Locally integrable function (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Locally integrable function" in English language version.

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According to Gel'fand & Shilov (1964, p. 3). Gel'fand, I. M.; Shilov, G. E. (1964) [1958], Generalized functions. Vol. I: Properties and operations, New York–London: Academic Press, pp. xviii+423, ISBN 978-0-12-279501-5, MR 0166596, Zbl 0115.33101 {{citation}}: ISBN / Date incompatibility (help). Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals.
See for example (Schwartz 1998, p. 18) and (Vladimirov 2002, p. 3). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
Another slight variant of this definition, chosen by Vladimirov (2002, p. 1), is to require only that $K ⋐ Ω$ (or, using the notation of Gilbarg & Trudinger (2001, p. 9), $K \subset\subset Ω$ ), meaning that $K$ is strictly included in $Ω$ i.e. it is a set having compact closure strictly included in the given ambient set. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002.
This is the approach developed for example by Cafiero (1959, pp. 285–342) and by Saks (1937, chapter I), without dealing explicitly with the locally integrable case. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
See for example (Strichartz 2003, pp. 12–13). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
This approach was praised by Schwartz (1998, pp. 16–17) who remarked also its usefulness, however using Definition 1 to define locally integrable functions. Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.
See for example (Vladimirov 2002, p. 3) and (Maz'ya & Poborchi 1997, p. 4). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033.
As remarked in the previous section, this is the approach adopted by Maz'ya & Shaposhnikova (2009), without developing the elementary details. Maz'ya, Vladimir G.; Shaposhnikova, Tatyana O. (2009), Theory of Sobolev multipliers. With applications to differential and integral operators, Grundlehren der Mathematischen Wissenschaft, vol. 337, Heidelberg: Springer-Verlag, pp. xiii+609, ISBN 978-3-540-69490-8, MR 2457601, Zbl 1157.46001.
See for example (Vladimirov 2002, p. 3), where a calligraphic ℒ is used. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Gilbarg & Trudinger 2001, p. 147), (Maz'ya & Poborchi 1997, p. 5) for a statement of this results, and also the brief notes in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
Gilbarg & Trudinger (2001, p. 147) and Maz'ya & Poborchi (1997, p. 5) only sketch very briefly the method of proof, while in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2) it is assumed as a known result, from which the subsequent development starts. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
According to Saks (1937, p. 36), "If $E$ is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure ( $μ$ ), then, in order that an additive function of a set ( $𝔛$ ) on $E$ be absolutely continuous on $E$ , it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of $E$ ". Assuming ( $μ$ ) to be the Lebesgue measure, the two statements can be seen to be equivalent. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
See for example (Hörmander 1990, p. 37). Hörmander, Lars (1990), The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft, vol. 256 (2nd ed.), Berlin-Heidelberg-New York City: Springer-Verlag, pp. xii+440, ISBN 0-387-52343-X, MR 1065136, Zbl 0712.35001 (available also as ISBN 3-540-52343-X).
See (Strichartz 2003, p. 12). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
See (Schwartz 1998, p. 19). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.
See (Vladimirov 2002, pp. 19–21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Vladimirov 2002, p. 21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
For a brief discussion of this example, see (Schwartz 1998, pp. 131–132). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.

books.google.com

According to Gel'fand & Shilov (1964, p. 3). Gel'fand, I. M.; Shilov, G. E. (1964) [1958], Generalized functions. Vol. I: Properties and operations, New York–London: Academic Press, pp. xviii+423, ISBN 978-0-12-279501-5, MR 0166596, Zbl 0115.33101 {{citation}}: ISBN / Date incompatibility (help). Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals.
See for example (Schwartz 1998, p. 18) and (Vladimirov 2002, p. 3). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
Another slight variant of this definition, chosen by Vladimirov (2002, p. 1), is to require only that $K ⋐ Ω$ (or, using the notation of Gilbarg & Trudinger (2001, p. 9), $K \subset\subset Ω$ ), meaning that $K$ is strictly included in $Ω$ i.e. it is a set having compact closure strictly included in the given ambient set. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002.
See for example (Strichartz 2003, pp. 12–13). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
See for example (Vladimirov 2002, p. 3) and (Maz'ya & Poborchi 1997, p. 4). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033.
As remarked in the previous section, this is the approach adopted by Maz'ya & Shaposhnikova (2009), without developing the elementary details. Maz'ya, Vladimir G.; Shaposhnikova, Tatyana O. (2009), Theory of Sobolev multipliers. With applications to differential and integral operators, Grundlehren der Mathematischen Wissenschaft, vol. 337, Heidelberg: Springer-Verlag, pp. xiii+609, ISBN 978-3-540-69490-8, MR 2457601, Zbl 1157.46001.
See for example (Vladimirov 2002, p. 3), where a calligraphic ℒ is used. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Gilbarg & Trudinger 2001, p. 147), (Maz'ya & Poborchi 1997, p. 5) for a statement of this results, and also the brief notes in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
Gilbarg & Trudinger (2001, p. 147) and Maz'ya & Poborchi (1997, p. 5) only sketch very briefly the method of proof, while in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2) it is assumed as a known result, from which the subsequent development starts. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
See (Strichartz 2003, p. 12). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
See (Vladimirov 2002, pp. 19–21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Vladimirov 2002, p. 21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.

icm.edu.pl

pldml.icm.edu.pl

This is the approach developed for example by Cafiero (1959, pp. 285–342) and by Saks (1937, chapter I), without dealing explicitly with the locally integrable case. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
According to Saks (1937, p. 36), "If $E$ is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure ( $μ$ ), then, in order that an additive function of a set ( $𝔛$ ) on $E$ be absolutely continuous on $E$ , it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of $E$ ". Assuming ( $μ$ ) to be the Lebesgue measure, the two statements can be seen to be equivalent. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.

zbmath.org

According to Gel'fand & Shilov (1964, p. 3). Gel'fand, I. M.; Shilov, G. E. (1964) [1958], Generalized functions. Vol. I: Properties and operations, New York–London: Academic Press, pp. xviii+423, ISBN 978-0-12-279501-5, MR 0166596, Zbl 0115.33101 {{citation}}: ISBN / Date incompatibility (help). Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals.
See for example (Schwartz 1998, p. 18) and (Vladimirov 2002, p. 3). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
Another slight variant of this definition, chosen by Vladimirov (2002, p. 1), is to require only that $K ⋐ Ω$ (or, using the notation of Gilbarg & Trudinger (2001, p. 9), $K \subset\subset Ω$ ), meaning that $K$ is strictly included in $Ω$ i.e. it is a set having compact closure strictly included in the given ambient set. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002.
This is the approach developed for example by Cafiero (1959, pp. 285–342) and by Saks (1937, chapter I), without dealing explicitly with the locally integrable case. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
See for example (Strichartz 2003, pp. 12–13). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
This approach was praised by Schwartz (1998, pp. 16–17) who remarked also its usefulness, however using Definition 1 to define locally integrable functions. Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.
See for example (Vladimirov 2002, p. 3) and (Maz'ya & Poborchi 1997, p. 4). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033.
As remarked in the previous section, this is the approach adopted by Maz'ya & Shaposhnikova (2009), without developing the elementary details. Maz'ya, Vladimir G.; Shaposhnikova, Tatyana O. (2009), Theory of Sobolev multipliers. With applications to differential and integral operators, Grundlehren der Mathematischen Wissenschaft, vol. 337, Heidelberg: Springer-Verlag, pp. xiii+609, ISBN 978-3-540-69490-8, MR 2457601, Zbl 1157.46001.
See for example (Vladimirov 2002, p. 3), where a calligraphic ℒ is used. Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Gilbarg & Trudinger 2001, p. 147), (Maz'ya & Poborchi 1997, p. 5) for a statement of this results, and also the brief notes in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
Gilbarg & Trudinger (2001, p. 147) and Maz'ya & Poborchi (1997, p. 5) only sketch very briefly the method of proof, while in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2) it is assumed as a known result, from which the subsequent development starts. Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002. Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033. Maz'ja, Vladimir G. (1985), Sobolev Spaces, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, ISBN 3-540-13589-8, MR 0817985, Zbl 0692.46023 (available also as ISBN 0-387-13589-8). Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations., Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
According to Saks (1937, p. 36), "If $E$ is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure ( $μ$ ), then, in order that an additive function of a set ( $𝔛$ ) on $E$ be absolutely continuous on $E$ , it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of $E$ ". Assuming ( $μ$ ) to be the Lebesgue measure, the two statements can be seen to be equivalent. Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warsaw-Lwów: G.E. Stechert & Co., pp. VI+347, JFM 63.0183.05, MR 0167578, Zbl 0017.30004. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
See for example (Hörmander 1990, p. 37). Hörmander, Lars (1990), The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft, vol. 256 (2nd ed.), Berlin-Heidelberg-New York City: Springer-Verlag, pp. xii+440, ISBN 0-387-52343-X, MR 1065136, Zbl 0712.35001 (available also as ISBN 3-540-52343-X).
See (Strichartz 2003, p. 12). Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2nd printing ed.), River Edge, NJ: World Scientific Publishers, pp. x+226, ISBN 981-238-430-8, MR 2000535, Zbl 1029.46039.
See (Schwartz 1998, p. 19). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.
See (Vladimirov 2002, pp. 19–21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
See (Vladimirov 2002, p. 21). Vladimirov, V. S. (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR 2012831, Zbl 1078.46029. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.
For a brief discussion of this example, see (Schwartz 1998, pp. 131–132). Schwartz, Laurent (1998) [1966], Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French) (Nouvelle ed.), Paris: Hermann Éditeurs, pp. xiii+420, ISBN 2-7056-5551-4, MR 0209834, Zbl 0149.09501.