Abstract
For function spaces equipped with Muckenhoupt weights, the validity of continuous Sobolev embeddings in case \({p_0 \leq p_1}\) is characterized. Extensions to Jawerth–Franke embeddings, vector-valued spaces, and examples involving some prominent weights are also provided.
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References
Brezis H., Mironescu P.: Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ., 1, 387–404 (2001)
Bui H.-Q.: Weighted Besov and Triebel spaces: interpolation by the real method. Hiroshima Math. J., 12, 581–605 (1982)
Bui H.-Q., Paluszyński M., Taibleson M. H.: A maximal function characterization of weighted Besov–Lipschitz and Triebel–Lizorkin spaces. Studia Math., 119, 219–246 (1996)
X. Cabré and X. Ros-Oton. Sobolev and isoperimetric inequalities with monomial weights. J. Differential Equations, 255 (2013), 4312–4336.
D.V. Cruz-Uribe, J. M. Martell, and C. Pérez. Weights, extrapolation and the theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel, 2011.
P.L. De Nápoli, I. Drelichman, and N. Saintier. Weighted embedding theorems for radial Besov and Triebel–Lizorkin spaces. arXiv preprint http://arxiv.org/pdf/1406.0542, 2014.
K. Disser, M. Meyries, and J. Rehberg. A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces. arXiv preprint http://arxiv.org/pdf/1312.5882, 2013.
R. Farwig and H. Sohr. Weighted L q-theory for the Stokes resolvent in exterior domains. . Math. Soc. Japan, 49 (1997), 251–288.
J. Franke. On the spaces \({{\bf{\rm F}}_{pq}^s}\) of Triebel-Lizorkin type: pointwise multipliers and spaces on domains. Math. Nachr., 125 (1997), 29–68.
J. García-Cuerva and J.L. Rubio de Francia. Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104.
L. Grafakos. Modern Fourier analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, second edition, 2009.
D. D. Haroske and L. Skrzypczak. Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights. I. Rev. Mat. Complut., 21 (2008), 135–177.
T.P. Hytönen. The two-weight inequality for the Hilbert transform with general measures. arXiv preprint http://arxiv.org/pdf/1312.0843, 2013.
Jawerth B.: Some observations on Besov and Lizorkin-Triebel spaces. Math. Scand., 40, 94–104 (1977)
T. Kühn, H.-G. Leopold, W. Sickel, And L. Skrzypczak. Entropy numbers of embeddings of weighted Besov spaces. II. Proc. Edinb. Math. Soc. (2), 49 (2006), 331–359.
M.T. Lacey, E.T. Sawyer, I. Uriarte-Tuero, And C.-Y. Shen. Two weight inequality for the Hilbert transform: A real variable characterization, Part I. Duke Math. J., to appear, available at http://arxiv.org/pdf/1201.4319, 2012.
M.T. Lacey and B.D. Wick. Two weight inequalities for Riesz transforms. arXiv preprint http://arxiv.org/pdf/1312.6163, 2013.
M. Meyries and M.C. Veraar. Sharp embedding results for spaces of smooth functions with power weights. Stud. Math., 208 (2012), 257–293.
M. Meyries and M.C. Veraar. Pointwise multiplication on vector-valued function spaces with power weights. Online first in J. Fourier Anal. Appl., 2014.
M. Meyries and M.C. Veraar. Traces and embeddings of anisotropic function spaces. Online first in Math. Ann., 2014.
F. Oru. Rôle des oscillations dans quelques problèmes d’analyse non-linéaire. PhD thesis, Doctorat de Ecole Normale Supérieure de Cachan.
Y. Rakotondratsimba. A two-weight inequality for the Bessel potential operator. Comment. Math. Univ. Carolin., 38 (1997), 497–511.
V.S. Rychkov. Littlewood-Paley theory and function spaces with \({A^{\rm loc}_p}\) weights. Math. Nachr., 224 (2001), 145–180.
Sawyer E.T.: A characterization of a two-weight norm inequality for maximal operators. Studia Math., 75, 1–11 (1982)
E.T. Sawyer, C.-Y. Shen, and I. Uriarte-Tuero. A geometric condition, necessity of energy, and two weight boundedness of fractional Riesz transforms. arXiv preprint http://arxiv.org/pdf/1310.4484, 2013.
Scharf B., Schmeisser H.-J., Sickel W.: Traces of vector-valued Sobolev spaces. Math. Nachr., 285, 1082–1106 (2012)
H.-J. Schmeisser and W. Sickel. Traces, Gagliardo-Nirenberg inequailties and Sobolev type embeddings for vector-valued function spaces. Jena manuscript, 2004.
H.-J. Schmeisser and W. Sickel. Vector-valued Sobolev spaces and Gagliardo-Nirenberg inequalities. In Nonlinear elliptic and parabolic problems, volume 64 of Progr. Nonlinear Differential Equations Appl., pages 463–472. Birkhäuser, Basel, 2005.
W. Sickel and H. Triebel. Hölder inequalities and sharp embeddings in function spaces of \({B^s_{pq}}\) and \({F^s_{pq}}\) type. Z. Anal. Anwendungen, 14 (1995), 105–140.
H. Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1983.
M.C. Veraar. Embedding results for \({\gamma}\)-spaces. Borichev, Alexander (ed.) et al., Recent trends in analysis. Proceedings of the conference in honor of Nikolai Nikolski on the occasion of his 70th birthday, Bordeaux, France, August 31 – September 2, 2011. Bucharest: The Theta Foundation. Theta Series in Advanced Mathematics (2013), 209–219.
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M. Veraar was supported by VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).
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Meyries, M., Veraar, M. Characterization of a class of embeddings for function spaces with Muckenhoupt weights. Arch. Math. 103, 435–449 (2014). https://doi.org/10.1007/s00013-014-0706-5
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DOI: https://doi.org/10.1007/s00013-014-0706-5