Abstract:
The paper puts forward new compactness criteria for spaces of summable and measurable functions on a metric space with measure satisfying the doubling condition. These criteria are formulated in terms of either local smoothness inequalities or maximal operators that measure local smoothness.
Bibliography: 28 titles.
Keywords:
compactness, total boundedness, space of summable functions, space of measurable functions, maximal operators, local smoothness.
\Bibitem{Kro12}
\by V.~G.~Krotov
\paper Criteria for compactness in $L^p$-spaces, $p\geqslant0$
\jour Sb. Math.
\yr 2012
\vol 203
\issue 7
\pages 1045--1064
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Linking options:
https://www.mathnet.ru/eng/sm7828
https://doi.org/10.1070/SM2012v203n07ABEH004253
https://www.mathnet.ru/eng/sm/v203/i7/p129
This publication is cited in the following 19 articles:
J. Huang, Y. Nessipbayev, F. Sukochev, D. Zanin, “Compactness criteria in quasi-Banach symmetric operator spaces associated with a non-commutative torus”, Journal of Functional Analysis, 2025, 110946
M. S. Ermakov, “O ravnomernoi sostoyatelnosti neparametricheskikh kriteriev”, Veroyatnost i statistika. 35, Posvyaschaetsya yubileyu Yany Isaevny BELOPOLSKOI, Zap. nauchn. sem. POMI, 526, POMI, SPb., 2023, 78–89
Michał Dymek, Przemysław Górka, “Compactness in the spaces of variable integrability and summability”, Mathematische Nachrichten, 296:9 (2023), 4317
Bedrossian J., Blumenthal A., Punshon-Smith S., “A Regularity Method For Lower Bounds on the Lyapunov Exponent For Stochastic Differential Equations”, Invent. Math., 227:2 (2022), 429–516
N. N. Romanovskii, “Sobolev embedding theorems and their generalizations for maps defined on topological spaces with measures”, Moscow University Mathematics Bulletin, 77:1 (2022), 27–40
Bandaliyev R.A., Gorka P., Guliyev V.S., Sawano Y., “Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces”, Mediterr. J. Math., 18:6 (2021), 232
J. Xu, “Precompact Sets in Bochner–Lebesgue Spaces
with Variable Exponen”, Math. Notes, 110:6 (2021), 932–941
Guo W., Zhao G., “On Relatively Compact Sets in Quasi-Banach Function Spaces”, Proc. Amer. Math. Soc., 148:8 (2020), 3359–3373
Gorka P., Pospiech P., “Banach Function Spaces on Locally Compact Groups”, Ann. Funct. Anal., 10:4 (2019), 460–471
Bandaliyev R., Gorka P., “Relatively Compact Sets in Variable-Exponent Lebesgue Spaces”, Banach J. Math. Anal., 12:2 (2018), 331–346
R. A. Bandaliev, S. G. Hasanov, “On denseness of $C_0^\infty(\Omega)$ and compactness in $L_{p(x)}(\Omega)$ for $0<p(x)<1$”, Mosc. Math. J., 18:1 (2018), 1–13
N. N. Romanovskiǐ, “Sobolev embedding theorems and generalizations for functions on a metric measure space”, Siberian Math. J., 59:1 (2018), 126–135
A. I. Porabkovich, “Samouluchshenie $L^p$-neravenstva Puankare pri $p>0$”, Chebyshevskii sb., 17:1 (2016), 187–200
P. Gorka, H. Rafeiro, “From Arzelà–Ascoli to Riesz–Kolmogorov”, Nonlinear Anal., 144 (2016), 23–31
S. A. Bondarev, V. G. Krotov, “Fine properties of functions from Hajłasz–Sobolev classes $M_{\alpha}^p$, $p>0$. I. Lebesgue points”, J. Contemp. Math. Anal., 51:6 (2016), 282–295
V. G. Krotov, A. I. Porabkovich, “Estimates of $L^p$-Oscillations of Functions for $p>0$”, Math. Notes, 97:3 (2015), 384–395
N. N. Romanovskiǐ, “Embedding theorems and a variational problem for functions on a metric measure space”, Siberian Math. J., 55:3 (2014), 511–529
N. N. Romanovskiǐ, “Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings”, Siberian Math. J., 54:2 (2013), 353–367
Veniamin G. Krotov, Springer Proceedings in Mathematics & Statistics, 25, Recent Advances in Harmonic Analysis and Applications, 2012, 197
Citation in format AMSBIB
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