Abstract: We introduce "local grand" Lebesgue spaces \(L^{p),\theta}_{x_0,a}(\Omega)\), \(0 < p < \infty,\) \(\Omega \subseteq \mathbb{R}^n\), where the process of "grandization" relates to a single point \(x_0\in \Omega\), contrast to the case of usual known grand spaces \(L^{p),\theta}(\Omega)\), where "grandization" relates to all the points of \(\Omega\).
We define the space \(L^{p),\theta}_{x_0,a}(\Omega)\) by means of the weight \(a(|x-x_0|)^{\varepsilon p}\) with small exponent, \(a(0)=0\).
Under some rather wide assumptions on the choice of the local "grandizer" \(a(t)\), we
prove some properties of these spaces including their equivalence under different choices of the grandizers \(a(t)\) and show that the maximal, singular and Hardy operators preserve such a "single-point grandization" of Lebesgue spaces \(L^p(\Omega)\), \(1<p<\infty\), provided that the lower Matuszewska-Orlicz index of the function \(a\) is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.
Keywords: grand space, Lebesgue space, Muckenhoupt weight, maximal operator, singular operator, Hardy operator, Stein-Weiss interpolation theorem, Matuszewska-Orlicz indices
For citation: Samko, S. G. and Umarkhadzhiev, S. M. Local Grand Lebesgue Spaces, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 96-108.
DOI 10.46698/e4624-8934-5248-n
1. Iwaniec, T. and Sbordone, C. On the Integrability of the Jacobian
under Minimal Hypotheses, Arch. Rational Mech. Anal., 1992 , vol. 119,
no. 2, pp. 129-143. DOI: 10.1007/BF00375119.
2. Greco, L., Iwaniec, T. and Sbordone, C. Inverting the \(p\)-Harmonic Operator,
Manuscripta Math., 1997, vol. 92, no. 1, pp. 249-258.
DOI: 10.1007/BF02678192.
3. Fiorenza, A., Gupta, B. and Jain, P. The Maximal Theorem in Weighted
Grand Lebesgue Spaces, Studia Math., 2008, vol. 188, no. 2, pp. 123-133.
DOI: 10.4064/sm188-2-2.
4. Jain, P., Singh, A. P., Singh, M. and Stepanov, V. D. Sawyer's Duality
Principle for Grand Lebesgue Spaces, Math. Nachr., 2019, vol. 292, no. 4, pp. 841-849.
DOI: 10.1002/mana.201700312.
5. Kokilashvili, V. and Meskhi, A. A Note on the Boundedness of the Hilbert Transform
in Weighted Grand Lebesgue Spaces, Georgian Math. J., 2009, vol. 16, no. 3, pp. 547-551.
DOI: 10.1515/GMJ.2009.547.
6. Fiorenza, A., Formica, M. R., Gogatishvili, A., Kopaliani, T. and Rakotoson, J. M.
Characterization of Interpolation Between Grand, Small or Classical Lebesgue Spaces,
Nonlinear Analysis, 2018, vol. 177, pp. 422-453. DOI: 10.1016/j.na.2017.09.005.
7. Kokilashvili, V. and Meskhi, A. Fractional Integrals with Measure
in Grand Lebesgue and Morrey Spaces, Integral Transforms and Special Functions, 2020, pp. 1-15.
DOI: 10.1080/10652469.2020.1833003.
8. Edmunds, D. E., Kokilashvili, V. and Meskhi, A. Sobolev-Type Inequalities
for Potentials in Grand Variable Exponent Lebesgue Spaces,
Mathematische Nachrichten, 2019, vol. 292, no. 10, pp. 2174-2188.
DOI: 10.1002/mana.201800239.
9. Samko, S. G. and Umarkhadzhiev, S. M. On Iwaniec-Sbordone Spaces on Sets
which May Have Infinite Measure, Azerb. J. Math., 2011, vol. 1, no. 1, pp. 67-84.
URL: https://www.azjm.org/volumes/1-1.html.
10. Samko, S. G. and Umarkhadzhiev, S. M. Grand Morrey Type Spaces,
Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp. 104-118.
DOI: 10.46698/c3825-5071-7579-i.
11. Samko, S. and Umarkhadzhiev, S. Riesz Fractional Integrals in Grand Lebesgue Spaces,
Fract. Calc. Appl. Anal., 2016, vol. 19, no. 3, pp. 608-624.
DOI: 10.1515/fca-2016-0033.
12. Samko, S. and Umarkhadzhiev, S. On Grand Lebesgue Spaces on Sets of Infinite Measure,
Mathematische Nachrichten, 2017, vol. 290, no. 5-6, pp. 913-919.
DOI: 10.1002/mana.201600136.
13. Umarkhadzhiev, S. Generalization of the Notion of Grand Lebesgue Space,
Russian Mathematics, 2014, vol. 58, no. 4, pp. 35-43,
DOI: 10.3103/S1066369X14040057.
14. Kokilashvili, V., Meskhi, A., Rafeiro, H. and Samko, S. Integral Operators
in Non-Standard Function Spaces. Vol. 2: Variable exponent Holder, Morrey-Campanato and Grand
Spaces, Birkhauser, 2016.
15. Matuszewska, W. and Orlicz, W. On Some Classes of Functions with Regard to their Orders of Growth,
Studia Math., 1965, vol. 26, pp. 11-24.
16. Samko, N. G. Weighted Hardy Operators in the Local Generalized Vanishing Morrey Spaces,
Positivity, 2013, vol. 17, no. 3, pp. 683-706. DOI: 10.1007/s11117-012-0199-z.
17. Besov, O. V., Il'in, V. P. and Nikol'skii, S. M. Integral Representations
of Functions and Embedding of Functions, Moscow, Nauka, 1975, 480 p. (in Russian).
18. Stein, E. M. and Weiss, G. Interpolation of Operators with Change of Measures,
Trans. Amer. Math. Soc., 1958, vol. 87, pp. 159-172.
19. Bergh, J. and Lofstrom, J. Interpolation Spaces. An Introduction,
Berlin, Springer, 1976.
20. Duoandikoetxea, J. Fourier Analysis,
Graduate Studies in Math., Vol. 29, Providence, Amer. Math. Soc., 2001.
21. Dyn'kin, E. M. and Osilenker, B. B. Weighted Norm Estimates
for Singular Integrals and their Applications,
J. Sov. Math., 1985, vol. 30, pp. 2094-2154.
22. Adams, D. R. and Hedberg, L. I. Function Spaces and Potential Theory,
Berlin, Springer, 1996.
23. Muckenhoupt, B. and Wheeden, R. L. Weighted Norm Inequalities for Fractional Integrals,
Trans. Amer. Math. Soc., 1974, vol. 192, pp. 261-274.
24. Kufner, A., Persson, L. E. and Samko, N. Weighted Inequalities of Hardy Type,
Second Edition, New Jersey, World Scientific, 2017.
25. Opic, B. and Kufner, A. Hardy-Type Inequalities,
Pitman Research Notes in Mathematics Series, Vol. 219,
Harlow, Longman Scientific & Technical, 1990.
26. Persson, E.-L. and Samko, S. G. A Note on the Best Constants in Some Hardy Inequalities,
J. Math. Inequal., 2015, vol. 9, no. 2, pp. 437-447. DOI: 10.7153/jmi-09-37.
27. Umarkhadzhiev, S. M. Integral Operators with Homogeneous Kernels in Grand Lebesgue Spaces,
Mathematical Notes, 2017, vol. 102, no. 5, pp. 710-721. DOI: 10.1134/S0001434617110104.