Laguerre calculus and its applications on the Heisenberg group (Q2758008)

This book provides an introduction to harmonic analysis, complex and real analysis, and partial differential equations on the Heisenberg group by reducing analysis to the Laguerre calculus.NEWLINENEWLINENEWLINEThe table of contents in the book is as follows.NEWLINENEWLINENEWLINE1. The Laguerre calculus.NEWLINENEWLINENEWLINE2. Estimates for powers of the sub-Laplacian.NEWLINENEWLINENEWLINE3. Estimates for the spectrum projection operators of the sub-Laplacian.NEWLINENEWLINENEWLINE4. Inverse of the operator \(\square_\alpha =\sum^n_{j=1} (X^2_j-X^2_{j+n}) -2i\alpha T\).NEWLINENEWLINENEWLINE5. The explicit solution of the \(\overline\partial\)-Neumann problem in a nonisotropic Siegel domain.NEWLINENEWLINENEWLINE6. Injectivity of the Pompeiu transform in the isotropic \(H_n\).NEWLINENEWLINENEWLINE7. Morera-type theorems for holomorphic \({\mathcal H}^p\) spaces in \(H_n(I)\). NEWLINENEWLINENEWLINE8. Morera-type theorems for holomorphic \({\mathcal H}^p\) spaces in \(H_n\)(II). NEWLINENEWLINENEWLINEThe aim of Chapter 1 is to introduce the background that the authors need in the rest of the book to study analysis on the Heisenberg group. Especially, the authors explain the Laguerre calculus which is the symbolic tensor calculus induced by the Laguerre functions on the Heisenberg group. The Laguerre functions play an important role in the Fock-Bargman and Schrödinger representations of the Heisenberg group. The Laguerre calculus is used to study some partial differential operators related to the Kohn Laplacian on the Heisenberg group.NEWLINENEWLINENEWLINEIn Chapter 2, the authors introduce the Heisenberg Laplacian which is closely linked with the \(\overline\partial\)-Neumann problem and \(\overline\partial_b\) complex on a non-degenerate CR manifold. The authors consider the regularity of the fundamental solution for the power of the Heisenberg Laplacian. The authors also compute the kernel of the fundamental solution for powers of the Kohn Laplacian via the heat kernel which can be obtained easily via the Laguerre calculus.NEWLINENEWLINENEWLINEIn Chapter 3, the authors summarize some of \textit{R. Strichartz}'s results [J. Funct. Anal. 96, 350-406 (1991; Zbl 0734.43004)], which are devoted to the theory of harmonic analysis on the isotropic Heisenberg group from the Laguerre calculus point of view.NEWLINENEWLINENEWLINEIn Chapter 4, the authors find the symbol of the inverse of the sub-Laplacian and the corresponding kernel. The authors also derive the solving operator for the wave operator.NEWLINENEWLINENEWLINEIn Chapter 5, the authors introduce the non-isotropic Siegel domain whose boundary is identified with the Heisenberg group. The authors find an explicit formula of the fundamental solutions of the \(\overline \partial\)-Neumann problem and the corresponding heat equation for the non-isotropic Siegel domain by the method of Laguerre calculus.NEWLINENEWLINENEWLINEIn Chapter 6, the authors consider Pompeiu problems in \(L^\infty(H_n)\) on the Heisenberg group \(H_n\) which are either \(U(n)\)-invariant or \(\mathbb{T}^n\)-invariant, and the authors characterize exactly the condition on the radii that are necessary and sufficient for the injectivity of the Pompeiu transform in \(L^\infty (H_n)\) and, a fortiori, in \(L^p(H_n)\), \(1\leq p\leq \infty\).NEWLINENEWLINENEWLINEIn Chapter 7, the authors explain some of the results concerning the Morera-type theorem and show that they are consequences of the Fourier-Laguerre representation of square integrable functions on the Heisenberg group.NEWLINENEWLINENEWLINEIn Chapter 8, the authors obtain a Morera theorem for \(L^\infty\) functions on the Heisenberg group and, a fortiori, for \(L^p\) functions, \(1\leq p\leq\infty\).