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# Mathematica Slovaca

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# The L p primitive integral

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

## Abstract

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.

• [1] ALVAREZ, J.— GUZMÁN-PARTIDA, M.— PÉREZ-ESTEVA, S.: Harmonic extensions of distributions, Math. Nachr. 280 (2007), 1443–1466. http://dx.doi.org/10.1002/mana.200510558

• [2] ANG, D. D.— SCHMITT, K.— VY, L. K.: A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 355–371. Google Scholar

• [3] AXLER, S.— BOURDON, P.— RAMEY, W.: Harmonic Function Theory, Springer-Verlag, New York, 2001. http://dx.doi.org/10.1007/978-1-4757-8137-3

• [4] BARROS-NETO, J.: An Introduction to the Theory of Distributions, Marcel Dekker, New York, 1973. Google Scholar

• [5] BARTLE, R. G.: Elements of Integration, Wiley, New York, 1966. Google Scholar

• [6] ČELIDZE, V. G.— DŽVARŠEĭŠVILI, A. G.: The Theory of the Denjoy Integral and some Applications, World Scientific, Singapore, 1989. http://dx.doi.org/10.1142/0935

• [7] CICHOCKA, A.— KIERAT, W.: An application of the Wiener functions to the Dirichlet problem of the Laplace equation, Integral Transforms Spec. Funct. 7 (1998), 13–20. http://dx.doi.org/10.1080/10652469808819182

• [8] CLARKSON, J. A.: Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414. http://dx.doi.org/10.1090/S0002-9947-1936-1501880-4

• [9] FOLLAND, G. B.: Real Analysis, Wiley, New York, 1999. Google Scholar

• [10] FRIEDLANDER, F. G.— JOSHI, M.: Introduction to the Theory of Distributions, Cambridge University Press, Cambridge, 1999. Google Scholar

• [11] LEADER, S.: The Kurzweil-Henstock Integral and Its Differentials, Marcel Dekker, New York, 2001. Google Scholar

• [12] LIEB, E. H.— LOSS, M.: Analysis, Amer. Math. Soc., Providence, RI, 2001. Google Scholar

• [13] MIKUSIŃSKI, P.— OSTASZEWSKI, K.: Embedding Henstock integrable functions into the space of Schwartz distributions, Real Anal. Exchange 14 (1988–89), 24–29. Google Scholar

• [14] RUDIN, W.: Real and Complex Analysis, McGraw-Hill, New York, 1987. Google Scholar

• [15] SCHWARTZ, L.: Th`eorie des distributions, Hermann, Paris, 1966. Google Scholar

• [16] TALVILA, E.: The Distributional Denjoy Integral, Real Anal. Exchange 33 (2008), 51–82. Google Scholar

• [17] TALVILA, E.: Convolutions with the continuous primitive integral, Abstr. Appl. Anal. 2009 (2009), Art. ID 307404.

• [18] TALVILA, E.: The regulated primitive integral, Illinois J. Math. 53 (2009), 1187–1219. Google Scholar

• [19] TALVILA, E.: Integrals and Banach spaces for finite order distributions, Czechoslovak Math. J. 62 (2012), 77–104. http://dx.doi.org/10.1007/s10587-012-0018-5

• [20] YOSIDA, K.: Functional Analysis, Springer-Verlag, Berlin, 1980. http://dx.doi.org/10.1007/978-3-642-61859-8

• [21] ZEMANIAN, A. H.: Distribution Theory and Transform Analysis, Dover, New York, 1987. Google Scholar

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,

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