Topic
Locally integrable function
About: Locally integrable function is a research topic. Over the lifetime, 831 publications have been published within this topic receiving 11391 citations.
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Book•
17 Nov 1995
TL;DR: In this paper, the interplay between function space theory and potential theory is discussed. And a surprisingly large part of classical potential theory has been extended to this nonlinear setting, sometimes surprising, usually they are nontrivial and have required new methods.
Abstract: The subject of this book is the interplay between function space theory and potential theory. A crucial step in classical potential theory is the identification of the potential energy of a charge with the square of a Hilbert space norm. This leads to the Dirichlet space of locally integrable functions whose gradients are square integrable. More recently, a generalized potential theory has been developed, which has an analogous relationship to the standard Banach function spaces, Sobolev spaces, Besov spaces etc., that appear naturally in the study of partial differential equations. A surprisingly large part of classical potential theory has been extended to this nonlinear setting. The extensions are sometimes surprising, usually they are nontrivial and have required new methods.
1,681 citations
TL;DR: In this paper, the authors investigated the properties of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact group and every reasonable function space on G. They showed that inclusions, the quality of embeddings, reflexivity and minimality and maximality of co-orbit spaces can be completely characterized by the same properties of corresponding sequence spaces.
Abstract: We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact groupG and every reasonable function space onG. Whereas Part I was devoted to atomic decompositions of such spaces, Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples (cf. Part III) one recovers several and often difficult theorems with ease.
378 citations
1 Jan 1994
TL;DR: In this paper, the basic ideas of the theory of distributions are presented and a brief but solid introduction to the theory is given, particularly to those aspects that are important in the theory for asymptotic expansions.
Abstract: The purpose of this chapter is to present the basic ideas of the theory of distributions. Distributions or generalized functions, as they are also known, have proved to be very useful in many branches of pure and applied mathematics. Many textbooks, monographs and articles have been written on their theory and their applications [12], [23], [53], [63], [64], [69], [71], [80], [97], [111]. Our present aim is to give a brief but solid introduction to the theory of distributions, particularly to those aspects that are important in the theory of asymptotic expansions.
373 citations
TL;DR: In this paper, the authors prove the L 1 -contraction principle and uniqueness of solutions for quasilinear elliptic-parabolic equations of the form[formula] where b is monotone nondecreasing and continuous.
340 citations
TL;DR: In this article, it was shown that the convergence rate of λ 0(n) to inf ǫ depends only on the order ρ (not necessarily even or integer or finite).
157 citations
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| Year | Papers |
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| 2025 | 2 |
| 2024 | 3 |
| 2023 | 9 |
| 2022 | 29 |
| 2021 | 29 |
| 2020 | 32 |