February  2021, 20(2): 697-735. doi: 10.3934/cpaa.2020286

Elliptic problems with rough boundary data in generalized Sobolev spaces

1. 

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01024, Ukraine

2. 

University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany

* Corresponding author

Received  May 2020 Revised  September 2020 Published  February 2021 Early access  December 2020

Fund Project: The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686. This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology). The first author was supported by President of Ukraine's grant for competitive project F82/45932

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

Citation: Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286
References:
[1]

M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79 (1997), 1-144.  doi: 10.1007/978-3-662-06721-5_1.  Google Scholar

[2]

A. V. Anop and T. M. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology, 22 (2016), 295-310.   Google Scholar

[3]

A. V. AnopT. M. Kasirenko and O. O. Murach, Irregular elliptic boundary-value problems and Hörmander spaces, Ukranian Math. J., 70 (2018), 341-361.  doi: 10.1007/s11253-018-1504-1.  Google Scholar

[4]

A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J., 66 (2014), 969-985.  doi: 10.1007/s11253-014-0988-6.  Google Scholar

[5]

A. V. Anop and A. A. Murach, To the theory of elliptic boundary-value problems in Hörmander spaces, Transactions of Institute of Mathematics of NAS of Ukraine, 12 (2015), 39-64.   Google Scholar

[6]

A. V. Anop and A. A. Murach, Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale, Transactions of Institute of Mathematics of NAS of Ukraine, 13 (2016), 27-54.   Google Scholar

[7]

A. V. Anop and A. A. Murach, Homogeneous elliptic equations in an extended Sobolev scale, Dopov. Nac. Akad. Nauk Ukr., (2018), 3–11. doi: 10.15407/dopovidi2018.03.003.  Google Scholar

[8]

V. G. Avakumović, O jednom O-inverznom stavu, Rad Jugoslav. Akad. Znan. Umjet., 254 (1936), 167-186.   Google Scholar

[9]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968. Google Scholar

[10]

J. Behrndt, S. Hassi and H. de. Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Springer, Cham, 2020. doi: 10.1007/978-3-030-36714-5.  Google Scholar

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V. V. Buldygin, K. H. Indlekofer, O. I. Klesov and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-99537-3.  Google Scholar

[13]

J. FageotA. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Process. Appl., 127 (2017), 1599-1621.  doi: 10.1016/j.spa.2016.08.011.  Google Scholar

[14]

M. Faierman, Fredholm theory for an elliptic differential operator defined on $\mathbb R^n$ and acting on generalized Sobolev spaces, Commun. Pure Appl. Anal., 19 (2020), 1463-1483.  doi: 10.3934/cpaa.2020074.  Google Scholar

[15]

C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282.   Google Scholar

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Proc. Sympos. Pure Math., 79 (2008), 105-173.  doi: 10.1090/pspum/079/2500491.  Google Scholar

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G. Geymonat, Sul problema di Dirichlet per le equazoni lineari ellittiche, Ann. Sci. Norm. Sup. Pisa, 16 (1962), 225-284.   Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (revised 3rd edition), Springer, Berlin, 1998.  Google Scholar

[19]

M. L. Gol'dman, Imbedding theorems for anisotropic Nikol'skiǐ–Besov spaces with moduli of continuity of a general type, Proc. Steklov Inst. Math., 170 (1987), 95-116.   Google Scholar

[20]

L. Hörmander, On the theory of general partial differential equations, Acta Math., 94 (1955), 161-248.  doi: 10.1007/BF02392492.  Google Scholar

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L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅱ. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

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L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅲ. Pseudo-Differential Operators, Springer, Berlin, 1985.  Google Scholar

[24]

T. KasirenkoV. Mikhailets and A. Murach, Sobolev-like Hilbert spaces induced by elliptic operators, Complex Anal. Oper. Theor., 13 (2019), 1431-1440.  doi: 10.1007/s11785-018-00886-8.  Google Scholar

[25]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/052.  Google Scholar

[26]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, Ⅱ, Ann. Inst. Fourier (Grenoble), 11 (1961), 137-178.   Google Scholar

[27]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, V, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 1-44.   Google Scholar

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J. L. Lions and E. Magenes, Problémes aux limites non homogénes, VI, J. Analyse Math., 11 (1963), 165-188.  doi: 10.1007/BF02789983.  Google Scholar

[29]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I, Springer, Berlin, 1972.  Google Scholar

[30]

V. LosV. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.  Google Scholar

[31]

B. Malgrange, Sur une classe d'opératuers différentiels hypoelliptiques, Bull. Soc. Math., 85 (1957), 283-306.   Google Scholar

[32]

E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali, Atti Ⅶ Congr. Un. Mat., (1965), 134–197.  Google Scholar

[33]

W. Matuszewska, On a generalization of regularly increasing functions, Studia Math., 24 (1964), 271-279.  doi: 10.4064/sm-24-3-271-279.  Google Scholar

[34]

V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian. Math. J., 57 (2005), 817-825.  doi: 10.1007/s11253-005-0231-6.  Google Scholar

[35]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅱ, Ukrainian Math. J., 58 (2006), 398-417.  doi: 10.1007/s11253-006-0074-9.  Google Scholar

[36]

V. A. Mikhailets and A. A Murach, Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces, Ukrainian Math. J., 58 (2006), 1748-1767.  doi: 10.1007/s11253-006-0166-6.  Google Scholar

[37]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅲ, Ukrainian Math. J., 59 (2007), 744-765.  doi: 10.1007/s11253-007-0048-6.  Google Scholar

[38]

V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.  Google Scholar

[39]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar

[40]

V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.  Google Scholar

[41]

A. A. Murach, Extension of some Lions–Magenes theorems, Methods Funct. Anal. Topology, 15 (2009), 152-167.   Google Scholar

[42]

S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [2nd edition], Springer-Verlag, New York–Heidelberg, 1975.  Google Scholar

[43]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, in Mathematical Reports. Vol. 1, Part. 2 (ed. J. Peetre), Harwood Academic Publishers, London, 1984.  Google Scholar

[44]

J. Peetre, Another approach to elliptic boundary problems, Commun. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.  Google Scholar

[45]

J. Peetre, On interpolation functions. Ⅱ, Acta Sci. Math., 29 (1968), 91-92.   Google Scholar

[46]

A. Pliś, A smooth linear elliptic differential equation without any solution in a sphere, Commun. Pure Appl. Math., 14 (1961), 599-617.  doi: 10.1002/cpa.3160140331.  Google Scholar

[47]

Ya. A. Roitberg, Elliptic problems with nonhomogeneous boundary conditions and local increase of smoothness up to the boundary for generalized solutions, Dokl. Math., 5 (1964), 1034-1038.   Google Scholar

[48]

Ya. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-011-5410-9.  Google Scholar

[49]

Ya. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9275-8.  Google Scholar

[50]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progr. Nonlinear Differ. Equ. Appl., 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.  Google Scholar

[51]

R. T. Seeley, Singular integrals and boundary-value problems, Amer. J. Math., 88 (1966), 781-809.  doi: 10.2307/2373078.  Google Scholar

[52]

E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.  Google Scholar

[53]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [2nd edition], Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[54]

M. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Banach Center Publ., 95 (2011), 385-398.  doi: 10.4064/bc95-0-24.  Google Scholar

[55]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk, 20 (1965), 3-74.   Google Scholar

[56]

T. N. Zinchenko and A. A. Murach, Douglis–Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2012), 1672-1687.  doi: 10.1007/s11253-013-0743-4.  Google Scholar

show all references

References:
[1]

M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79 (1997), 1-144.  doi: 10.1007/978-3-662-06721-5_1.  Google Scholar

[2]

A. V. Anop and T. M. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology, 22 (2016), 295-310.   Google Scholar

[3]

A. V. AnopT. M. Kasirenko and O. O. Murach, Irregular elliptic boundary-value problems and Hörmander spaces, Ukranian Math. J., 70 (2018), 341-361.  doi: 10.1007/s11253-018-1504-1.  Google Scholar

[4]

A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J., 66 (2014), 969-985.  doi: 10.1007/s11253-014-0988-6.  Google Scholar

[5]

A. V. Anop and A. A. Murach, To the theory of elliptic boundary-value problems in Hörmander spaces, Transactions of Institute of Mathematics of NAS of Ukraine, 12 (2015), 39-64.   Google Scholar

[6]

A. V. Anop and A. A. Murach, Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale, Transactions of Institute of Mathematics of NAS of Ukraine, 13 (2016), 27-54.   Google Scholar

[7]

A. V. Anop and A. A. Murach, Homogeneous elliptic equations in an extended Sobolev scale, Dopov. Nac. Akad. Nauk Ukr., (2018), 3–11. doi: 10.15407/dopovidi2018.03.003.  Google Scholar

[8]

V. G. Avakumović, O jednom O-inverznom stavu, Rad Jugoslav. Akad. Znan. Umjet., 254 (1936), 167-186.   Google Scholar

[9]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968. Google Scholar

[10]

J. Behrndt, S. Hassi and H. de. Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Springer, Cham, 2020. doi: 10.1007/978-3-030-36714-5.  Google Scholar

[11] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.   Google Scholar
[12]

V. V. Buldygin, K. H. Indlekofer, O. I. Klesov and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-99537-3.  Google Scholar

[13]

J. FageotA. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Process. Appl., 127 (2017), 1599-1621.  doi: 10.1016/j.spa.2016.08.011.  Google Scholar

[14]

M. Faierman, Fredholm theory for an elliptic differential operator defined on $\mathbb R^n$ and acting on generalized Sobolev spaces, Commun. Pure Appl. Anal., 19 (2020), 1463-1483.  doi: 10.3934/cpaa.2020074.  Google Scholar

[15]

C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282.   Google Scholar

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Proc. Sympos. Pure Math., 79 (2008), 105-173.  doi: 10.1090/pspum/079/2500491.  Google Scholar

[17]

G. Geymonat, Sul problema di Dirichlet per le equazoni lineari ellittiche, Ann. Sci. Norm. Sup. Pisa, 16 (1962), 225-284.   Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (revised 3rd edition), Springer, Berlin, 1998.  Google Scholar

[19]

M. L. Gol'dman, Imbedding theorems for anisotropic Nikol'skiǐ–Besov spaces with moduli of continuity of a general type, Proc. Steklov Inst. Math., 170 (1987), 95-116.   Google Scholar

[20]

L. Hörmander, On the theory of general partial differential equations, Acta Math., 94 (1955), 161-248.  doi: 10.1007/BF02392492.  Google Scholar

[21]

L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.  Google Scholar

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅱ. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[23]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅲ. Pseudo-Differential Operators, Springer, Berlin, 1985.  Google Scholar

[24]

T. KasirenkoV. Mikhailets and A. Murach, Sobolev-like Hilbert spaces induced by elliptic operators, Complex Anal. Oper. Theor., 13 (2019), 1431-1440.  doi: 10.1007/s11785-018-00886-8.  Google Scholar

[25]

V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/052.  Google Scholar

[26]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, Ⅱ, Ann. Inst. Fourier (Grenoble), 11 (1961), 137-178.   Google Scholar

[27]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, V, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 1-44.   Google Scholar

[28]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, VI, J. Analyse Math., 11 (1963), 165-188.  doi: 10.1007/BF02789983.  Google Scholar

[29]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I, Springer, Berlin, 1972.  Google Scholar

[30]

V. LosV. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.  Google Scholar

[31]

B. Malgrange, Sur une classe d'opératuers différentiels hypoelliptiques, Bull. Soc. Math., 85 (1957), 283-306.   Google Scholar

[32]

E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali, Atti Ⅶ Congr. Un. Mat., (1965), 134–197.  Google Scholar

[33]

W. Matuszewska, On a generalization of regularly increasing functions, Studia Math., 24 (1964), 271-279.  doi: 10.4064/sm-24-3-271-279.  Google Scholar

[34]

V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian. Math. J., 57 (2005), 817-825.  doi: 10.1007/s11253-005-0231-6.  Google Scholar

[35]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅱ, Ukrainian Math. J., 58 (2006), 398-417.  doi: 10.1007/s11253-006-0074-9.  Google Scholar

[36]

V. A. Mikhailets and A. A Murach, Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces, Ukrainian Math. J., 58 (2006), 1748-1767.  doi: 10.1007/s11253-006-0166-6.  Google Scholar

[37]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅲ, Ukrainian Math. J., 59 (2007), 744-765.  doi: 10.1007/s11253-007-0048-6.  Google Scholar

[38]

V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.  Google Scholar

[39]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar

[40]

V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.  Google Scholar

[41]

A. A. Murach, Extension of some Lions–Magenes theorems, Methods Funct. Anal. Topology, 15 (2009), 152-167.   Google Scholar

[42]

S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [2nd edition], Springer-Verlag, New York–Heidelberg, 1975.  Google Scholar

[43]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, in Mathematical Reports. Vol. 1, Part. 2 (ed. J. Peetre), Harwood Academic Publishers, London, 1984.  Google Scholar

[44]

J. Peetre, Another approach to elliptic boundary problems, Commun. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.  Google Scholar

[45]

J. Peetre, On interpolation functions. Ⅱ, Acta Sci. Math., 29 (1968), 91-92.   Google Scholar

[46]

A. Pliś, A smooth linear elliptic differential equation without any solution in a sphere, Commun. Pure Appl. Math., 14 (1961), 599-617.  doi: 10.1002/cpa.3160140331.  Google Scholar

[47]

Ya. A. Roitberg, Elliptic problems with nonhomogeneous boundary conditions and local increase of smoothness up to the boundary for generalized solutions, Dokl. Math., 5 (1964), 1034-1038.   Google Scholar

[48]

Ya. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-011-5410-9.  Google Scholar

[49]

Ya. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9275-8.  Google Scholar

[50]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progr. Nonlinear Differ. Equ. Appl., 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.  Google Scholar

[51]

R. T. Seeley, Singular integrals and boundary-value problems, Amer. J. Math., 88 (1966), 781-809.  doi: 10.2307/2373078.  Google Scholar

[52]

E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.  Google Scholar

[53]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [2nd edition], Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[54]

M. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Banach Center Publ., 95 (2011), 385-398.  doi: 10.4064/bc95-0-24.  Google Scholar

[55]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk, 20 (1965), 3-74.   Google Scholar

[56]

T. N. Zinchenko and A. A. Murach, Douglis–Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2012), 1672-1687.  doi: 10.1007/s11253-013-0743-4.  Google Scholar

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