Abstract
Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the \(L^p\)-boundedness property in the context of interpolation inequalities between Sobolev and Morrey spaces, which may be useful for those working in this domain. We also construct a nontrivial counterexample, which shows that the range of admissible values of \(p\) is optimal in a certain sense. Our proofs rely on integral representations and on the theory of maximal and sharp maximal functions.
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References
D. R. Adams, Lecture Notes on \(L^p\)-Potential Theory, Dept. of Math., University of Umea, 1981.
D. R. Adams and L. I. Hedberg, Functions Spaces and Potential Theory, Springer-Verlag, Berlin–Heidelberg, 1996.
R. A. Adams and J. J. F. Fourier, Sobolev spaces, 2nd edition, Pure and Applied Mathematics, vol. 140, Elsevier/Academic, Amsterdam, 2003.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
N.-A. Dao, J.-I. Díaz, and Q.-H. Nguyen, “Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces”, Nonlinear Anal., 173 (2018), 146–153.
D. S. McCormick, J. C. Robinson, and J. L. Rodrigo, “Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO”, Milan J. Math., 81:2 (2013), 265–289.
L. Nirenberg, “On elliptic partial differential equations”, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115–162.
G. Patalucci and A. Pisante, “Improved Sobolev embeddings, profile decomposition, and concentration compactness for fractional Sobolev spaces”, Calc. Var. Partial Differ. Equ., 50:3–4 (2014), 799–829.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.
E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton Univ. Press, Princeton, NJ, 2011.
J. Van Schaftingen, “Interpolation inequalities between Sobolev and Morrey–Campanato spaces: A common gateway to concentration-compactness and Gagliardo–Nirenberg interpolation inequalities”, Port. Math., 71:3 (2014), 159–175.
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Translated from Funktsional'nyi Analiz i Ego Prilozheniya, 2020, Vol. 54, No. 3, pp. 63-72 https://doi.org/10.4213/faa3628 .
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Tran, MP., Nguyen, TN. A Remark on the Interpolation Inequality between Sobolev Spaces and Morrey Spaces. Funct Anal Its Appl 54, 200–207 (2020). https://doi.org/10.1134/S0016266320030053
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DOI: https://doi.org/10.1134/S0016266320030053