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Commutative Harmonic Analysis I pp 167–259Cite as

Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory

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Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 15))

Abstract

The integral convolution operator in ℝn

$$ Tf\left( x \right) = \int\limits_{{\mathbb{R}^n}} {k\left( {x - y} \right)f\left( y \right)dy} $$
((0.1))

is well defined and bounded in L p(ℝn), 1≤p ≤ ∞, as k∈ L l(ℝn). However, operators very similar to (0.1), but having a non-integrable kernel, appear quite frequently in applications. The Hilbert transform on the real line l is a typical example.

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Editors and Affiliations

  1. Department of Mathematics, Leningrad University, Staryj Peterhof, 198094, Leningrad, USSR

    V. P. Khavin

  2. Steklov Mathematical Institute, Fontanka 27, 191011, Leningrad, USSR

    N. K. Nikol’skij

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Dyn’kin, E.M. (1991). Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory. In: Khavin, V.P., Nikol’skij, N.K. (eds) Commutative Harmonic Analysis I. Encyclopaedia of Mathematical Sciences, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02732-5_3

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