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The description for this book, Calculus on Heisenberg Manifolds. (AM-119), Volume 119, will be forthcoming.

Hypoelliptic operators. --- Calculus. --- Differentiable manifolds. --- Adjoint. --- Affine transformation. --- Approximation. --- Asymptotic expansion. --- Calculation. --- Codimension. --- Complex geometry. --- Complex manifold. --- Computation. --- Convolution. --- De Rham cohomology. --- Derivative. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Estimation. --- Fourier integral operator. --- Fourier transform. --- Function space. --- Heat equation. --- Heisenberg group. --- Hilbert space. --- Homogeneous function. --- Hypoelliptic operator. --- Identity element. --- Integration by parts. --- Invertible matrix. --- Manifold. --- Nilpotent group. --- Parametrix. --- Partial differential equation. --- Pointwise product. --- Pointwise. --- Polynomial. --- Principal part. --- Pseudo-differential operator. --- Riemannian manifold. --- Self-adjoint. --- Several complex variables. --- Singular integral. --- Smoothing. --- Structure constants. --- Subset. --- Summation. --- Tangent bundle. --- Theorem. --- Transpose. --- Unit circle. --- Vector field.

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The description for this book, Recent Developments in Several Complex Variables. (AM-100), Volume 100, will be forthcoming.

Complex analysis --- Functions of several complex variables. --- Complex variables --- Several complex variables, Functions of --- Functions of complex variables --- Analytic continuation. --- Analytic function. --- Analytic set. --- Analytic space. --- Asymptotic expansion. --- Automorphic function. --- Axiom. --- Base change. --- Bergman metric. --- Betti number. --- Big O notation. --- Bilinear form. --- Boundary value problem. --- CR manifold. --- Canonical bundle. --- Cauchy problem. --- Cauchy–Riemann equations. --- Characteristic variety. --- Codimension. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Commutative property. --- Commutator. --- Compactification (mathematics). --- Complete intersection. --- Complete metric space. --- Complex dimension. --- Complex manifold. --- Complex number. --- Complex plane. --- Complex projective space. --- Complex space. --- Complex-analytic variety. --- Degeneracy (mathematics). --- Dense set. --- Determinant. --- Diffeomorphism. --- Differentiable function. --- Dimension (vector space). --- Dimension. --- Eigenvalues and eigenvectors. --- Embedding. --- Existential quantification. --- Explicit formulae (L-function). --- Fermat curve. --- Fiber bundle. --- Fundamental solution. --- Gorenstein ring. --- Hartogs' extension theorem. --- Hilbert space. --- Hilbert transform. --- Holomorphic function. --- Homotopy. --- Hyperfunction. --- Hypersurface. --- Hypoelliptic operator. --- Interpolation theorem. --- Irreducible component. --- Isometry. --- Linear map. --- Manifold. --- Maximal ideal. --- Monic polynomial. --- Monotonic function. --- Multiple integral. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Open set. --- Orthogonal group. --- Parametrization. --- Permutation. --- Plurisubharmonic function. --- Polynomial. --- Principal bundle. --- Principal part. --- Principal value. --- Projection (linear algebra). --- Projective line. --- Proper map. --- Quadratic function. --- Real projective space. --- Resolution of singularities. --- Riemann surface. --- Riemannian manifold. --- Sectional curvature. --- Sheaf cohomology. --- Special case. --- Submanifold. --- Subset. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Uniqueness theorem. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Fonctions de variables complexes --- Colloque

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Many of the operators one meets in several complex variables, such as the famous Lewy operator, are not locally solvable. Nevertheless, such an operator L can be thoroughly studied if one can find a suitable relative parametrix--an operator K such that LK is essentially the orthogonal projection onto the range of L. The analysis is by far most decisive if one is able to work in the real analytic, as opposed to the smooth, setting. With this motivation, the author develops an analytic calculus for the Heisenberg group. Features include: simple, explicit formulae for products and adjoints; simple representation-theoretic conditions, analogous to ellipticity, for finding parametrices in the calculus; invariance under analytic contact transformations; regularity with respect to non-isotropic Sobolev and Lipschitz spaces; and preservation of local analyticity. The calculus is suitable for doing analysis on real analytic strictly pseudoconvex CR manifolds. In this context, the main new application is a proof that the Szego projection preserves local analyticity, even in the three-dimensional setting. Relative analytic parametrices are also constructed for the adjoint of the tangential Cauchy-Riemann operator.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Pseudodifferential operators. --- Functions of several complex variables. --- Solvable groups. --- Analytic function. --- Analytic set. --- Associative property. --- Asymptotic expansion. --- Atkinson's theorem. --- Banach space. --- Bilinear map. --- Boundary value problem. --- Bounded function. --- Bounded operator. --- Bump function. --- C space. --- CR manifold. --- Cauchy problem. --- Cauchy's integral formula. --- Cauchy–Schwarz inequality. --- Cayley transform. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Coefficient. --- Cokernel. --- Combinatorics. --- Complex conjugate. --- Complex number. --- Complexification (Lie group). --- Contact geometry. --- Convolution. --- Darboux's theorem (analysis). --- Darboux's theorem. --- Diagram (category theory). --- Diffeomorphism. --- Difference "ient. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorial. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fundamental solution. --- Heisenberg group. --- Hermitian adjoint. --- Hilbert space. --- Hodge theory. --- Hypoelliptic operator. --- Hölder's inequality. --- Implicit function theorem. --- Integral transform. --- Invertible matrix. --- Leibniz integral rule. --- Lie algebra. --- Mathematical induction. --- Mathematical proof. --- Mean value theorem. --- Multinomial theorem. --- Neighbourhood (mathematics). --- Neumann series. --- Nilpotent group. --- Orthogonal transformation. --- Orthonormal basis. --- Oscillatory integral. --- Paley–Wiener theorem. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Partition of unity. --- Plancherel theorem. --- Polynomial. --- Power function. --- Power series. --- Product rule. --- Property B. --- Pseudo-differential operator. --- Pullback (category theory). --- Quadratic form. --- Regularity theorem. --- Riesz transform. --- Schwartz space. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sesquilinear form. --- Several complex variables. --- Singular integral. --- Special case. --- Summation. --- Support (mathematics). --- Symmetrization. --- Theorem. --- Topology. --- Triangle inequality. --- Unbounded operator. --- Union (set theory). --- Unitary transformation. --- Variable (mathematics).

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This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Definite integrals. --- Orbit method. --- Bianchi identity. --- Brownian motion. --- Casimir operator. --- Clifford algebras. --- Clifford variables. --- Dirac operator. --- Euclidean vector space. --- Feynman-Kac formula. --- Gaussian integral. --- Gaussian type estimates. --- Heisenberg algebras. --- Kostant. --- Leftschetz formula. --- Littlewood-Paley decomposition. --- Malliavin calculus. --- Pontryagin maximum principle. --- Selberg's trace formula. --- Sobolev spaces. --- Toponogov's theorem. --- Witten complex. --- action functional. --- complexification. --- conjugations. --- convergence. --- convexity. --- de Rham complex. --- displacement function. --- distance function. --- elliptic Laplacian. --- elliptic orbital integrals. --- fixed point formulas. --- flat bundle. --- general kernels. --- general orbital integrals. --- geodesic flow. --- geodesics. --- harmonic oscillator. --- heat kernel. --- heat kernels. --- heat operators. --- hypoelliptic Laplacian. --- hypoelliptic deformation. --- hypoelliptic heat kernel. --- hypoelliptic heat kernels. --- hypoelliptic operators. --- hypoelliptic orbital integrals. --- index formulas. --- index theory. --- infinite dimensional orbital integrals. --- keat kernels. --- local index theory. --- locally symmetric space. --- matrix part. --- model operator. --- nondegeneracy. --- orbifolds. --- orbital integrals. --- parallel transport trivialization. --- probabilistic construction. --- pseudodistances. --- quantitative estimates. --- quartic term. --- real vector space. --- refined estimates. --- rescaled heat kernel. --- resolvents. --- return map. --- rough estimates. --- scalar heat kernel. --- scalar heat kernels. --- scalar hypoelliptic Laplacian. --- scalar hypoelliptic heat kernels. --- scalar hypoelliptic operator. --- scalar part. --- semisimple orbital integrals. --- smooth kernels. --- standard elliptic heat kernel. --- supertraces. --- symmetric space. --- symplectic vector space. --- trace formula. --- unbounded operators. --- uniform bounds. --- uniform estimates. --- variational problems. --- vector bundles. --- wave equation. --- wave kernel. --- wave operator.