Vanishing mean oscillation and continuity of rearrangements.(English)Zbl 1531.42043
Summary: We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping \(f \mapsto f^\ast\) preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity holds at points in VMO (under certain conditions). This also applies to the symmetric decreasing rearrangement. Many examples are included to illustrate the results.
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Keywords:
vanishing mean oscillation;monotone rearrangement;symmetric decreasing rearrangement;continuity of nonlinear operatorsCite
References:
| [1] | Aldaz, J. M.; Pérez Lázaro, J., Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Am. Math. Soc., 5, 2443-2461 (2007) ·Zbl 1143.42021 |
| [2] | Almgren, F. J.; Lieb, E. H., Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc., 4, 683-773 (1989) ·Zbl 0688.46014 |
| [3] | Baernstein, A., Symmetrization in Analysis. New Mathematical Monographs (2019), Cambridge University Press: Cambridge University Press Cambridge, xviii+473 pp. ·Zbl 1509.32001 |
| [4] | Bourdaud, G., Remarques sur certains sous-espaces de \(\text{BMO}( \mathbb{R}^n)\) et de \(\operatorname{bmo}( \mathbb{R}^n)\). Ann. Inst. Fourier (Grenoble), 4, 1187-1218 (2002), (in French) ·Zbl 1061.46025 |
| [5] | Brezis, H.; Nirenberg, L., Degree theory and BMO. I. Compact manifolds without boundaries. Sel. Math. New Ser., 2, 197-263 (1995) ·Zbl 0852.58010 |
| [6] | Brezis, H.; Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Differ. Equ., 7, 773-789 (1980) ·Zbl 0437.35071 |
| [7] | Burchard, A., Steiner symmetrization is continuous in \(W^{1 , p}\). Geom. Funct. Anal., 5, 823-860 (1997) ·Zbl 0912.46034 |
| [8] | Burchard, A.; Dafni, G.; Gibara, R., Mean oscillation bounds on rearrangements. Trans. Am. Math. Soc., 6, 4429-4444 (2022) ·Zbl 1491.42033 |
| [9] | Butaev, A.; Dafni, G., Approximation and extension of functions of vanishing mean oscillation. J. Geom. Anal., 6892-6921 (2020) ·Zbl 1471.42049 |
| [10] | Carneiro, E.; Madrid, J.; Pierce, L. B., Endpoint Sobolev and BV continuity for maximal operators. J. Funct. Anal., 10, 3262-3294 (2017) ·Zbl 1402.42024 |
| [11] | Chiarenza, F.; Frasca, M.; Longo, P., \( W^{2 , p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc., 2, 841-853 (1993) ·Zbl 0818.35023 |
| [12] | Coron, J.-M., The continuity of the rearrangement in \(W^{1 , p}(\mathbb{R})\). Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 1, 57-85 (1984) ·Zbl 0574.46021 |
| [13] | Dafni, G.; Gibara, R., BMO on shapes and sharp constants, 1-33 ·Zbl 1468.46039 |
| [14] | John, F.; Nirenberg, L., On functions of bounded mean oscillation. Commun. Pure Appl. Math., 415-426 (1961) ·Zbl 0102.04302 |
| [15] | Hansson, K., Imbedding theorems of Sobolev type in potential theory. Math. Scand. (1979) ·Zbl 0437.31009 |
| [16] | Korenovskii, A., Mean Oscillations and Equimeasurable Rearrangements of Functions. Lecture Notes of the Unione Matematica Italiana (2007), Springer/UMI: Springer/UMI Berlin/Bologna, viii+188 pp. ·Zbl 1133.42035 |
| [17] | Krylov, N. V., Parabolic elliptic equations with VMO coefficients. Commun. Partial Differ. Equ., 453-475 (2007) ·Zbl 1114.35079 |
| [18] | Luiro, H., Continuity of the maximal operator in Sobolev spaces. Proc. Am. Math. Soc., 1, 243-251 (2007) ·Zbl 1136.42018 |
| [19] | Madrid, J., Endpoint Sobolev and BV continuity for maximal operators, II. Rev. Mat. Iberoam., 7, 2151-2168 (2019) ·Zbl 1429.42021 |
| [20] | Maz’ya, V.; Mitrea, M.; Shaposhnikova, T., The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients. J. Anal. Math., 167-239 (2010) ·Zbl 1199.35080 |
| [21] | Pólya, G., Problem. Arch. Math. Phys. Ser., 28, 174 (1920) |
| [22] | Pólya, G.; Szegő, G., Problems and Theorems in Analysis, vol. I (1998), Springer ·Zbl 0338.00001 |
| [23] | Sarason, D., Functions of vanishing mean oscillation. Trans. Am. Math. Soc., 391-405 (1975) ·Zbl 0319.42006 |
| [24] | Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series (1971), Princeton University Press: Princeton University Press Princeton, N.J., x+297 pp. ·Zbl 0232.42007 |
| [25] | Uchiyama, A., On the compactness of operators of Hankel type. Tohoku Math. J. (2), 1, 163-171 (1978) ·Zbl 0384.47023 |
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.
