Fourier multipliers on anisotropic mixed-norm spaces of distributions
DOI:
https://doi.org/10.7146/math.scand.a-113031Abstract
A new general Hörmander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multipliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.
References
Antonić, N. and Ivec, I., On the Hörmander-Mihlin theorem for mixed-norm Lebesgue spaces, J. Math. Anal. Appl. 433 (2016), no. 1, 176–199. https://doi.org/10.1016/j.jmaa.2015.07.002
Bagby, R. J., An extended inequality for the maximal function, Proc. Amer. Math. Soc. 48 (1975), 419–422. https://doi.org/10.2307/2040276
Benedek, A. and Panzone, R., The space $L^p$, with mixed norm, Duke Math. J. 28 (1961), 301–324.
Bényi, Á. and Bownik, M., Anisotropic classes of homogeneous pseudodifferential symbols, Studia Math. 200 (2010), no. 1, 41–66. https://doi.org/10.4064/sm200-1-3
Bényi, Á. and Bownik, M., Anisotropic classes of inhomogeneous pseudodifferential symbols, Collect. Math. 64 (2013), no. 2, 155–173. https://doi.org/10.1007/s13348-011-0056-6
Borup, L. and Nielsen, M., On anisotropic Triebel-Lizorkin type spaces, with applications to the study of pseudo-differential operators, J. Funct. Spaces Appl. 6 (2008), no. 2, 107–154. https://doi.org/10.1155/2008/510584
Bownik, M., Anisotropic Hardy spaces and wavelets, vol. 164, Mem. Amer. Math. Soc., no. 781, American Mathematical Society, 2003. https://doi.org/10.1090/memo/0781
Bownik, M., Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z. 250 (2005), no. 3, 539–571. https://doi.org/10.1007/s00209-005-0765-1
Bownik, M. and Ho, K.-P., Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510. https://doi.org/10.1090/S0002-9947-05-03660-3
Cho, Y.-K. and Kim, D., A Fourier multiplier theorem on the Besov-Lipschitz spaces, Korean J. Math. 16 (2008), no. 1, 85–90.
Cleanthous, G., Georgiadis, A. G., and Nielsen, M., Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal., 34 pp., to appear, doi:10.1016/j.acha.2017.10.001. https://doi.org/10.1016/j.acha.2017.10.001
Cleanthous, G., Georgiadis, A. G., and Nielsen, M., Anisotropic mixed-norm Hardy spaces, J. Geom. Anal. 27 (2017), no. 4, 2758–2787. https://doi.org/10.1007/s12220-017-9781-8
Duong, X. T., Ouhabaz, E. M., and Sikora, A., Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), no. 2, 443–485. https://doi.org/10.1016/S0022-1236(02)00009-5
Duong, X. T. and Yan, L., Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, J. Math. Soc. Japan 63 (2011), no. 1, 295–319.
Fernandez, D. L., Vector-valued singular integral operators on $L^p$-spaces with mixed norms and applications, Pacific J. Math. 129 (1987), no. 2, 257–275.
Georgiadis, A. G., $H^p$-bounds for spectral multipliers on Riemannian manifolds, Bull. Sci. Math. 134 (2010), no. 7, 750–766. https://doi.org/10.1016/j.bulsci.2010.01.001
Georgiadis, A. G. and Nielsen, M., Pseudodifferential operators on mixed-norm Besov and Triebel-Lizorkin spaces, Math. Nachr. 289 (2016), no. 16, 2019–2036. https://doi.org/10.1002/mana.201500390
Hörmander, L., Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93–140. https://doi.org/10.1007/BF02547187
Hytönen, T. P., Anisotropic Fourier multipliers and singular integrals for vector-valued functions, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 455–468. https://doi.org/10.1007/s10231-006-0014-1
Johnsen, J., Munch Hansen, S., and Sickel, W., Anisotropic, mixed-norm Lizorkin-Triebel spaces and diffeomorphic maps, J. Funct. Spaces (2014), Art. 964794, 15 pp. https://doi.org/10.1155/2014/964794
Johnsen, J. and Sickel, W., On the trace problem for Lizorkin-Triebel spaces with mixed norms, Math. Nachr. 281 (2008), no. 5, 669–696. https://doi.org/10.1002/mana.200610634
Kerkyacharian, G. and Petrushev, P., Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, Trans. Amer. Math. Soc. 367 (2015), no. 1, 121–189. https://doi.org/10.1090/S0002-9947-2014-05993-X
Kumano-go, H., Pseudodifferential operators, MIT Press, Cambridge, Mass.-London, 1981.
Kyrezi, I. and Marias, M., $H^p$-bounds for spectral multipliers on graphs, Trans. Amer. Math. Soc. 361 (2009), no. 2, 1053–1067. https://doi.org/10.1090/S0002-9947-08-04596-0
Kyriazis, G., Petrushev, P., and Xu, Y., Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 477–513. https://doi.org/10.1112/plms/pdn010
Liu, L., Yang, D., and Yuan, W., Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels, Collect. Math. 67 (2016), no. 2, 247–310. https://doi.org/10.1007/s13348-015-0142-2
Lizorkin, P. I., Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm: applications, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 218–247.
Lohoué, N. and Marias, M., Multipliers on locally symmetric spaces, J. Geom. Anal. 24 (2014), no. 2, 627–648. https://doi.org/10.1007/s12220-012-9348-7
Marcinkiewicz, J., Sur les multiplicateurs des séries de fourier, Studia Math. 8 (1939), 78–91.
Mihlin, S. G., On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703.
Nikol\cprime ski\u ı, S. M., Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975.
Schmeisser, H.-J., Maximal inequalities and Fourier multipliers for spaces with mixed quasinorms. Applications, Z. Anal. Anwendungen 3 (1984), no. 2, 153–166. https://doi.org/10.4171/ZAA/97
Schmeisser, H.-J. and Triebel, H., Topics in Fourier analysis and function spaces, Mathematik und ihre Anwendungen in Physik und Technik, vol. 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
Triebel, H., Complex interpolation and Fourier multipliers for the spaces $B^s_p,q$ and $F^s_p,q$ of Besov-Hardy-Sobolev type: the case $0<p\leq \infty $, $0<q\leq \infty $, Math. Z. 176 (1981), no. 4, 495–510. https://doi.org/10.1007/BF01214760
Triebel, H., Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. https://doi.org/10.1007/978-3-0346-0416-1
Yang, D. and Yuan, W., New Besov-type spaces and Triebel-Lizorkin-type spaces including $Q$ spaces, Math. Z. 265 (2010), no. 2, 451–480. https://doi.org/10.1007/s00209-009-0524-9
Yang, D., Yuan, W., and Zhuo, C., Fourier multipliers on Triebel-Lizorkin-type spaces, J. Funct. Spaces Appl. (2012), Art. 431016, 37 pp. https://doi.org/10.1155/2012/431016
Bagby, R. J., An extended inequality for the maximal function, Proc. Amer. Math. Soc. 48 (1975), 419–422. https://doi.org/10.2307/2040276
Benedek, A. and Panzone, R., The space $L^p$, with mixed norm, Duke Math. J. 28 (1961), 301–324.
Bényi, Á. and Bownik, M., Anisotropic classes of homogeneous pseudodifferential symbols, Studia Math. 200 (2010), no. 1, 41–66. https://doi.org/10.4064/sm200-1-3
Bényi, Á. and Bownik, M., Anisotropic classes of inhomogeneous pseudodifferential symbols, Collect. Math. 64 (2013), no. 2, 155–173. https://doi.org/10.1007/s13348-011-0056-6
Borup, L. and Nielsen, M., On anisotropic Triebel-Lizorkin type spaces, with applications to the study of pseudo-differential operators, J. Funct. Spaces Appl. 6 (2008), no. 2, 107–154. https://doi.org/10.1155/2008/510584
Bownik, M., Anisotropic Hardy spaces and wavelets, vol. 164, Mem. Amer. Math. Soc., no. 781, American Mathematical Society, 2003. https://doi.org/10.1090/memo/0781
Bownik, M., Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z. 250 (2005), no. 3, 539–571. https://doi.org/10.1007/s00209-005-0765-1
Bownik, M. and Ho, K.-P., Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510. https://doi.org/10.1090/S0002-9947-05-03660-3
Cho, Y.-K. and Kim, D., A Fourier multiplier theorem on the Besov-Lipschitz spaces, Korean J. Math. 16 (2008), no. 1, 85–90.
Cleanthous, G., Georgiadis, A. G., and Nielsen, M., Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal., 34 pp., to appear, doi:10.1016/j.acha.2017.10.001. https://doi.org/10.1016/j.acha.2017.10.001
Cleanthous, G., Georgiadis, A. G., and Nielsen, M., Anisotropic mixed-norm Hardy spaces, J. Geom. Anal. 27 (2017), no. 4, 2758–2787. https://doi.org/10.1007/s12220-017-9781-8
Duong, X. T., Ouhabaz, E. M., and Sikora, A., Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), no. 2, 443–485. https://doi.org/10.1016/S0022-1236(02)00009-5
Duong, X. T. and Yan, L., Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, J. Math. Soc. Japan 63 (2011), no. 1, 295–319.
Fernandez, D. L., Vector-valued singular integral operators on $L^p$-spaces with mixed norms and applications, Pacific J. Math. 129 (1987), no. 2, 257–275.
Georgiadis, A. G., $H^p$-bounds for spectral multipliers on Riemannian manifolds, Bull. Sci. Math. 134 (2010), no. 7, 750–766. https://doi.org/10.1016/j.bulsci.2010.01.001
Georgiadis, A. G. and Nielsen, M., Pseudodifferential operators on mixed-norm Besov and Triebel-Lizorkin spaces, Math. Nachr. 289 (2016), no. 16, 2019–2036. https://doi.org/10.1002/mana.201500390
Hörmander, L., Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93–140. https://doi.org/10.1007/BF02547187
Hytönen, T. P., Anisotropic Fourier multipliers and singular integrals for vector-valued functions, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 455–468. https://doi.org/10.1007/s10231-006-0014-1
Johnsen, J., Munch Hansen, S., and Sickel, W., Anisotropic, mixed-norm Lizorkin-Triebel spaces and diffeomorphic maps, J. Funct. Spaces (2014), Art. 964794, 15 pp. https://doi.org/10.1155/2014/964794
Johnsen, J. and Sickel, W., On the trace problem for Lizorkin-Triebel spaces with mixed norms, Math. Nachr. 281 (2008), no. 5, 669–696. https://doi.org/10.1002/mana.200610634
Kerkyacharian, G. and Petrushev, P., Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, Trans. Amer. Math. Soc. 367 (2015), no. 1, 121–189. https://doi.org/10.1090/S0002-9947-2014-05993-X
Kumano-go, H., Pseudodifferential operators, MIT Press, Cambridge, Mass.-London, 1981.
Kyrezi, I. and Marias, M., $H^p$-bounds for spectral multipliers on graphs, Trans. Amer. Math. Soc. 361 (2009), no. 2, 1053–1067. https://doi.org/10.1090/S0002-9947-08-04596-0
Kyriazis, G., Petrushev, P., and Xu, Y., Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 477–513. https://doi.org/10.1112/plms/pdn010
Liu, L., Yang, D., and Yuan, W., Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels, Collect. Math. 67 (2016), no. 2, 247–310. https://doi.org/10.1007/s13348-015-0142-2
Lizorkin, P. I., Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm: applications, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 218–247.
Lohoué, N. and Marias, M., Multipliers on locally symmetric spaces, J. Geom. Anal. 24 (2014), no. 2, 627–648. https://doi.org/10.1007/s12220-012-9348-7
Marcinkiewicz, J., Sur les multiplicateurs des séries de fourier, Studia Math. 8 (1939), 78–91.
Mihlin, S. G., On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703.
Nikol\cprime ski\u ı, S. M., Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975.
Schmeisser, H.-J., Maximal inequalities and Fourier multipliers for spaces with mixed quasinorms. Applications, Z. Anal. Anwendungen 3 (1984), no. 2, 153–166. https://doi.org/10.4171/ZAA/97
Schmeisser, H.-J. and Triebel, H., Topics in Fourier analysis and function spaces, Mathematik und ihre Anwendungen in Physik und Technik, vol. 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
Triebel, H., Complex interpolation and Fourier multipliers for the spaces $B^s_p,q$ and $F^s_p,q$ of Besov-Hardy-Sobolev type: the case $0<p\leq \infty $, $0<q\leq \infty $, Math. Z. 176 (1981), no. 4, 495–510. https://doi.org/10.1007/BF01214760
Triebel, H., Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. https://doi.org/10.1007/978-3-0346-0416-1
Yang, D. and Yuan, W., New Besov-type spaces and Triebel-Lizorkin-type spaces including $Q$ spaces, Math. Z. 265 (2010), no. 2, 451–480. https://doi.org/10.1007/s00209-009-0524-9
Yang, D., Yuan, W., and Zhuo, C., Fourier multipliers on Triebel-Lizorkin-type spaces, J. Funct. Spaces Appl. (2012), Art. 431016, 37 pp. https://doi.org/10.1155/2012/431016
Downloads
Published
2019-06-17
How to Cite
Cleanthous, G., Georgiadis, A. G., & Nielsen, M. (2019). Fourier multipliers on anisotropic mixed-norm spaces of distributions. MATHEMATICA SCANDINAVICA, 124(2), 289–304. https://doi.org/10.7146/math.scand.a-113031
Issue
Section
Articles
