Research and Innovation Articles
Published 2018-12-12
Keywords
- Pseudo-differential operator,
- discrete operator,
- periodic operator,
- nuclearity,
- boundedness
- Fourier integral operator,
- multilinear analysis ...More
How to Cite
Cardona, D., & Kumar, V. (2018). Multilinear analysis for discrete and periodic pseudo-differential operators in Lp-spaces. Revista Integración, Temas De matemáticas, 36(2), 151–164. https://doi.org/10.18273/revint.v36n2-2018006
Abstract
In this note we announce our investigation on the Lp properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators by
showing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials
as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.
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References
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[20] Ghaemi M. B., Jamalpour Birgani M. and Wong M.W., “Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products”, J. Pseudo-Differ. Oper. Appl. 8 (2017), No. 2, 191–201.
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[28] Kumar V., “Pseudo-differential operators on homegeneous spaces of Compact and Hausdorff groups”, to appear in Forum Mathematicum, (2018).
[29] Mclean W.M., “Local and Global description of periodic pseudo-differential operators”, Math. Nachr. 150 (191), 151–161.
[30] Michalowski N., Rule D. and Staubach W., “Multilinear pseudodifferential operators beyond Calderón-Zygmund operators”, J. Math. Anal. Appl. 414 (2014), 149–165.
[31] Miyachi A. and Tomita N., “Minimal smoothness conditions for bilinear Fourier multipliers”, Rev. Mat. Iberoam. 29 (2013), 495–530.
[32] Miyachi A. and Tomita N. “Calderón-Vaillancourt type theorem for bilinear operators”, Indiana Univ. Math. J. 62 (2013), 1165–1201.
[33] Miyachi A. and Tomita N. “Bilinear pseudo-differential operators with exotic symbols”, to appear in Ann. Inst. Fourier (Grenoble), arXiv:180106744.
[34] Molahajloo S., “A characterization of compact pseudo-differential operators on S1”, Oper. Theory Adv. Appl. 213 (2011), 25–29.
[35] Molahajloo S. and Wong M.W., “Pseudo-differential Operators on S1”, New developments in pseudo-differential operators, Eds. L. Rodino and M.W. Wong. 297–306 (2008).
[36] Molahajloo S. and Wong M.W., “Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on S1” J. Pseudo-Differ. Oper. Appl. 1 (2010), No. 2, 183–205.
[37] Muscalu C., Tao T. and Thiele C., “Multilinear operators given by singular multipliers”, J. Amer. Math. Soc. 15 (2002), 469–496.
[38] Rabinovich V.S., “Exponential estimates of solutions of pseudo-differential equations on the lattice (μZ)n, applications to the lattice Schrödinger and Dirac operators”, J. Pseudo-Differ. Oper. Appl. 1 (2010), No. 2, 233–253.
[39] Rabinovich V.S., “Wiener algebra of operators on the lattice (μZ)n depending on the small parameter μ > 0”, Complex Var. Elliptic Equ. 58 (2013), No. 6, 751–766.
[40] Rabinovich V.S. and Roch S., “The essential spectrum of Schrödinger operators on lattices”, J. Phys. A. 39 (2006), No. 26, 8377–8394.
[41] Rabinovich V.S. and Roch S., “Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics”, J. Phys. A. 42 (2009), No. 38, 385207.
[42] Rempala J.A., “On a proof of the boundedness and nuclearity of pseudodifferential operators in Rn”, Annales Polonici Mathematici 52 (1990), 59–65.
[43] Rodriguez C.A., “Lp−estimates for pseudo-differential operators on Zn”, J. Pseudo-Differ. Oper. Appl. 2 (2011), 367–365.
[44] Ruzhansky M. and Turunen V., “Quantization of Pseudo-Differential Operators on the Torus”, J. Fourier Anal. Appl. 16 (2010), 943–982.
[45] Ruzhansky M. and Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhaüser-Verlag, Basel, (2010).
[46] Tomita N., “A Hörmander type multiplier theorem for multilinear operators”, J. Funct. Anal. 259 (2010), 2028–2044.
[47] Turunen V. and Vainikko G., “On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwendungen 17 (1998), 9–22.
[48] Stein E. and Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J. (1971).
[2] Aoki S., “On the boundedness and the nuclearity of pseudo-differential operators”, Comm. Partial Diff. Equations 6 (1981), No. 8, 849–881.
[3] Bényi A., Bernicot F., Maldonado D., Naibo V. and Torres R., “On the Hörmander classes of bilinear pseudo-differential operators II”, Indiana Univ. Math. J. 62 (2013), 1733–1764.
[4] Bényi A., Maldonado D., Naibo V. and Torres R., “On the Hörmander classes of bilinear pseudodifferential operators”, Integral Equ. Oper. Theory 67 (2010), 341–364.
[5] Botchway L., Kibiti G. and Ruzhansky M., “Difference equations and pseudo-differential operators on Zn”, arXiv:1705.07564.
[6] Cardona D. “Estimativos L2 para una clase de operadores pseudodiferenciales definidos en el toro”, Rev. Integr. Temas Mat. 31 (2013), No. 2, 142–157.
[7] Cardona D., “Weak type (1, 1) bounds for a class of periodic pseudo-differential operators”, J. Pseudo-Differ. Oper. Appl. 5 (2014), No. 4, 507–515.
[8] Cardona D., “On the boundedness of periodic pseudo-differential operators”, Monatsh. Math. 185 (2017), No. 2, 189–206.
[9] Cardona D., “Pseudo-differential operators on Zn with applications to discrete fractional integral operators”, arXiv:1803.00231.
[10] Cardona D. and Kumar V., “Lp-boundedness and Lp-nuclearity of multilinear pseudodifferential operators on Zn and the torus T n”, arXiv:1809.08380.
[11] Cardona D., Messiouene R. and Senoussaoui A., “Lp-bounds for periodic Fourier integral operators (Fourier series operators)”, arXiv:1807.09892.
[12] Catana V., “Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn”, Math. Model. Nat. Phenom. 9 (2014), No. 5, 17–38.
[13] Coifman R. and Meyer Y., “On commutators of singular integrals and bilinear singular integrals”, Trans. Amer. Math. Soc. 212 (1975), 315–331.
[14] Coifman R. and Meyer Y., Ondelettes et operateurs III. Operateurs multilineaires, Hermann, Paris, (1991).
[15] Delgado J., “Lp bounds for pseudo-differential operators on the torus”, Oper. Theory Adv. Appl. 231 (2012), 103–116.
[16] Delgado J., “A trace formula for nuclear operators on Lp”, Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications, in: Schulze, B.W., Wong, M.W. (eds.) 205, Birkhäuser, Basel (2010), 181–193.
[17] Delgado J. and Wong M.W., “Lp-nuclear pseudo-differential operators on Z and S1”, Proc. Amer. Math. Soc. 141 (2013), No. 11, 3935–3944.
[18] Delgado J., “The trace of nuclear operators on Lp(μ) for -finite Borel measures on second countable spaces”, Integr. Equ. Oper. Theory 68 (2010), No. 1, 61–74.
[19] Fujita M. and Tomita N., “Weighted norm inequalities for multilinear Fourier multipliers”, Trans. Amer. Math. Soc. 364 (2012), No. 12, 6335–6353.
[20] Ghaemi M. B., Jamalpour Birgani M. and Wong M.W., “Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products”, J. Pseudo-Differ. Oper. Appl. 8 (2017), No. 2, 191–201.
[21] Grafakos L., Miyachi A. and Tomita N., “On multilinear Fourier multipliers of limited smoothness”, Canad. J. Math. 65 (2013), 299–330.
[22] Grafakos L. and Si Z., “The Hörmander multiplier theorem for multilinear operators”, J. Reine Angew. Math. 668 (2012), 133–147.
[23] Grafakos L. and Torres R., “Discrete decompositions for bilinear operators and almost diagonal conditions”, Trans. Amer. Math. Soc. 354 (2012), 1153–1176.
[24] Grafakos L. and Torres R., “Multilinear Calderón-Zygmund theory”, Adv. Math. 165 (2002), 124–164.
[25] Hörmander L., The Analysis of the linear partial differential operators Vol. III, IV Springer-Verlag, (1985).
[26] Jamalpour Birgani M., “Characterizations of Nuclear Pseudo-differential Operators on Z with some Applications”, Math. Model. Nat. Phenom. 13 (2018), 13–30.
[27] Kenig C. and Stein E., “Multilinear estimates and fractional integration”, Math. Res. Lett. 6 (1999), 1–15.
[28] Kumar V., “Pseudo-differential operators on homegeneous spaces of Compact and Hausdorff groups”, to appear in Forum Mathematicum, (2018).
[29] Mclean W.M., “Local and Global description of periodic pseudo-differential operators”, Math. Nachr. 150 (191), 151–161.
[30] Michalowski N., Rule D. and Staubach W., “Multilinear pseudodifferential operators beyond Calderón-Zygmund operators”, J. Math. Anal. Appl. 414 (2014), 149–165.
[31] Miyachi A. and Tomita N., “Minimal smoothness conditions for bilinear Fourier multipliers”, Rev. Mat. Iberoam. 29 (2013), 495–530.
[32] Miyachi A. and Tomita N. “Calderón-Vaillancourt type theorem for bilinear operators”, Indiana Univ. Math. J. 62 (2013), 1165–1201.
[33] Miyachi A. and Tomita N. “Bilinear pseudo-differential operators with exotic symbols”, to appear in Ann. Inst. Fourier (Grenoble), arXiv:180106744.
[34] Molahajloo S., “A characterization of compact pseudo-differential operators on S1”, Oper. Theory Adv. Appl. 213 (2011), 25–29.
[35] Molahajloo S. and Wong M.W., “Pseudo-differential Operators on S1”, New developments in pseudo-differential operators, Eds. L. Rodino and M.W. Wong. 297–306 (2008).
[36] Molahajloo S. and Wong M.W., “Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on S1” J. Pseudo-Differ. Oper. Appl. 1 (2010), No. 2, 183–205.
[37] Muscalu C., Tao T. and Thiele C., “Multilinear operators given by singular multipliers”, J. Amer. Math. Soc. 15 (2002), 469–496.
[38] Rabinovich V.S., “Exponential estimates of solutions of pseudo-differential equations on the lattice (μZ)n, applications to the lattice Schrödinger and Dirac operators”, J. Pseudo-Differ. Oper. Appl. 1 (2010), No. 2, 233–253.
[39] Rabinovich V.S., “Wiener algebra of operators on the lattice (μZ)n depending on the small parameter μ > 0”, Complex Var. Elliptic Equ. 58 (2013), No. 6, 751–766.
[40] Rabinovich V.S. and Roch S., “The essential spectrum of Schrödinger operators on lattices”, J. Phys. A. 39 (2006), No. 26, 8377–8394.
[41] Rabinovich V.S. and Roch S., “Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics”, J. Phys. A. 42 (2009), No. 38, 385207.
[42] Rempala J.A., “On a proof of the boundedness and nuclearity of pseudodifferential operators in Rn”, Annales Polonici Mathematici 52 (1990), 59–65.
[43] Rodriguez C.A., “Lp−estimates for pseudo-differential operators on Zn”, J. Pseudo-Differ. Oper. Appl. 2 (2011), 367–365.
[44] Ruzhansky M. and Turunen V., “Quantization of Pseudo-Differential Operators on the Torus”, J. Fourier Anal. Appl. 16 (2010), 943–982.
[45] Ruzhansky M. and Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhaüser-Verlag, Basel, (2010).
[46] Tomita N., “A Hörmander type multiplier theorem for multilinear operators”, J. Funct. Anal. 259 (2010), 2028–2044.
[47] Turunen V. and Vainikko G., “On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwendungen 17 (1998), 9–22.
[48] Stein E. and Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J. (1971).