Abstract
The paper is dedicated to the study of embeddings of the anisotropic Besov spaces \(B_{p,{\theta _1}, \ldots ,{\theta _n}}^{{\beta _1}, \ldots ,{\beta _n}}\) (ℝn) into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents βk tend to 1 (βk < 1). In particular, these results give an extension of the estimate proved by Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces.
One of the key results of the paper is an anisotropic type estimate of rearrangements in terms of partial moduli of continuity.
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References
S. Barza, A. Kamińska, L. E. Persson and J. Soria, Mixed norm and multidimensional Lorentz spaces, Positivity, 10, (2006), 539–554.
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press (Boston, 1988).
O. V. Besov, V. P. Il’in and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, Vols. 1, 2, Winston $ Sons (Washington, DC), Halsted Press (New York, Toronto, London, 1978–1979).
A. P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc., 263, (1981), 149–167.
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in: Optimal Control and Partial Differential Equations, in honour of Professor Alain Ben-soussan’s 60th Birthday. J. L. Menaldi, E. Rofman, A. Sulem (eds), IOS Press (Amsterdam, 2001), pp. 439–455.
J. Bourgain, H. Brezis and P. Mironescu, Limiting Embedding Theorems for Ws,p when s ↑ 1 and applications, J. Anal. Math., 87, (2002), 77–101.
H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat.Nauk, 57, (2002), 59–74 (in Russian); English translation in Russian Math. Surveys, 57 (2002), 693–708.
K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Appl. Math., vol. 28, Queen’s University (Kingston, ON, 1971).
M. L. Gol’dman, Embedding of generalized Nikol’skiĭ–Besov spaces into Lorentz spaces, Trudy Mat. Inst. Steklov, 172, (1985), 128–139 (in Russian); English translation in Proc. Stekolov Inst. Math., 172 (1985), 143–154.
G. E. Karadzhov, M. Milman, and J. Xiao, Limits of higher-order Besov spaces and sharp reiteration theorems, J. Funct. Anal., 221 (2005), 323–339.
V. I. Kolyada, The imbedding of certain classes of functions of several variables, Sib. Mat. Zh., 14, (1973), 766–790 (in Russian); English translation in Siberian Math. J., 14 (1973).
V. I. Kolyada, On embedding H \(H_p^{{\omega _1}, \ldots ,{\omega _\nu }}\) classes, Mat. Sb., 127 (1985), 352–383 (in Russian); English translation in Math. USSR-Sb., 55, (1986), 351–381.
V. I. Kolyada, Estimates of rearrangements and embedding theorems, Mat. Sb., 136, (1988), 3–23 (in Russian); English translation in Math. USSR-Sb., 64 (1989), 1–21.
V. I. Kolyada, Rearrangement of functions and embedding of anisotropic spaces of Sobolev type, East J. Approx., 4 (1998), 111–199.
V. I. Kolyada, Mixed norms and Sobolev type inequalities, Banach Center Publ., 72, (2006), 141–160.
V. I. Kolyada, On embedding theorems, in: Nonlinear Analysis, Function Spaces and Applications, vol. 8, Czechoslovak Academy of Sciences, Mathematical Institute (Prague, 2007), pp. 34–94.
V. I. Kolyada, Iterated rearrangements and Gagliardo-Sobolev type inequalities, J. Math. Anal. Appl., 387, (2012), 335–348.
V. I. Kolyada, On limiting relations for capacities, Real Anal. Exchange, 38, (2013), 211–240.
V. I. Kolyada and A. K. Lerner, On limiting embeddings of Besov spaces, Studia Math., 171, (2005), 1–13.
G. Leoni and D. Spector, Characterization of Sobolev and BV spaces, J. Funct. Anal., 261 (2011), 2926–2958.
E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math., vol. 14, Amer. Math. Soc. (Providence, RI, 2001).
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., 55 (1949), 961–962.
M. Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc., 357, (2005), 3425–3442.
V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195, (2002), 230–238.
S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag (Berlin, Heidelberg, New York, 1975).
P. Oswald, On the moduli of continuity of equimeasurable functions in the classes ϕ(L), Mat. Zametki, 17, (1975), 231–244 (in Russian); English translation in Math. Notes, 17 (1975), 134–141.
P. Oswald, Moduli of continuity of equimeasurable functions and approximation of functions by algebraic polynomials in Lp, Ph.D. Thesis, Odessa State University (Odessa, 1978) (in Russian).
J. Peetre, Espaces d’interpolation et espaces de Soboleff, Ann. Inst. Fourier (Grenoble), 16, (1966), 279–317.
F. J. Pérez Lázaro, Embeddings for anisotropic Besov spaces, Acta Math. Hungar., 119, (2008), 25–40.
H. Triebel, Limits of Besov norms, Arch. Math., 96, (2011), 169–175.
P. L. Ul’yanov, Embedding of certain function classes H ωp , Izv. Akad. Nauk SSSR Ser. Mat., 32, (1968), 649–686 (in Russian); English translation in Math. USSR Izv., 2 (1968), 601–637.
I. Wik, The non-increasing rearrangement as extremal function, Report no. 7, Univ. Umea, Dept. of Math. (1974).
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The author is grateful to the referee for the careful revision and useful remarks.
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Dedicated to Professor O. V. Besov on the occasion of his 90th birthday
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Kolyada, V.I. Rearrangement Estimates and Limiting Embeddings for Anisotropic Besov Spaces. Anal Math 49, 1053–1071 (2023). https://doi.org/10.1007/s10476-023-0241-3
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DOI: https://doi.org/10.1007/s10476-023-0241-3