Titlebook: An Introductory Course in Lebesgue Spaces; Rene Erlin Castillo,Humberto Rafeiro Textbook 2016 Springer International Publishing Switzerlan

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Convex Functions and Inequalities,important tools in Analysis in general and not only in Convex Analysis. In this chapter we will introduce the notion of convexity in its various formulations, and we give some characterizations of convex functions and a few applications of convexity, namely, some classical inequalities as well as no

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Lebesgue Sequence Spaceserties of the spaces, e.g., completeness, separability, duality, and embedding. We also examine the validity of Hölder, Minkowski, Hardy, and Hilbert inequality which are related to the aforementioned spaces. Although Lebesgue sequence spaces can be obtained from Lebesgue spaces using a discrete mea

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Lebesgue Spacesh function spaces. In this chapter we will study these spaces and this study will be used in the subsequent chapters. After introducing the space as a normed space, we also obtain denseness results, embedding properties and study the Riesz representation theorem using two different proofs. Weak conv

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Lorentz Spacesaces which depend now on two parameters. Our first task therefore will be to define the Lorentz spaces and derive some of their properties, like completeness, separability, normability, duality among other topics, e.g., Hölder’s type inequality, Lorentz sequence spaces, and the spaces .exp and .log.

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Nonstandard Lebesgue Spaces., in the modeling of electrorheological fluids, thermorheological fluids, in the study of image processing, in differential equations with nonstandard growth, among others. Thus, naturally, new fine scales of function spaces have been introduced, namely variable exponent spaces and grand spaces. In

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Interpolation of Operators The underlying idea is to obtain boundedness of an operator based on the available information in the endpoints. In the first section we will deal with the Riesz-Thorin interpolation theorem, also known as the complex method, and give some applications, viz. Hausdorff-Young inequality and Young’s i

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Maximal Operatorve covering lemmas of Vitali type. After the covering lemmas we will study in some detail the maximal operator in Lebesgue spaces and show the Lebesgue differentiation theorem as well as a Theorem of Cotlar. We introduce and study the class of locally log-Hölder continuous functions in order to show

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