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In particular, the text addresses an introductory study of the toroidal Besov spaces, which appear in many applications to partial di erential equations with periodic conditions and in harmonic analysis. This work can be very useful for undergraduate and graduate students in Mathematics as well as for researchers interested in the topics mentioned above. ) ) [1] => Array ( [type] => 04 [typeonixlist] => 153 [audience] => 00 [audienceonixlist] => 154 [content] => Array ( [spa] => ContentsPrefacio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . xi1 Spaces of functions on Rn. .. . . . . . .. . . . . . . . . . . . .. . . . 11.1 The space of test functions . . . . . . . . . . . . . . . . . . . . 21.2 The space of rapidly decreasing functions. . . . . . . . 41.3 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . 51.4 Continuous embeddings . . . . . . . . . . . . . . . . . . . . . 61.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Spaces of functions on the n-dimensional torus . . . . . . . . . . . . . . . .92.1 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The n-dimensional torus . . . . . . . . . . . . . . . . . . . . . 132.3 Function spaces on Tn and identifications . . . . . . . . . . . . 152.4 Lp(Tn, E) spaces . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 The space C∞(Tn, E) of test functions on the torus . . . . . . 253 Distributions on Tn and Zn . . . . . . . . . . . . . . . . 273.1 Toroidal distributions . . . . . . . . . . . . . . . . . . . . . . . 273.2 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . 323.3 Toroidal Fourier transform . . . . . . . . . . . . . . . . . . . . 403.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Toroidal Besov spaces . . . . . . . . . . . . . . . . 514.1 Resolution of unity . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Definition and properties of Bs pq(Tn, E) . . . . . . . . . . . . . 564.3 Lifting property . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Embeddings in Besov spaces . . . . . . . . . . . . . . . . . . . 755 Interpolation of toroidal Besov spaces . . . . . . . . . . . . . . . . 795.1 Compatible spaces . . . . . . . . . . . . . . . . . . . . . . . . 805.2 The K-functional . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Real interpolation space and properties . . . . . . . . . . . . . 845.4 Real interpolation for the spaces Bs pq(Tn, E) . . . . . . . . . . 90Bibliografy . . . . . . . . . . . . . . . . . . . . . . . . 101List of symbols . . . . . . . . . . . . . . . . . . . . . . 105Index . . . . . . . . . . . . . . . . . . . . . . . . 107 ) ) ) [contributor] => Array ( [0] => Array ( [noshare] => [simehid] => 7ab1d6919524e8e6472fe2430052d30c [role] => A01 [roleonixlist] => 17 [name] => Bienvenido Barraza Martínez [nameinverted] => Barraza Martínez, Bienvenido [professionalaffiliation] => Array ( [0] => Array ( [professionalposition] => Array ( [0] => Array ( [value] => [lang] => spa ) ) ) ) [prize] => Array ( [0] => Array ( [name] => Array ( [0] => Array ( [value] => [lang] => spa ) ) ) ) [biography] => Array ( [0] => Array ( [value] => Doctor en Ciencias Naturales (Dr.rer.nat) del Fachbereich Mathematik de la Johannes Gutenberg – Universität Mainz, Alemania. Maestría en Matemáticas Universidad Nacional de Colombia sede Medellín Lic. en Matemáticas y Física de la Universidad del Atlántico, Barranquilla-Colombia. Profesor Asociado del departamento de Matemáticas y Estadística. Líneas de investigación: Análisis Funcional y Ecuaciones Diferenciales, en particular, teoría operadores pseudodiferenciales y ecuaciones de evolución. 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Matemático de la Universidad del Atlántico. Profesor Instructor del Departamento de Matemáticas y Estadística. Áreas e intereses de investigación: Análisis Funcional, Ecuaciones Diferenciales Parciales, Teoría de Control, Didáctica e investigación de aula en matemáticas. 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Maestría en Matemáticas-Universidad del Valle -Uninorte (Colombia). Licenciado en Matemáticas y Física de la Universidad del Atlántico, Barranquilla-Colombia. Profesor Asociado del departamento de Matemáticas y Estadística. Líneas de investigación: Análisis Funcional y Ecuaciones Diferenciales, en particular, teoría operadores pseudodiferenciales y ecuaciones de evolución. 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Contents
Prefacio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . xi
1 Spaces of functions on Rn. .. . . . . . .. . . . . . . . . . . . .. . . . 1
1.1 The space of test functions . . . . . . . . . . . . . . . . . . . . 2
1.2 The space of rapidly decreasing functions. . . . . . . . 4
1.3 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Continuous embeddings . . . . . . . . . . . . . . . . . . . . . 6
1.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Spaces of functions on the n-dimensional torus . . . . . . . . . . . . . . . .9
2.1 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The n-dimensional torus . . . . . . . . . . . . . . . . . . . . . 13
2.3 Function spaces on Tn and identifications . . . . . . . . . . . . 15
2.4 Lp(Tn, E) spaces . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The space C∞(Tn, E) of test functions on the torus . . . . . . 25
3 Distributions on Tn and Zn . . . . . . . . . . . . . . . . 27
3.1 Toroidal distributions . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Toroidal Fourier transform . . . . . . . . . . . . . . . . . . . . 40
3.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Toroidal Besov spaces . . . . . . . . . . . . . . . . 51
4.1 Resolution of unity . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Definition and properties of Bs pq(Tn, E) . . . . . . . . . . . . . 56
4.3 Lifting property . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Embeddings in Besov spaces . . . . . . . . . . . . . . . . . . . 75
5 Interpolation of toroidal Besov spaces . . . . . . . . . . . . . . . . 79
5.1 Compatible spaces . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 The K-functional . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Real interpolation space and properties . . . . . . . . . . . . . 84
5.4 Real interpolation for the spaces Bs pq(Tn, E) . . . . . . . . . . 90
Bibliografy . . . . . . . . . . . . . . . . . . . . . . . . 101
List of symbols . . . . . . . . . . . . . . . . . . . . . . 105
Index . . . . . . . . . . . . . . . . . . . . . . . . 107
Doctor en Ciencias Naturales (Dr.rer.nat) del Fachbereich Mathematik de la Johannes Gutenberg – Universität Mainz, Alemania. Maestría en Matemáticas Universidad Nacional de Colombia sede Medellín Lic. en Matemáticas y Física de la Universidad del Atlántico, Barranquilla-Colombia. Profesor Asociado del departamento de Matemáticas y Estadística. Líneas de investigación: Análisis Funcional y Ecuaciones Diferenciales, en particular, teoría operadores pseudodiferenciales y ecuaciones de evolución.
Magister en Matemáticas de la Universidad del Norte. Matemático de la Universidad del Atlántico. Profesor Instructor del Departamento de Matemáticas y Estadística. Áreas e intereses de investigación: Análisis Funcional, Ecuaciones Diferenciales Parciales, Teoría de Control, Didáctica e investigación de aula en matemáticas.
Doctor en Ciencias Naturales (Dr.rer.nat) del Fachbereich Mathematik de la Johannes Gutenberg – Universität Mainz, Alemania. Maestría en Matemáticas-Universidad del Valle -Uninorte (Colombia). Licenciado en Matemáticas y Física de la Universidad del Atlántico, Barranquilla-Colombia. Profesor Asociado del departamento de Matemáticas y Estadística. Líneas de investigación: Análisis Funcional y Ecuaciones Diferenciales, en particular, teoría operadores pseudodiferenciales y ecuaciones de evolución.
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