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Parabolic equations involving Laguerre operators and weighted mixed-norm estimates
A truncated real interpolation method and characterizations of screened Sobolev spaces
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA |
In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.
References:
[1] |
R. A. Adams and John J. F. Fournier, Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, second edition, 2003.
![]() |
[2] |
Sergey V. Astashkin, Konstantin V. Lykov and Mario Milman,
Limiting interpolation spaces via extrapolation, J. Approx. Theory, 240 (2019), 16-70.
doi: 10.1016/j.jat.2018.09.007. |
[3] |
Jean Bourgain, H. Brezis and Petru Mironescu, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001.
![]() |
[4] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s, p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[5] |
Hajer Bahouri, Jean-Yves Chemin and Raphaël Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[6] |
V. Benci and D. Fortunato,
Weighted Sobolev spaces and the nonlinear Dirichlet problem in unbounded domains, Ann. Mat. Pura Appl., 121 (1979), 319-336.
doi: 10.1007/BF02412010. |
[7] |
Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiǐ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1978.
![]() |
[8] |
Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiǐ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1979.
![]() |
[9] |
Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[10] |
H. Brezis and Hoai-Minh Nguyen, Non-local functionals related to the total variation and connections with image processing, Ann. PDE, 4 (2018), 77pp.
doi: 10.1007/s40818-018-0044-1. |
[11] |
Melvyn S. Berger and Martin Schechter,
Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains, Trans. Am. Math. Soc., 172 (1972), 261-278.
doi: 10.2307/1996347. |
[12] |
Victor I. Burenkov, Sobolev Spaces on Domains, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998.
doi: 10.1007/978-3-663-11374-4. |
[13] |
Fernando Cobos, Luz M. Fernández-Cabrera, Thomas Kühn and Tino Ullrich,
On an extreme class of real interpolation spaces, J. Funct. Anal., 256 (2009), 2321-2366.
doi: 10.1016/j.jfa.2008.12.013. |
[14] |
Qiang Du, Max Gunzburger, R. B. Lehoucq and Kun Zhou,
Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.
doi: 10.1137/110833294. |
[15] |
Eleonora DiNezza, Giampiero Palatucci and Enrico Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
D. E. Edmunds and W. D. Evans,
Elliptic and degenerate-elliptic operators in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1973), 591-640.
|
[17] |
Matthieu Felsinger, Moritz Kassmann and Paul Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
doi: 10.1007/s00209-014-1394-3. |
[18] |
Patrick T. Flynn and Huy Q. Nguyen, The vanishing surface tension limit of the Muskat problem, arXiv: 2001.10473. Google Scholar |
[19] |
M. E. Gomez and M. Milman,
Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc., 34 (1986), 305-316.
doi: 10.1112/jlms/s2-34.2.305. |
[20] |
Loukas Grafakos, Classical Fourier Analysis, Springer, New York, third edition, 2014.
doi: 10.1007/978-1-4939-1194-3. |
[21] |
Loukas Grafakos, Modern Fourier Analysis, Springer, New York, third edition, 2014.
doi: 10.1007/978-1-4939-1230-8. |
[22] |
Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
doi: 10.1137/1.9781611972030.ch1. |
[23] |
Jan Gustavsson, Interpolation of semi-norms, Technical report, Lund University, 1970. Google Scholar |
[24] |
Piotr Hajł asz and Agnieszka Kał amajska,
Polynomial asymptotics and approximation of Sobolev functions, Studia Math., 113 (1995), 55-64.
|
[25] |
R. Hanks,
Interpolation by the real method between BMO, $L^{\alpha }(0 < \alpha < \infty)$ and $H^{\alpha }(0 < \alpha < \infty)$, Indiana Univ. Math. J., 26 (1977), 679-689.
doi: 10.1512/iumj.1977.26.26054. |
[26] |
Björn Jawerth and Mario Milman, Extrapolation theory with applications, Mem. Am. Math. Soc., 89 (1991), 82pp.
doi: 10.1090/memo/0440. |
[27] |
Alois Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. |
[28] |
Giovanni Leoni, A first course in Sobolev spaces, American Mathematical Society, Providence, RI, second edition, 2017. |
[29] |
Giovanni Leoni and Ian Tice,
Traces for homogeneous sobolev spaces in infinite strip-like domains, J. Funct. Anal., 277 (2019), 2288-2380.
doi: 10.1016/j.jfa.2019.01.005. |
[30] |
Alessandra Lunardi, Interpolation Theory, Edizioni della Normale, Pisa, 2018.
doi: 10.1007/978-88-7642-638-4. |
[31] |
Vladimir Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, Heidelberg, augmented edition, 2011.
doi: 10.1007/978-3-642-15564-2. |
[32] |
V. Maz'ya, M. Mitrea and T. Shaposhnikova,
The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients, J. Anal. Math., 110 (2010), 167-239.
doi: 10.1007/s11854-010-0005-4. |
[33] |
Jindřich Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-10455-8. |
[34] |
Huy Q. Nguyen, On well-posedness of the Muskat problem with surface tension, arXiv: 1907.11552. Google Scholar |
[35] |
Huy Q. Nguyen and Benoît Pausader, A paradifferential approach for well-posedness of the muskat problem, Arch. Ration. Mech. Anal., 237 (2020) 25–100.
doi: 10.1007/s00205-020-01494-7. |
[36] |
J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de Matemática, No. 39. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968. |
[37] |
Jaak Peetre, New thoughts on Besov Spaces, Mathematics Department, Duke University, Durham, N. C., 1976. |
[38] |
Augusto C. Ponce,
A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differ. Equ., 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[39] |
Augusto C. Ponce and Daniel Spector,
On formulae decoupling the total variation of BV functions, Nonlinear Anal., 154 (2017), 241-257.
doi: 10.1016/j.na.2016.08.028. |
[40] |
Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
![]() |
[41] |
Robert S. Strichartz,
"Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260.
doi: 10.1007/s11854-016-0008-x. |
[42] |
Gudrun Thater,
Neumann problem in domains with outlets of bounded diameter, Acta Appl. Math., 73 (2002), 251-274.
doi: 10.1023/A:1019736224759. |
[43] |
Angus Ellis Taylor and David C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, second edition, 1980. |
[44] |
Hans Triebel, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978. |
[45] |
Hans Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. |
[46] |
Kosaku Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-61859-8. |
[47] |
Kun Zhou and Qiang Du,
Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal., 48 (2010), 1759-1780.
doi: 10.1137/090781267. |
show all references
References:
[1] |
R. A. Adams and John J. F. Fournier, Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, second edition, 2003.
![]() |
[2] |
Sergey V. Astashkin, Konstantin V. Lykov and Mario Milman,
Limiting interpolation spaces via extrapolation, J. Approx. Theory, 240 (2019), 16-70.
doi: 10.1016/j.jat.2018.09.007. |
[3] |
Jean Bourgain, H. Brezis and Petru Mironescu, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001.
![]() |
[4] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s, p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[5] |
Hajer Bahouri, Jean-Yves Chemin and Raphaël Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[6] |
V. Benci and D. Fortunato,
Weighted Sobolev spaces and the nonlinear Dirichlet problem in unbounded domains, Ann. Mat. Pura Appl., 121 (1979), 319-336.
doi: 10.1007/BF02412010. |
[7] |
Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiǐ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1978.
![]() |
[8] |
Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiǐ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1979.
![]() |
[9] |
Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[10] |
H. Brezis and Hoai-Minh Nguyen, Non-local functionals related to the total variation and connections with image processing, Ann. PDE, 4 (2018), 77pp.
doi: 10.1007/s40818-018-0044-1. |
[11] |
Melvyn S. Berger and Martin Schechter,
Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains, Trans. Am. Math. Soc., 172 (1972), 261-278.
doi: 10.2307/1996347. |
[12] |
Victor I. Burenkov, Sobolev Spaces on Domains, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998.
doi: 10.1007/978-3-663-11374-4. |
[13] |
Fernando Cobos, Luz M. Fernández-Cabrera, Thomas Kühn and Tino Ullrich,
On an extreme class of real interpolation spaces, J. Funct. Anal., 256 (2009), 2321-2366.
doi: 10.1016/j.jfa.2008.12.013. |
[14] |
Qiang Du, Max Gunzburger, R. B. Lehoucq and Kun Zhou,
Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.
doi: 10.1137/110833294. |
[15] |
Eleonora DiNezza, Giampiero Palatucci and Enrico Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
D. E. Edmunds and W. D. Evans,
Elliptic and degenerate-elliptic operators in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1973), 591-640.
|
[17] |
Matthieu Felsinger, Moritz Kassmann and Paul Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
doi: 10.1007/s00209-014-1394-3. |
[18] |
Patrick T. Flynn and Huy Q. Nguyen, The vanishing surface tension limit of the Muskat problem, arXiv: 2001.10473. Google Scholar |
[19] |
M. E. Gomez and M. Milman,
Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc., 34 (1986), 305-316.
doi: 10.1112/jlms/s2-34.2.305. |
[20] |
Loukas Grafakos, Classical Fourier Analysis, Springer, New York, third edition, 2014.
doi: 10.1007/978-1-4939-1194-3. |
[21] |
Loukas Grafakos, Modern Fourier Analysis, Springer, New York, third edition, 2014.
doi: 10.1007/978-1-4939-1230-8. |
[22] |
Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
doi: 10.1137/1.9781611972030.ch1. |
[23] |
Jan Gustavsson, Interpolation of semi-norms, Technical report, Lund University, 1970. Google Scholar |
[24] |
Piotr Hajł asz and Agnieszka Kał amajska,
Polynomial asymptotics and approximation of Sobolev functions, Studia Math., 113 (1995), 55-64.
|
[25] |
R. Hanks,
Interpolation by the real method between BMO, $L^{\alpha }(0 < \alpha < \infty)$ and $H^{\alpha }(0 < \alpha < \infty)$, Indiana Univ. Math. J., 26 (1977), 679-689.
doi: 10.1512/iumj.1977.26.26054. |
[26] |
Björn Jawerth and Mario Milman, Extrapolation theory with applications, Mem. Am. Math. Soc., 89 (1991), 82pp.
doi: 10.1090/memo/0440. |
[27] |
Alois Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. |
[28] |
Giovanni Leoni, A first course in Sobolev spaces, American Mathematical Society, Providence, RI, second edition, 2017. |
[29] |
Giovanni Leoni and Ian Tice,
Traces for homogeneous sobolev spaces in infinite strip-like domains, J. Funct. Anal., 277 (2019), 2288-2380.
doi: 10.1016/j.jfa.2019.01.005. |
[30] |
Alessandra Lunardi, Interpolation Theory, Edizioni della Normale, Pisa, 2018.
doi: 10.1007/978-88-7642-638-4. |
[31] |
Vladimir Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, Heidelberg, augmented edition, 2011.
doi: 10.1007/978-3-642-15564-2. |
[32] |
V. Maz'ya, M. Mitrea and T. Shaposhnikova,
The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients, J. Anal. Math., 110 (2010), 167-239.
doi: 10.1007/s11854-010-0005-4. |
[33] |
Jindřich Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-10455-8. |
[34] |
Huy Q. Nguyen, On well-posedness of the Muskat problem with surface tension, arXiv: 1907.11552. Google Scholar |
[35] |
Huy Q. Nguyen and Benoît Pausader, A paradifferential approach for well-posedness of the muskat problem, Arch. Ration. Mech. Anal., 237 (2020) 25–100.
doi: 10.1007/s00205-020-01494-7. |
[36] |
J. Peetre, A Theory of Interpolation of Normed Spaces, Notas de Matemática, No. 39. Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968. |
[37] |
Jaak Peetre, New thoughts on Besov Spaces, Mathematics Department, Duke University, Durham, N. C., 1976. |
[38] |
Augusto C. Ponce,
A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differ. Equ., 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[39] |
Augusto C. Ponce and Daniel Spector,
On formulae decoupling the total variation of BV functions, Nonlinear Anal., 154 (2017), 241-257.
doi: 10.1016/j.na.2016.08.028. |
[40] |
Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
![]() |
[41] |
Robert S. Strichartz,
"Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260.
doi: 10.1007/s11854-016-0008-x. |
[42] |
Gudrun Thater,
Neumann problem in domains with outlets of bounded diameter, Acta Appl. Math., 73 (2002), 251-274.
doi: 10.1023/A:1019736224759. |
[43] |
Angus Ellis Taylor and David C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, second edition, 1980. |
[44] |
Hans Triebel, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978. |
[45] |
Hans Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. |
[46] |
Kosaku Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-61859-8. |
[47] |
Kun Zhou and Qiang Du,
Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal., 48 (2010), 1759-1780.
doi: 10.1137/090781267. |
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