December  2020, 19(12): 5509-5566. doi: 10.3934/cpaa.2020250

A truncated real interpolation method and characterizations of screened Sobolev spaces

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

* Corresponding author

Received  April 2020 Revised  July 2020 Published  December 2020 Early access  October 2020

Fund Project: I. Tice was supported by an NSF CAREER Grant (DMS #1653161). N. Stevenson was supported by the summer research support provided by this grant

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.

Citation: Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
References:
[1] R. A. Adams and John J. F. Fournier, Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, second edition, 2003. 
[2]

Sergey V. AstashkinKonstantin 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 BourgainH. Brezis and Petru Mironescu, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001. 
[4]

J. BourgainH. 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. BesovValentin 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. BesovValentin 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 CobosLuz M. Fernández-CabreraThomas 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 DuMax GunzburgerR. 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 DiNezzaGiampiero 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 FelsingerMoritz 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.

[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.

[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'yaM. 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.

[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. AstashkinKonstantin 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 BourgainH. Brezis and Petru Mironescu, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001. 
[4]

J. BourgainH. 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. BesovValentin 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. BesovValentin 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 CobosLuz M. Fernández-CabreraThomas 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 DuMax GunzburgerR. 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 DiNezzaGiampiero 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 FelsingerMoritz 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.

[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.

[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'yaM. 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.

[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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