In the theory of pseudodifferential operators, one of the most essential topics is the study of mapping properties of pseudodifferential operators between various kinds of function spaces. The investigation of $L_p$-boundedness of pseudodifferential operators is particularly important, considering its consequences for the regularity and existence of solutions of PDEs.
The purpose of this talk is to discuss the counterpart of this problem on noncommutative tori. Noncommutative tori are the most intensively studied noncommutative spaces in noncommutative geometry and arise in various parts of mathematics and mathematical physics. Pseudodifferential calculus on noncommutative tori was introduced in early 1980s by A. Connes, and it has emerged as an indispensable tool in the recent study of differential geometry of noncommutative tori. Meanwhile, J. Rosenberg introduced the notion of Riemannian metric on noncommutative tori a decade ago. In this talk, I will first recall the notion of a curved noncommutative torus, i.e., a noncommutative torus endowed with a Riemannian metric in the sense of J. Rosenberg. I will then show the boundedness of pseudodifferential operators on noncommutative $L_p$-spaces associated with the volume form induced by a Riemannian metric. Based on joint work with V. Kumar.