Singular Integrals in Quantum Euclidean Spaces (Memoirs of the American Mathematical Society)

Gonzalez-Perez, Adrian M./ Junge, Marius/ Parcet, Javier

American Mathematical Society（2022/05発売）

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製本 Paperback:紙装版/ペーパーバック版
言語 ENG
商品コード 9781470449377
DDC分類 515.4

Full Description

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 results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calder'on-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce Lp-boundedness and Sobolev p-estimates for regular, exotic and forbidden symbols in the expected ranks. In the L2 level both Calder'on-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove Lp-regularity of solutions for elliptic PDEs.