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Volume 64, Issue 6


The L p primitive integral

Erik Talvila
Published Online: 2015-01-10 | DOI: https://doi.org/10.2478/s12175-014-0288-5


For each 1 ≤ p < ∞, a space of integrable Schwartz distributions L′p, is defined by taking the distributional derivative of all functions in L p. Here, L p is with respect to Lebesgue measure on the real line. If f ∈ L′p such that f is the distributional derivative of F ∈ L p, then the integral is defined as $\int\limits_{ - \infty }^\infty {fG} = - \int\limits_{ - \infty }^\infty {F(x)g(x)dx} $, where g ∈ L q, $G(x) = \int\limits_0^x {g(t)dt} $ and 1/p + 1/q =1. A norm is ‖f‖p′ = ‖F‖p. The spaces L′p and L p are isometrically isomorphic. Distributions in L′p share many properties with functions in L p. Hence, L′p is reflexive, its dual space is identified with L q, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract L-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes L′1 into a Banach algebra isometrically isomorphic to the convolution algebra on L 1. Spaces of higher order derivatives of L p functions are defined. These are also Banach spaces isometrically isomorphic to L p.

MSC: Primary 46E30, 46F10, 46G12; Secondary 42A38, 42A85, 46B42, 46C05

Keywords: Lebesgue space; Banach space; Schwartz distribution; generalised function; primitive; integral; Fourier transform; convolution; Banach lattice; Hilbert space; Poisson integral

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About the article

Published Online: 2015-01-10

Published in Print: 2014-12-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 6, Pages 1497–1524, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-014-0288-5.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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