Harmonic analysis | Operator theory | Singular integrals

Singular integral operators of convolution type

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory. (Wikipedia).

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From playlist The Integral

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From playlist The Integral

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From playlist The Integral

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From playlist The Integral

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From playlist The Integral

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From playlist Centro di Ricerca Matematica Ennio De Giorgi

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From playlist Transcendental Functions

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From playlist Probability and Statistics

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From playlist Analysis and its Applications

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From playlist ICTS Colloquia

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From playlist MIT 18.085 Computational Science & Engineering I, Fall 2008

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From playlist Colloquiums MathAlp

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From playlist Integrals

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From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"

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Poisson kernel | Operator norm | Fourier transform | G. H. Hardy | Marcel Riesz | Marcinkiewicz interpolation theorem | Riesz transform | Indicator function | Strong operator topology | Cauchy's integral formula | Lebesgue point | Antoni Zygmund | Rising sun lemma | Lebesgue differentiation theorem | Unit disk | Cauchy principal value | Hölder's inequality | Mathematics | Dual space | Carleson measure | Distribution (mathematics) | Hilbert transform | Riesz–Thorin theorem | Dominated convergence theorem | Hardy space | Hilbert–Schmidt operator | Harmonic conjugate | Harmonic analysis | Hardy–Littlewood maximal function