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doi: 10.3934/cpaa.2020250

A truncated real interpolation method and characterizations of screened Sobolev spaces

Noah Stevenson and  Ian Tice ,

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

* Corresponding author

Received  April 2020 Revised  July 2020 Published  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.

Keywords: Interpolation of seminormed spaces, screened Sobolev and Besov spaces, Littlewood-Paley.
Mathematics Subject Classification: Primary: 46M35, 46E35; Secondary: 46A04, 46F05, 46N20.
Citation: Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020250
References:
[1] R. A. Adams and John J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, second edition, 2003.   Google Scholar
[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.  Google Scholar

[3]

Jean Bourgain, H. Brezis and Petru Mironescu. Another look at Sobolev spaces. Optimal control and partial differential equations, pages 439–455. IOS, Amsterdam, 2001.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1978.  Google Scholar

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Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

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H. Brezis and Hoai-Minh Nguyen. Non-local functionals related to the total variation and connections with image processing, Ann. PDE, 4(1): Art. 9, 77, 2018. doi: 10.1007/s40818-018-0044-1.  Google Scholar

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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.  Google Scholar

[12]

Victor I. Burenkov, Sobolev Spaces on Domains, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.  Google Scholar

[13]

Fernando Cobos, Luz M. Fernández-Cabrera, Thomas Kühn and Tinono Ullrich, On an extreme class of real interpolation spaces, J. Funct. Anal, 256 (2009), 2321-2366.  doi: 10.1016/j.jfa.2008.12.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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[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.  Google Scholar

[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. (2), 34(2): 305–316, 1986. doi: 10.1112/jlms/s2-34.2.305.  Google Scholar

[20]

Loukas Grafakos, Classical Fourier Analysis, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[21]

Loukas Grafakos, Modern Fourier Analysis, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[22]

Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

Björn Jawerth and Mario Milman. Extrapolation theory with applications, Mem. Am. Math. Soc., 89 (1991), ⅳ+82pp. doi: 10.1090/memo/0440.  Google Scholar

[27]

Alois Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication. John Wiley & Sons, Inc. New York, 1985.  Google Scholar

[28]

Giovanni Leoni, A first course in Sobolev spaces, American Mathematical Society, Providence, RI, second edition, 2017.  Google Scholar

[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.  Google Scholar

[30]

Alessandra Lunardi, Interpolation Theory, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Jindřich Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Jaak Peetre, New thoughts on Besov Spaces, Mathematics Department, Duke University, Durham, N. C. 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40] Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.   Google Scholar
[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.  Google Scholar

[42]

Gudrun Thater, Neumann problem in domains with outlets of bounded diameter, Acta Appl. Math, 73 (2002), 251-274.  doi: 10.1023/A:1019736224759.  Google Scholar

[43]

Angus Ellis Taylor and David C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, second edition, 1980.  Google Scholar

[44]

Hans Triebel, Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library, North-Holland Publishing Co. Amsterdam-New York, 1978.  Google Scholar

[45]

Hans Triebel, Theory of Function Spaces, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010.  Google Scholar

[46]

Kosaku Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams and John J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, second edition, 2003.   Google Scholar
[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.  Google Scholar

[3]

Jean Bourgain, H. Brezis and Petru Mironescu. Another look at Sobolev spaces. Optimal control and partial differential equations, pages 439–455. IOS, Amsterdam, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, V. H. Winston & Sons, Washington, D. C., Halsted Press, New York-Toronto, Ont. London, 1978.  Google Scholar

[8]

Oleg V. Besov, Valentin P. Il'in and Sergey M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. Ⅱ. V. H. Winston & Sons, Washington, D. C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont. London, 1979.  Google Scholar

[9]

Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[10]

H. Brezis and Hoai-Minh Nguyen. Non-local functionals related to the total variation and connections with image processing, Ann. PDE, 4(1): Art. 9, 77, 2018. doi: 10.1007/s40818-018-0044-1.  Google Scholar

[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.  Google Scholar

[12]

Victor I. Burenkov, Sobolev Spaces on Domains, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.  Google Scholar

[13]

Fernando Cobos, Luz M. Fernández-Cabrera, Thomas Kühn and Tinono Ullrich, On an extreme class of real interpolation spaces, J. Funct. Anal, 256 (2009), 2321-2366.  doi: 10.1016/j.jfa.2008.12.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. (2), 34(2): 305–316, 1986. doi: 10.1112/jlms/s2-34.2.305.  Google Scholar

[20]

Loukas Grafakos, Classical Fourier Analysis, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[21]

Loukas Grafakos, Modern Fourier Analysis, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[22]

Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

Björn Jawerth and Mario Milman. Extrapolation theory with applications, Mem. Am. Math. Soc., 89 (1991), ⅳ+82pp. doi: 10.1090/memo/0440.  Google Scholar

[27]

Alois Kufner, Weighted Sobolev Spaces, A Wiley-Interscience Publication. John Wiley & Sons, Inc. New York, 1985.  Google Scholar

[28]

Giovanni Leoni, A first course in Sobolev spaces, American Mathematical Society, Providence, RI, second edition, 2017.  Google Scholar

[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.  Google Scholar

[30]

Alessandra Lunardi, Interpolation Theory, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Jindřich Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Jaak Peetre, New thoughts on Besov Spaces, Mathematics Department, Duke University, Durham, N. C. 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40] Elias M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.   Google Scholar
[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.  Google Scholar

[42]

Gudrun Thater, Neumann problem in domains with outlets of bounded diameter, Acta Appl. Math, 73 (2002), 251-274.  doi: 10.1023/A:1019736224759.  Google Scholar

[43]

Angus Ellis Taylor and David C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, second edition, 1980.  Google Scholar

[44]

Hans Triebel, Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library, North-Holland Publishing Co. Amsterdam-New York, 1978.  Google Scholar

[45]

Hans Triebel, Theory of Function Spaces, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010.  Google Scholar

[46]

Kosaku Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

[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.  Google Scholar

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