# Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals

@article{Ha2018PseudodifferentialCO, title={Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals}, author={Hyunsu Ha and Gihyun Lee and Raphael Ponge}, journal={International Journal of Mathematics}, year={2018} }

This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes’ pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detailed description is still missing. In this paper, we focus on constructing an oscillating integral for noncommutative tori and laying down the main functional analysis ground for understanding Connes’ pseudodifferential calculus. In particular, this allows us…

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## References

SHOWING 1-10 OF 82 REFERENCES

### Noncommutative variations on Laplace’s equation

- Mathematics
- 2008

As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the…

### Modular curvature for toric noncommutative manifolds

- MathematicsJournal of Noncommutative Geometry
- 2018

A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun…

### The theory of pseudo-differential operators on the noncommutative n-torus

- Mathematics
- 2018

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a…

### Twisted Crossed Products and Magnetic Pseudodieren tial Operators

- Mathematics
- 2004

There is a connection between the Weyl pseudodieren tial calculus and crossed product C -algebras associated with certain dynamical systems. And in fact both topics are involved in the quantization…

### ANALOGUES OF THE EXPONENTIAL MAP ASSOCIATED WITH COMPLEX STRUCTURES ON NONCOMMUTATIVE TWO-TORI

- Mathematics
- 2004

We define and study analogues of exponentials for functions on noncommutative two-tori that depend on the choice of a complex structure. The major difference with the commutative case is that our…

### Magnetic pseudodifferential operators with coefficients in C*-algebras

- Mathematics
- 2009

In previous articles, a magnetic pseudodifferential calculus and a family of C*-algebras associated with twisted dynamical systems were introduced and the connections between them have been…

### Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work

- Mathematics
- 2018

In this paper, we initiate a systematic approach for the variational calculus used in the study of modular geometry on noncommutative (two) tori. We introduce several transformations on the space of…

### Noncommutative residues for pseudo-differential operators on the noncommutative two-torus

- Mathematics
- 2011

We initiate the notion of a noncommutative residue on classical pseudo-differential operators on the noncommutative two-torus and prove that up to a constant multiple, it is the unique trace on the…

### The Gauss-Bonnet Theorem for the noncommutative two torus

- Mathematics
- 2009

In this paper we shall show that the value at the origin, �(0), of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent…

### Singular integrals in quantum Euclidean spaces

- MathematicsMemoirs of the American Mathematical Society
- 2021

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our…