inequality, Bishop- Gromov inequality, Bonnet-Myers theorem etc. The extensions of such results to the setting of sub-Riemannian geometries are challenging problems. A possible approach is to exploit the generalized Bakry-Emery curvature criterion.

**Question 4 :**

We aim to investigate the properties of the wave propagator in more general situations on Riemannian manifolds and try to get a better understanding of how it is related to the corresponding heat semigroup. This could lead to weak dispersive estimates for the Schrödinger group.

**Question 5 :**

The notion of fundamental solutions of parabolic operators with real coefficients (and elliptic principal part) is quite understood. We plan to investigate the case of complex coefficients (in which case the solution is a distribution and not a function). On the other hand, we aim to present a unified and consistent theory of fundamental solutions for parabolic problems with non-smooth coefficients that allows to deal with boundary value problems (Dirichlet or Neumann).

**Question 6:**

We wish to understand what could be a bi-parameter analogue of the sparse control. Up to our knowledge, nothing is known in the biparameter world in such a direction ... and we would also like to give some new results. It is unclear if the linear $*A_*2$ bounds are true for a biparameter (even simple) operator and we would like to find a counterexample. We would like also to understand what kind of geometrical control, we could expect. By geometrical, we mean a sum over a collection of rectangles of some elementary operators.