Weighted periodic and discrete pseudo-differential operators. (English) Zbl 1542.35466
Summary: In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class \(M_{\rho, \Lambda}^{m} (\mathbb{T} \times \mathbb{Z})\) (associated to a suitable weight function \(\Lambda\) on \(\mathbb{Z})\) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of \(M\)-elliptic pseudo-differential operators on \(\mathbb{T}\). Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class \(M_{\rho, \Lambda}^{0} (\mathbb{T} \times \mathbb{Z})\) and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on \(L^{2} (\mathbb{T})\). Finally, we provide Gårding’s and Sharp Gårding’s inequality for \(M\)-elliptic operators on \(\mathbb{Z}\) and \(\mathbb{T}\), respectively, and present an application in the context of strong solution of the pseudo-differential equation \(T_{\sigma} u = f\) in \(L^{2} \left( \mathbb{T} \right)\).
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 47G30 | Pseudodifferential operators |
| 43A85 | Harmonic analysis on homogeneous spaces |
| 35A17 | Parametrices in context of PDEs |
Keywords:
\(M\)-ellipticity; symbolic calculus; compact operators; Gohberg’s lemma; Gårding’s inequality; strong solutionReferences:
| [1] | Atkinson, FV, The normal solubility of linear equations in normed spaces, Mat. Sbornik N.S., 28, 70, 3-14, 1951 · Zbl 0042.12001 |
| [2] | Beals, R., A general class of pseudo-differential operators, Duke Math. J., 42, 1-42, 1975 · Zbl 0343.35078 · doi:10.1215/S0012-7094-75-04201-5 |
| [3] | Beals, R.; Fefferman, C., Spatially inhomogeneous pseudodifferential operators, I, Comm. Pure Appl. Math., 27, 1-24, 1974 · Zbl 0279.35071 · doi:10.1002/cpa.3160270102 |
| [4] | Botchway, LNA; Kibiti, PG; Ruzhansky, M., Difference equations and pseudo-differential operators on \({\mathbb{Z} }^n\), J. Funct. Anal., 278, 11, 108473, 2017 · Zbl 1434.58011 · doi:10.1016/j.jfa.2020.108473 |
| [5] | Cardona, D., Pseudo-differential Operators on \({\mathbb{Z} }^n\) with applications to discrete fractional integral operators, Bull. Iran. Math. Soc., 45, 4, 1227-1241, 2019 · Zbl 1423.42014 · doi:10.1007/s41980-018-00195-y |
| [6] | Cardona, D.; del Corral, C.; Kumar, V., Dixmier traces for discrete pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., 11, 647-656, 2020 · Zbl 1503.47065 · doi:10.1007/s11868-020-00335-1 |
| [7] | Cardona, D., Delgado, J., Ruzhansky, M.: Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations, arXiv:2111.07469 (2021) |
| [8] | Cardona, D., Federico, S., Ruzhansky, M.: Subelliptic sharp Gårding inequality on compact Lie groups, arXiv:2110.00838 (2021) |
| [9] | Cardona, D.; Kumar, V., Multilinear analysis for discrete and periodic pseudo-differential operators in \(L^p\) spaces, Rev. Integr. temas Mat., 36, 2, 151-164, 2018 · Zbl 1426.58007 · doi:10.18273/revint.v36n2-2018006 |
| [10] | Cardona, D.; Kumar, V., \(L^p\)-boundedness and \(L^p\)-nuclearity of multilinear pseudo-differential operators on \({\mathbb{Z} }^n\) and the torus \({\mathbb{T} }^n,\), J. Fourier Anal. Appl., 25, 6, 2973-3017, 2019 · Zbl 1429.58036 · doi:10.1007/s00041-019-09689-7 |
| [11] | Cardona, D.; Kumar, V.; Ruzhansky, M.; Tokmagambetov, N., Global Functional calculus, lower/upper bounds and evolution equations on manifolds with boundary, Adv. Oper. Theory, 8, 50, 2023 · Zbl 07777061 · doi:10.1007/s43036-023-00254-0 |
| [12] | Dasgupta, A.; Ruzhansky, M., The Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups, J. Anal. Math., 128, 179-190, 2016 · Zbl 1342.35462 · doi:10.1007/s11854-016-0005-0 |
| [13] | Fefferman, C., \(L^p\)-bounds for pseudo-differential operators, Isr. J. Math., 14, 413-417, 1973 · Zbl 0259.47045 · doi:10.1007/BF02764718 |
| [14] | Fefferman, C.; Phong, DH, On positivity of pseudo-differential operators, Proc. Natl. Acad. Sci. USA, 75, 4673-4674, 1978 · Zbl 0391.35062 · doi:10.1073/pnas.75.10.4673 |
| [15] | Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups, Progress in Mathematics, vol. 314. Birkhäuser, Basel (2016) · Zbl 1347.22001 |
| [16] | Gårding, L., Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand., 1, 55-72, 1953 · Zbl 0053.39101 · doi:10.7146/math.scand.a-10364 |
| [17] | Garello, G., Morando, A.: A class of \(L^p\) bounded pseudodifferential operators. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (Eds.) Progress in Analysis, Vol. I, pp. 689-696. World Scientific (2003) · Zbl 1061.35185 |
| [18] | Garello, G.; Morando, A., \(L^p\)-bounded pseudo-differential opearors and regularity for multi-quasi elliptic equations, Integr. Equ. Oper. Theory, 51, 501-517, 2005 · Zbl 1082.35175 · doi:10.1007/s00020-001-1262-5 |
| [19] | Gohberg, IC, On the theory of multi-dimensional singular integral equations, Sov. Math. Dokl., 1, 960-963, 1960 · Zbl 0161.32101 |
| [20] | Grušhin, VV, Pseudo-differential operators in \({\mathbb{R}}^n\) with bounded symbols, Funkcional. Anal. i Priložen, 4, 3, 37-50, 1970 · Zbl 0223.35084 |
| [21] | Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems, Ann. Math., 83, 2, 129-209, 1966 · Zbl 0132.07402 · doi:10.2307/1970473 |
| [22] | Hörmander, L., The Cauchy problem for differential equations with double characteristics, J. Anal. Math., 32, 118-196, 1977 · Zbl 0367.35054 · doi:10.1007/BF02803578 |
| [23] | Hörmander, L., The Analysis of Linear Partial Differential Operators III, 1985, Berlin: Springer, Berlin · Zbl 0601.35001 |
| [24] | Kalleji, MK, Essential spectrum of \(M\)-hypoelliptic pseudo-differential on the torus, J. Pseudo-Differ. Oper. Appl., 6, 439-459, 2016 · Zbl 1339.58014 · doi:10.1007/s11868-015-0134-8 |
| [25] | Kohn, JJ; Nirenberg, L., An algebra of pseudo-differential operators, Comm. Pure Appl. Math., 18, 269-305, 1965 · Zbl 0171.35101 · doi:10.1002/cpa.3160180121 |
| [26] | Kumar, V.; Mondal, SS, Symbolic calculus and m-ellipticity of pseudo-differential operators on \(\mathbb{Z}^n\), Anal. Appl. (Singap.), 21, 6, 1447-1475, 2023 · Zbl 1526.35347 · doi:10.1142/S0219530523500215 |
| [27] | Molahajloo, S.; Wong, MW, Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \({\mathbb{S}}^1\), J. Pseudo-Differ. Oper. Appl., 1, 183-205, 2010 · Zbl 1222.47075 · doi:10.1007/s11868-010-0010-5 |
| [28] | Pirhayati, M.: Spectral theory of pseudo-differential operators on \({\mathbb{S} }^1\). In: Pseudo-Differential Operators: Analysis, Applications and Computations. Birkhäuser, Basel (2008) |
| [29] | Ruzhansky, M.; Velasquez-Rodriguez, JP, Non-harmonic Gohberg’s lemma, Gershgorin theory and heat equation on manifolds with boundary, Math. Nachr., 294, 9, 1783-1820, 2021 · Zbl 1531.42050 · doi:10.1002/mana.201900072 |
| [30] | Ruzhansky, M., Turunen, V.: Pseudo-differential Operators and Symmetries: background analysis and advanced topics. In: Volume 2 of Pseudo-differential Operators. Theory and Applications. Birkhäuser-Verlag, Basel (2010) · Zbl 1193.35261 |
| [31] | Ruzhansky, M.; Turunen, V., Sharp Gårding inequality on compact Lie groups, J. Funct. Anal., 260, 10, 2881-2901, 2011 · Zbl 1238.22004 · doi:10.1016/j.jfa.2011.02.014 |
| [32] | Ruzhansky, M.; Turunen, V.; Wirth, J., Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl., 20, 3, 476-499, 2014 · Zbl 1432.35270 · doi:10.1007/s00041-014-9322-9 |
| [33] | Ruzhansky, M.; Wirth, J., Global functional calculus for operators on compact Lie groups, J. Funct. Anal., 267, 772-798, 2014 · Zbl 1295.35388 · doi:10.1016/j.jfa.2014.04.009 |
| [34] | Schechter, M., On the essential spectrum of an arbitrary operator I, J. Math. Anal. Appl., 13, 205-215, 1966 · Zbl 0147.12101 · doi:10.1016/0022-247X(66)90085-0 |
| [35] | Schechter, M.: Spectra of Partial Differential Operators, 2nd edn. North-Holland (1986) · Zbl 0607.35005 |
| [36] | Taylor, ME, Pseudo-differential Operators, 1981, Princton: Princton University Press, Princton · Zbl 0453.47026 |
| [37] | Velasquez-Rodriguez, JP, On some spectral properties of pseudo-differential operators on \({\mathbb{T}} \), J. Fourier Anal. Appl., 25, 5, 2703-2732, 2019 · Zbl 1447.58028 · doi:10.1007/s00041-019-09680-2 |
| [38] | Wolf, F., On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math., 12, 211-228, 1959 · Zbl 0087.30501 · doi:10.1002/cpa.3160120202 |
| [39] | Wong, MW, \(M\)-elliptic pseudo-differential operators on \(L^p\left({\mathbb{R}}^n\right)\), Math. Nachr., 279, 319-326, 2006 · Zbl 1102.47038 · doi:10.1002/mana.200310363 |
| [40] | Wong, M.W.: An Introduction to Pseudo-differential Operators, Volume 6 of Series on Analysis, Applications and Computation, 3rd ed. World Scientific, Hackensack (2014) · Zbl 1296.35226 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
