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Maximal functions. --- Littlewood-Paley theory. --- Fourier transformations. --- Riesz spaces. --- Flag manifolds. --- Hardy spaces. --- Functions of several real variables. --- Functions of complex variables.

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"In this monograph, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood- Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderon reproducing formulae in the flag setting and a version of the Plancherel-Polya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure"--

Hardy spaces. --- Maximal functions. --- Littlewood-Paley theory. --- Singular integrals. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- $H^p$-spaces. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Maximal functions, Littlewood-Paley theory. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Singular and oscillatory integrals (Calderón-Zygmund, etc.).

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Various applications of equimeasurable function rearrangements to the ''best constant"-type problems are considered in this volume. Several classical theorems are presented along with some very recent results. In particular, the text includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for bounded mean oscillation (BMO) functions with sharp exponent, a refinement of the Gurov-Reshetnyak lemma, sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, reverse Hölder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians involved with these topics. .

Maximal functions. --- Fourier analysis. --- Spaces of measures. --- Analysis, Fourier --- Mathematical analysis --- Measures, Spaces of --- Function spaces --- Measure theory --- Topological spaces --- Fourier analysis --- Functions of several real variables --- Maxima and minima --- Global analysis (Mathematics). --- Functional analysis. --- Analysis. --- Fourier Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis

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"Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1"--

Differential equations, Elliptic. --- Degenerate differential equations. --- Harmonic functions. --- Harmonic analysis. --- Boundary value problems. --- Potential theory -- Higher-dimensional theory -- Harmonic, subharmonic, superharmonic functions. --- Partial differential equations -- Elliptic equations and systems -- Boundary value problems for second-order elliptic equations. --- Measure and integration -- Classical measure theory -- Length, area, volume, other geometric measure theory. --- Measure and integration -- Classical measure theory -- Hausdorff and packing measures. --- Partial differential equations -- Elliptic equations and systems -- Degenerate elliptic equations. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Singular and oscillatory integrals (Calderón-Zygmund, etc.). --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Maximal functions, Littlewood-Paley theory. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Harmonic analysis and PDE.