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This work is a fresh presentation of the Ahlfors-Weyl theory of holomorphic curves that takes into account some recent developments in Nevanlinna theory and several complex variables. The treatment is differential geometric throughout, and assumes no previous acquaintance with the classical theory of Nevanlinna. The main emphasis is on holomorphic curves defined over Riemann surfaces, which admit a harmonic exhaustion, and the main theorems of the subject are proved for such surfaces. The author discusses several directions for further research.

Analytic functions. --- Functions, Meromorphic. --- Value distribution theory. --- Meromorphic functions --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Distribution of values theory --- Functions, Entire --- Functions, Meromorphic --- Addition. --- Algebraic curve. --- Algebraic number. --- Atlas (topology). --- Binomial coefficient. --- Cauchy–Riemann equations. --- Compact Riemann surface. --- Compact space. --- Complex manifold. --- Complex projective space. --- Computation. --- Continuous function (set theory). --- Covariant derivative. --- Critical value. --- Curvature form. --- Diagram (category theory). --- Differential form. --- Differential geometry of surfaces. --- Differential geometry. --- Dimension. --- Divisor. --- Essential singularity. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Gaussian curvature. --- Geodesic curvature. --- Geometry. --- Grassmannian. --- Harmonic function. --- Hermann Weyl. --- Hermitian manifold. --- Holomorphic function. --- Homology (mathematics). --- Hyperbolic manifold. --- Hyperplane. --- Hypersurface. --- Improper integral. --- Intersection number (graph theory). --- Isometry. --- Line integral. --- Manifold. --- Meromorphic function. --- Minimal surface. --- Nevanlinna theory. --- One-form. --- Open problem. --- Open set. --- Orthogonal complement. --- Parameter. --- Picard theorem. --- Product metric. --- Q.E.D. --- Remainder. --- Riemann sphere. --- Riemann surface. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent space. --- Tangent. --- Theorem. --- Three-dimensional space (mathematics). --- Unit circle. --- Unit vector. --- Vector field. --- Volume element. --- Volume form. --- Fonctions de plusieurs variables complexes

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This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student.

Harmonic analysis. Fourier analysis --- Harmonic analysis --- Semigroups --- 517.986.6 --- Lie groups --- Littlewood-Paley theory --- #WWIS:d.d. Prof. L. Bouckaert/BOUC --- Fourier analysis --- Functions of several real variables --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Harmonic analysis of functions of groups and homogeneous spaces --- Harmonic analysis. --- Littlewood-Paley theory. --- Lie groups. --- Semigroups. --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces --- Addition. --- Analytic function. --- Axiom. --- Boundary value problem. --- Central limit theorem. --- Change of variables. --- Circle group. --- Classification theorem. --- Commutative property. --- Compact group. --- Complex analysis. --- Convex set. --- Coset. --- Covering space. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- E6 (mathematics). --- E7 (mathematics). --- E8 (mathematics). --- Elementary proof. --- Equation. --- Equivalence class. --- Existence theorem. --- Existential quantification. --- Fourier analysis. --- Fourier series. --- Fourier transform. --- Function space. --- General linear group. --- Haar measure. --- Harmonic function. --- Hermite polynomials. --- Hilbert transform. --- Homogeneous space. --- Homomorphism. --- Ideal (ring theory). --- Identity matrix. --- Indecomposability. --- Integral transform. --- Invariant measure. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Lebesgue measure. --- Legendre polynomials. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local diffeomorphism. --- Markov process. --- Martingale (probability theory). --- Matrix group. --- Measurable function. --- Measure (mathematics). --- Multiple integral. --- Normal subgroup. --- One-dimensional space. --- Open set. --- Ordinary differential equation. --- Orthogonality. --- Orthonormality. --- Parseval's theorem. --- Partial differential equation. --- Probability space. --- Quadratic form. --- Rank of a group. --- Regular representation. --- Riemannian manifold. --- Riesz transform. --- Schur orthogonality relations. --- Scientific notation. --- Semigroup. --- Sequence. --- Special case. --- Stone–Weierstrass theorem. --- Sturm–Liouville theory. --- Subgroup. --- Subset. --- Summation. --- Tensor algebra. --- Tensor product. --- Theorem. --- Theory. --- Topological group. --- Topological space. --- Torus. --- Trigonometric polynomial. --- Trivial representation. --- Uniform convergence. --- Unitary operator. --- Unitary representation. --- Vector field. --- Vector space. --- Lie, Groupes de --- Analyse harmonique