Bourgain-Morrey spaces and their applications to boundedness of operators. (English) Zbl 1508.42027
In this paper, the authors study the so-called Bourgain-Morrey spaces. Precisely, the authors study the fundamental properties of the Bourgain-Morrey space, including approximation and interpolation properties. Moreover, the boundedness of operators, such as the Hardy-Littlewood maximal operator, fractional integral operators, fractional maximal operators, and singular integral operators, on the Bourgain-Morrey space is also established. In particular, the embedding result by P. Bégout and A. Vargas [Trans. Am. Math. Soc. 359, No. 11, 5257–5282 (2007; Zbl 1171.35109)] is refined. Furthermore, the authors describe the dual of Bourgain-Morrey spaces and combine this with existing results to conclude the reflexivity of these spaces.
Reviewer: Sibei Yang (Lanzhou)
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B70 | Interpolation between normed linear spaces |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Bourgain-Morrey spaces; fractional integral operators; singular integral operators; dualityCitations:
Zbl 1171.35109References:
| [1] | Alabalik, A.Ç.; Almeida, A.; Samko, S., On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators, Banach J. Math. Anal., 14, 1-14 (2020) · Zbl 1465.46028 |
| [2] | Bégout, P.; Vargas, A., Mass concentration phenomena for the \(L^2\)-critical nonlinear Schrödinger equation, Trans. Am. Math. Soc., 359, 11, 5257-5282 (2007) · Zbl 1171.35109 |
| [3] | Bergh, J.; Löfström, J., Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223 (1976), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0344.46071 |
| [4] | Bergh, J., Relation between the 2 complex methods of interpolation, Indiana Univ. Math. J., 28, 5, 775-778 (1979) · Zbl 0394.41004 |
| [5] | Bourgain, J., On the restriction and multiplier problems in \(\mathbb{R}^3\), (Geometric Aspects of Functional Analysis (1989-90). Geometric Aspects of Functional Analysis (1989-90), Lecture Notes in Math., vol. 1469 (1991), Springer: Springer Berlin), 179-191 · Zbl 0792.42004 |
| [6] | Calderón, A. P., Intermediate spaces and interpolation, the complex method, Stud. Math., 24, 113-190 (1964) · Zbl 0204.13703 |
| [7] | Chiarenza, F.; Frasca, M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7), 7, 3-4, 273-279 (1987), (1988) · Zbl 0717.42023 |
| [8] | Cruz-Uribe, D.; Moen, K.; Rodney, S., Matrix \(A_p\) weights, degenerate Sobolev spaces, and mappings of finite distortion, J. Geom. Anal., 26, 2797-2830 (2016) · Zbl 1357.30015 |
| [9] | Grafakos, L., Classical Fourier Analysis, Texts in Mathematics, vol. 249 (2014), Springer: Springer New York · Zbl 1304.42001 |
| [10] | Hakim, D. I.; Sawano, Y., Calderón’s first and second complex interpolation of closed subspaces of Morrey spaces, J. Fourier Anal. Appl., 23, 5, 1195-1226 (2017) · Zbl 1393.46015 |
| [11] | Hunt, R. A., On \(L(p, q)\) spaces, Ens. Mat., 12, 249-276 (1966) · Zbl 0181.40301 |
| [12] | Izumi, T.; Sato, E.; Yabuta, K., Remarks on a subspace of Morrey spaces, Tokyo J. Math., 37, 1, 185-197 (2014) · Zbl 1304.42059 |
| [13] | Lerner, A. K.; Nazarov, F., Intuitive dyadic calculus: the basics, Expo. Math., 37, 3, 225-265 (2019) · Zbl 1440.42062 |
| [14] | Lemarié-Rieusset, P. G., Multipliers and Morrey spaces, Potential Anal., 38, 3, 741-752 (2013) · Zbl 1267.42024 |
| [15] | Lemarié-Rieusset, P. G., Erratum to: multipliers and Morrey spaces, Potential Anal., 41, 3, 1359-1362 (2014) · Zbl 1302.42031 |
| [16] | Liang, Y.; Sawano, Y.; Ullrich, T.; Yang, D.; Yuan, W., A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Diss. Math., 489 (2013), 114 pp · Zbl 1283.46027 |
| [17] | Masaki, S., Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, Preprint |
| [18] | Masaki, S.; Segata, J., Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 35, 2, 283-326 (2018) · Zbl 1383.35196 |
| [19] | Masaki, S.; Segata, J., Refinement of Strichartz estimates for Airy equation in nondiagonal case and its application, SIAM J. Math. Anal., 50, 3, 2839-2866 (2018) · Zbl 1392.35265 |
| [20] | Mastyło, M.; Sawano, Y., Complex interpolation and Calderón-Mityagin couples of Morrey spaces, Anal. PDE, 12, 7, 1711-1740 (2019) · Zbl 1435.46018 |
| [21] | Mastyło, M.; Sawano, Y., Applications of interpolation methods and Morrey spaces to elliptic PDEs (interpolation methods and Morrey spaces), Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), XXI, 999-1021 (2020) · Zbl 1480.46032 |
| [22] | Merle, F.; Vega, L., Compactness at blow-up time for \(L^2\) solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not., 8, 399-425 (1998) · Zbl 0913.35126 |
| [23] | Moyua, A.; Vargas, A.; Vega, L., Restriction theorems and maximal operators related to oscillatory integrals in \(\mathbb{R}^3\), Duke Math. J., 96, 3, 547-574 (1999) · Zbl 0946.42011 |
| [24] | Ragusa, M. A., Embeddings for Morrey-Lorentz spaces, J. Optim. Theory Appl., 154, 2, 491-499 (2012) · Zbl 1270.46025 |
| [25] | Sawano, Y., A non-dense subspace in \(\mathcal{M}_q^p\) with \(1 < q < p < \infty \), Trans. A. Razmadze Math. Inst., 171, 3, 379-380 (2017) · Zbl 1385.46019 |
| [26] | Sawano, Y.; Di Fazio, G.; Hakim, D. I., Morrey Spaces. Introduction and Applications to Integral Operators and PDE’s, Monographs and Research Notes in Mathematics, vol. 1 (2020), Chapman & Hall CRC Press: Chapman & Hall CRC Press Boca Raton, FL · Zbl 1482.42002 |
| [27] | Sawano, Y.; Ho, K. P.; Yang, D.; Yang, S., Hardy spaces for ball quasi-Banach function spaces, Diss. Math., 525, 1-102 (2017) · Zbl 1392.42021 |
| [28] | Sawano, Y.; Sugano, S., Complex interpolation and the Adams theorem, Potential Anal., 54, 2, 299-305 (2021) · Zbl 1477.42025 |
| [29] | Sawano, Y.; Sugano, S.; Tanaka, H., Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Am. Math. Soc., 363, 12, 6481-6503 (2011) · Zbl 1229.42024 |
| [30] | Sawano, Y.; Tanaka, H., The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo, 22, 663-683 (2015) · Zbl 1334.42051 |
| [31] | Su, W.; Yang, D.; Yuan, W., Interpolations of mixed-norm function spaces, Bull. Malays. Math. Sci. Soc., 45, 1, 153-175 (2022) · Zbl 1486.42039 |
| [32] | Tsutsui, Y., Sharp maximal inequalities and bilinear estimates, (Harmonic Analysis and Nonlinear Partial Differential Equations. Harmonic Analysis and Nonlinear Partial Differential Equations, Suurikaiseki Kôkyûroku Bessatsu, vol. B18 (2010), Res. Inst. Math. Sci. (RIMS): Res. Inst. Math. Sci. (RIMS) Kyoto), 133-146 · Zbl 1218.46021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
