Compactness Criteria for Variable $L^p$ Spaces on Metric Measure Spaces with Non-Doubling Measures
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| Format: | Recurso digital |
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2025
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| _version_ | 1866901840734453760 |
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper establishes novel compactness criteria for sets in variable $L^p$ spaces defined on metric measure spaces equipped with non-doubling measures. Classical compactness theorems, such as Fréchet-Kolmogorov, rely heavily on translation invariance and doubling conditions of the underlying measure, which are absent in the setting of non-doubling metric measure spaces. We introduce a generalized framework that accounts for the intrinsic geometric and measure-theoretic irregularities of these spaces. Our criteria are formulated using concepts of uniform integrability with respect to the variable exponent and a novel form of measure-theoretic tightness, adapted to the lack of doubling. The results provide essential tools for analyzing the existence and regularity of solutions to partial differential equations, variational problems, and problems in harmonic analysis in highly irregular domains where standard Lebesgue space theory is insufficient. The presented methodology overcomes significant analytical challenges posed by the non-doubling property, contributing to a deeper understanding of function spaces in complex geometric environments. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17710330 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Compactness Criteria for Variable $L^p$ Spaces on Metric Measure Spaces with Non-Doubling Measures Revista, Zen MATH, 10 This paper establishes novel compactness criteria for sets in variable $L^p$ spaces defined on metric measure spaces equipped with non-doubling measures. Classical compactness theorems, such as Fréchet-Kolmogorov, rely heavily on translation invariance and doubling conditions of the underlying measure, which are absent in the setting of non-doubling metric measure spaces. We introduce a generalized framework that accounts for the intrinsic geometric and measure-theoretic irregularities of these spaces. Our criteria are formulated using concepts of uniform integrability with respect to the variable exponent and a novel form of measure-theoretic tightness, adapted to the lack of doubling. The results provide essential tools for analyzing the existence and regularity of solutions to partial differential equations, variational problems, and problems in harmonic analysis in highly irregular domains where standard Lebesgue space theory is insufficient. The presented methodology overcomes significant analytical challenges posed by the non-doubling property, contributing to a deeper understanding of function spaces in complex geometric environments. |
| title | Compactness Criteria for Variable $L^p$ Spaces on Metric Measure Spaces with Non-Doubling Measures |
| url | https://doi.org/10.5281/zenodo.17710330 |