Linear and bilinear Fourier multipliers on Orlicz modulation spaces. (English) Zbl 07891495
The aim of this paper is to investigate the properties with respect to linear Fourier multipliers and bilinear Fourier multipliers from Orlicz modulation spaces to Orlicz modulation spaces.
Let \(m(\xi)\) be a bounded measurable function on \(\mathbb{R}^d\) (resp. \(m(\xi,\eta)\) on \(\mathbb{R}^{2d}\)) such that \[ T_m(f)(x)=\int_{\mathbb{R}^d}\hat{f}(\xi)m(\xi)e^{2\pi i<\xi,x>}d\xi \] \[ (\mathrm{resp.}\quad B_m(f_1,f_2)(x)=\int_{\mathbb{R}^d}\int_{\mathbf{R}^d}\hat{f_1}(\xi)\hat{f_2}(\eta)m(\xi,\eta)e^{2\pi i<\xi+\eta,x>}d\xi d\eta), \] where \(\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi i<x,\xi>}dx\) is the Fourier transform of \(f\). The study of linear Fourier multipliers in the setting of Lebesgue spaces is classical, and the investigation of bilinear multipliers was originated by R. R. Coifman and Y. Meyer [Lect. Notes Math. 779, 104–122 (1980; Zbl 0427.42006)], and was continued by L. Grafakos and R. H. Torres [Adv. Math. 165, No. 1, 124–164 (2002; Zbl 1032.42020)], C. E. Kenig and E. M. Stein [Math. Res. Lett. 6, No. 1, 1–15 (1999); erratum ibid. 6, No. 3–4, 467 (1999; Zbl 0952.42005)], M. Lacey and C. Thiele [Ann. Math. (2) 146, No. 3, 693–724 (1997; Zbl 0914.46034)] and others.
Orlicz spaces generalize \(L^p\) spaces and contain Sobolev spaces as subspaces. The linear Fourier multipliers and the bilinear Fourier multipliers on Orlicz spaces are investigated by O. Blasco and A. Osançlıol [Math. Nachr. 292, No. 12, 2522–2536 (2019; Zbl 1440.42041)], O. Blasco and R. Üster [Math. Nachr. 296, No. 12, 5400–5425 (2023; Zbl 1536.43002); Mediterr. J. Math. 21, No. 1, Paper No. 41, 21 p. (2024; Zbl 07830963)], and R. Üster [Result. Math. 76, No. 4, Paper No. 183, 15 p. (2021; Zbl 1481.43001)].
On the other hand, modulation spaces arising from the time-frequency analysis were introduced by H. G. Feichtinger [Modulation spaces on locally compact abelian groups. Techn. Rep., University of Vienna (1983)], and appear in the theory of pseudo-differential operators.
In this paper, the authors define Orlicz modulation spaces and investigate the properties of the operators \(T_m\) and \(B_m\) on the Orlicz modulation spaces.
Let \(\Phi\) be a Young function, \(\omega\) a continuous positive function with polynomial growth such that \(\omega(x+y)\leq \omega(x)\omega(y) \quad (x,y\in\mathbb{R}^d)\), and \(L^{\Phi}(\mathbb{R}^d)\) the Orlicz space defined by \(\Phi\) with the Luxemburg norm: \[ N_\Phi(f)=\inf\{\lambda>0:\int_{\mathbb{R}^d}\Phi(\frac{|f(x)|}{\lambda})dx\leq1\}. \] \(L^\Phi_\omega(\mathbb{R}^d)\) is also defined by the set of all measurable functions \(f\) such that \(f\omega\in L^\Phi(\mathbb{R}^d)\). For \(\Phi_j\) \((j=1,2)\) Young functions, the spaces \(L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) are defined as follows:
\(F\) in \(L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})=L^{\Phi_2}(\mathbb{R}^d,L^{\Phi_1}(\mathbb{R}^d))\) has the following properties:
\(F\) is a complex-valued function on \(\mathbb{R}^{2d}\), and \[ \xi\longrightarrow\ F(\cdot,\xi)\in L^{\Phi_1}(\mathbb{R}^d) \] is a Banach space valued function with the Luxemburg norm \(N_{\Phi_1}(F(\cdot,\xi))\) in \(L^{\Phi_1}(\mathbb{R}^d)\) and \[ N_{\Phi_1,\Phi_2}(F)=N_{\Phi_2}(N_{\Phi_1}(F(\cdot,\xi)))<\infty. \] For a positive continuous function \(\omega\) on \(\mathbb{R}^{2d}\), \(L^{\Phi_1,\Phi_2}_\omega(\mathbb{R}^{2d})\) is defined by the set of all functions \(F\) such that \(F\omega\in L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) with the norm \[ N_{\Phi_1,\Phi_2,\omega}(F)=N_{\Phi_1,\Phi_2}(F\omega). \] Moreover, for \(\phi\in\mathcal{S}(\mathbb{R}^d)\) and \(f\in\mathcal{S}^\prime(\mathbb{R}^d), \ V_\phi(f)\) is defined by \[ V_\phi(f)(x,s)=\int f(u)\overline{\phi(u-x)}e^{-2\pi i<u,s>}du. \] \(\mathcal{W}_d\) is also defined by the set of all positive continuous functions \(\omega\) on \(\mathbb{R}^d\) with polynomial growth such that \(\omega(x+y)\leq\omega(x)\omega(y)\ (x,y\in\mathbb{R}^d)\). Then for \(0\neq\psi\in\mathcal{S}(\mathbb{R}^d)\),\(\omega\in\mathcal{W}_{2d}\) and \(\Phi,\ \Phi_1,\ \Phi_2\) Young functions, the Orlicz modulation spaces \(M_\omega^\Phi\) and \(M_\omega^{\Phi_1,\Phi_2}\) are defined by the following: \[ M_\omega^\Phi=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^\Phi(\mathbb{R}^{2d})\}. \] \[ M_\omega^{\Phi_1,\Phi_2}=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\}. \] \[ ||f||_{M_\omega^\Phi}=N_{\Phi,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^\Phi(\mathbb{R}^d). \] \[ ||f||_{M_\omega^{\Phi_1,\Phi_2}}=N_{\Phi_1,\Phi_2,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^d). \] Let \(\Phi_i,\Psi_i\) be Young functions, \(\omega_i\in\mathcal{W}_{2d}\), and \(M_{\omega_i}^{\Phi_i,\Psi_i}(\mathbb{R}^d)\) be the corresponding Orlicz modulation spaces for \(i=1,2,3\).
In this paper, the authors investigate the properties of the linear operators \(T_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\) to \(M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) and the bilinear operators \(B_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\times M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) to \(M_{\omega_3}^{\Phi_3,\Psi_3}(\mathbb{R}^d)\).
Let \(m(\xi)\) be a bounded measurable function on \(\mathbb{R}^d\) (resp. \(m(\xi,\eta)\) on \(\mathbb{R}^{2d}\)) such that \[ T_m(f)(x)=\int_{\mathbb{R}^d}\hat{f}(\xi)m(\xi)e^{2\pi i<\xi,x>}d\xi \] \[ (\mathrm{resp.}\quad B_m(f_1,f_2)(x)=\int_{\mathbb{R}^d}\int_{\mathbf{R}^d}\hat{f_1}(\xi)\hat{f_2}(\eta)m(\xi,\eta)e^{2\pi i<\xi+\eta,x>}d\xi d\eta), \] where \(\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi i<x,\xi>}dx\) is the Fourier transform of \(f\). The study of linear Fourier multipliers in the setting of Lebesgue spaces is classical, and the investigation of bilinear multipliers was originated by R. R. Coifman and Y. Meyer [Lect. Notes Math. 779, 104–122 (1980; Zbl 0427.42006)], and was continued by L. Grafakos and R. H. Torres [Adv. Math. 165, No. 1, 124–164 (2002; Zbl 1032.42020)], C. E. Kenig and E. M. Stein [Math. Res. Lett. 6, No. 1, 1–15 (1999); erratum ibid. 6, No. 3–4, 467 (1999; Zbl 0952.42005)], M. Lacey and C. Thiele [Ann. Math. (2) 146, No. 3, 693–724 (1997; Zbl 0914.46034)] and others.
Orlicz spaces generalize \(L^p\) spaces and contain Sobolev spaces as subspaces. The linear Fourier multipliers and the bilinear Fourier multipliers on Orlicz spaces are investigated by O. Blasco and A. Osançlıol [Math. Nachr. 292, No. 12, 2522–2536 (2019; Zbl 1440.42041)], O. Blasco and R. Üster [Math. Nachr. 296, No. 12, 5400–5425 (2023; Zbl 1536.43002); Mediterr. J. Math. 21, No. 1, Paper No. 41, 21 p. (2024; Zbl 07830963)], and R. Üster [Result. Math. 76, No. 4, Paper No. 183, 15 p. (2021; Zbl 1481.43001)].
On the other hand, modulation spaces arising from the time-frequency analysis were introduced by H. G. Feichtinger [Modulation spaces on locally compact abelian groups. Techn. Rep., University of Vienna (1983)], and appear in the theory of pseudo-differential operators.
In this paper, the authors define Orlicz modulation spaces and investigate the properties of the operators \(T_m\) and \(B_m\) on the Orlicz modulation spaces.
Let \(\Phi\) be a Young function, \(\omega\) a continuous positive function with polynomial growth such that \(\omega(x+y)\leq \omega(x)\omega(y) \quad (x,y\in\mathbb{R}^d)\), and \(L^{\Phi}(\mathbb{R}^d)\) the Orlicz space defined by \(\Phi\) with the Luxemburg norm: \[ N_\Phi(f)=\inf\{\lambda>0:\int_{\mathbb{R}^d}\Phi(\frac{|f(x)|}{\lambda})dx\leq1\}. \] \(L^\Phi_\omega(\mathbb{R}^d)\) is also defined by the set of all measurable functions \(f\) such that \(f\omega\in L^\Phi(\mathbb{R}^d)\). For \(\Phi_j\) \((j=1,2)\) Young functions, the spaces \(L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) are defined as follows:
\(F\) in \(L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})=L^{\Phi_2}(\mathbb{R}^d,L^{\Phi_1}(\mathbb{R}^d))\) has the following properties:
\(F\) is a complex-valued function on \(\mathbb{R}^{2d}\), and \[ \xi\longrightarrow\ F(\cdot,\xi)\in L^{\Phi_1}(\mathbb{R}^d) \] is a Banach space valued function with the Luxemburg norm \(N_{\Phi_1}(F(\cdot,\xi))\) in \(L^{\Phi_1}(\mathbb{R}^d)\) and \[ N_{\Phi_1,\Phi_2}(F)=N_{\Phi_2}(N_{\Phi_1}(F(\cdot,\xi)))<\infty. \] For a positive continuous function \(\omega\) on \(\mathbb{R}^{2d}\), \(L^{\Phi_1,\Phi_2}_\omega(\mathbb{R}^{2d})\) is defined by the set of all functions \(F\) such that \(F\omega\in L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) with the norm \[ N_{\Phi_1,\Phi_2,\omega}(F)=N_{\Phi_1,\Phi_2}(F\omega). \] Moreover, for \(\phi\in\mathcal{S}(\mathbb{R}^d)\) and \(f\in\mathcal{S}^\prime(\mathbb{R}^d), \ V_\phi(f)\) is defined by \[ V_\phi(f)(x,s)=\int f(u)\overline{\phi(u-x)}e^{-2\pi i<u,s>}du. \] \(\mathcal{W}_d\) is also defined by the set of all positive continuous functions \(\omega\) on \(\mathbb{R}^d\) with polynomial growth such that \(\omega(x+y)\leq\omega(x)\omega(y)\ (x,y\in\mathbb{R}^d)\). Then for \(0\neq\psi\in\mathcal{S}(\mathbb{R}^d)\),\(\omega\in\mathcal{W}_{2d}\) and \(\Phi,\ \Phi_1,\ \Phi_2\) Young functions, the Orlicz modulation spaces \(M_\omega^\Phi\) and \(M_\omega^{\Phi_1,\Phi_2}\) are defined by the following: \[ M_\omega^\Phi=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^\Phi(\mathbb{R}^{2d})\}. \] \[ M_\omega^{\Phi_1,\Phi_2}=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\}. \] \[ ||f||_{M_\omega^\Phi}=N_{\Phi,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^\Phi(\mathbb{R}^d). \] \[ ||f||_{M_\omega^{\Phi_1,\Phi_2}}=N_{\Phi_1,\Phi_2,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^d). \] Let \(\Phi_i,\Psi_i\) be Young functions, \(\omega_i\in\mathcal{W}_{2d}\), and \(M_{\omega_i}^{\Phi_i,\Psi_i}(\mathbb{R}^d)\) be the corresponding Orlicz modulation spaces for \(i=1,2,3\).
In this paper, the authors investigate the properties of the linear operators \(T_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\) to \(M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) and the bilinear operators \(B_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\times M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) to \(M_{\omega_3}^{\Phi_3,\Psi_3}(\mathbb{R}^d)\).
Reviewer: Enji Sato (Yamagata)
MSC:
| 42A45 | Multipliers in one variable harmonic analysis |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
References:
| [1] | Bényi, Á.; Grafakos, L.; Gröchenig, K.; Okoudjou, KA, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal., 19, 1, 131-139, 2005 · Zbl 1078.42022 · doi:10.1016/j.acha.2005.02.002 |
| [2] | Bényi, Á.; Gröchenig, K.; Okoudjou, KA; Rogers, LG, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246, 366-384, 2007 · Zbl 1120.42010 · doi:10.1016/j.jfa.2006.12.019 |
| [3] | Blasco, O., Notes on the spaces of bilinear multipliers, Rev. Un. Mat. Argentina, 50, 20-34, 2009 |
| [4] | Blasco, O.; Osançlıol, A., Notes on bilinear multipliers on Orlicz spaces, Math. Nachr., 292, 2522-2536, 2019 · Zbl 1440.42041 · doi:10.1002/mana.201800551 |
| [5] | Blasco, O.; Üster, R., Transference and restriction of Fourier multipliers on Orlicz spaces, Math. Nach. Math. Nachr., 296, 5400-5425, 2023 · Zbl 1536.43002 · doi:10.1002/mana.202200315 |
| [6] | Blasco, O., Üster, R.: Transference and restriction of bilinear Fourier multipliers on Orlicz spaces (Accepted) · Zbl 07830963 |
| [7] | Bukhvalov, A.V.: Spaces with mixed norm Vestnik Leningrad. Univ. 19 Mat. Meh. Astronom. Vyp. 4 [1973], pp. 5-12; English transl. in Vestn. Leningr. Univ. Math. 6 [1979], 303-311 · Zbl 0419.46010 |
| [8] | Cordero, E.; Nicola, F., Sharp continuity results for the short-time Fourier transforms and for localization operators, Monatshefte für Mathematik, 162, 251-276, 2011 · Zbl 1217.47091 · doi:10.1007/s00605-010-0210-3 |
| [9] | Coifman, R.R., Meyer, Y.: Fourier Analysis of Multilinear Convolution, Calderón Theorem and Analysis of Lipschitz Curves. Euclidean Harmonic Analysis (Proc. Sem. Univ. Maryland, College Univ., MD). Lecture Notes in Mathematics, vol. 779, pp. 104-122 (1979) · Zbl 0427.42006 |
| [10] | Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. Technical report. University of Vienna (1983) |
| [11] | Feichtinger, HG; Narimani, G., Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., 21, 349-359, 2006 · Zbl 1106.42005 · doi:10.1016/j.acha.2006.04.010 |
| [12] | Gilbert, JE; Nahmod, AR, Bilinear operators for non-smooth symbol I, J. Fourier Anal. Appl., 7, 435-467, 2001 · Zbl 0994.42014 · doi:10.1007/BF02511220 |
| [13] | Gröchenig, K., Foundations of Time Frequency Analysis, 2001, Boston: Birkhäuser Boston, Boston · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1 |
| [14] | Grafakos, L., Classical Fourier Analysis, 2008, Berlin: Springer, Berlin · Zbl 1220.42001 · doi:10.1007/978-0-387-09432-8 |
| [15] | Grafakos, L.; Torres, R., Multilinear Calderón-Zygmund theory, Adv. Math., 165, 124-164, 2002 · Zbl 1032.42020 · doi:10.1006/aima.2001.2028 |
| [16] | Harjulehto, P.; Hästö, P., Orlicz Spaces and Generalized Orlicz Spaces, 2019, Cham: Springer, Cham · Zbl 1436.46002 |
| [17] | Kenig, CE; Stein, EM, Multilinear estimates and fractional integration, Math. Res. Lett., 6, 1-15, 1999 · Zbl 0952.42005 · doi:10.4310/MRL.1999.v6.n1.a1 |
| [18] | Kobayashi, M., Multipliers on modulation spaces, SUT J. Math., 42, 2, 305-312, 2006 · Zbl 1131.42010 · doi:10.55937/sut/1262445124 |
| [19] | Krasnosel’skii, MA; Rutickii, JB, Convex Functions and Orlicz Spaces, 1961, Graningen: Noordhoff, Graningen · Zbl 0095.09103 |
| [20] | Kulak, O.; Gürkanlı, AT, Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces, J. Inequal. Appl., 8, 259, 2013 · Zbl 1296.42005 · doi:10.1186/1029-242X-2013-259 |
| [21] | Lacey, M.; Thiele, C., \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty \), Ann. Math., 146, 693-724, 1997 · Zbl 0914.46034 · doi:10.2307/2952458 |
| [22] | Lacey M.: Weak bounds for the bilinear Hilbert transform on \(L^p\). Documenta Mathematia, extra volume ICM 1-1000 (1997) |
| [23] | Lacey, M.: Weak bounds for the bilinear Hilbert transform on \(L^p\). J. Am. Math. Soc. 15 (2002) |
| [24] | Lacey M.: Weak bounds for the bilinear Hilbert transform on \(L^p\), 469-496 MR 2003b:42017 |
| [25] | Larsen, R., An Introduction to the Theory of Multipliers, 1971, Berlin: Springer, Berlin · Zbl 0213.13301 · doi:10.1007/978-3-642-65030-7 |
| [26] | O’Neil, R., Fractional integration in Orlicz spaces I, Trans. Am. Math. Soc., 115, 300-328, 1965 · Zbl 0132.09201 · doi:10.1090/S0002-9947-1965-0194881-0 |
| [27] | Osançlıol, A.; Öztop, S., Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc., 99, 399-414, 2015 · Zbl 1338.46039 · doi:10.1017/S1446788715000257 |
| [28] | Öztop, S.; Samei, E., Twisted Orlicz algebras I, Studia Math., 236, 271-296, 2017 · Zbl 1368.43006 · doi:10.4064/sm8562-9-2016 |
| [29] | Öztop, S.; Samei, E., Twisted Orlicz algebras II, Math. Nachr., 292, 1122-1136, 2019 · Zbl 1419.43001 · doi:10.1002/mana.201700362 |
| [30] | Rao, MM; Ren, ZD, Theory of Orlicz Spaces, 1991, CRM Press · Zbl 0724.46032 |
| [31] | Reiter, H.; Stegeman, JD, Classical Harmonic Analysis and Locally Compact Groups, 2000, Oxford: Clarendon Press, Oxford · Zbl 0965.43001 · doi:10.1093/oso/9780198511892.001.0001 |
| [32] | Rodriguez-López, S., A homomorphism theorem for bilinear multipliers, J. Lond. Math. Soc., 70, 2, 619-636, 2013 · Zbl 1290.43008 · doi:10.1112/jlms/jdt043 |
| [33] | Rudin, W., Real and Complex Analysis, 1974, New Delhi: Tata McGraw-Hill Publishing Company, New Delhi · Zbl 0278.26001 |
| [34] | Sandıkçı, A.; Gürkanlı, AT, Gabor analysis on the spaces \(M(p, q, w)(\mathbb{R}^d)\) and \(S(p, q, r, w)(\mathbb{R}^d)\), Acta Math. Sci. Ser. B Engl. Ed., 31, 1, 141-158, 2011 · Zbl 1240.43003 · doi:10.1016/S0252-9602(11)60216-6 |
| [35] | Schnackers, C., Führ Orlicz , H.: Modulation Spaces. In: Proceedings of the 10th International Conference on Sampling Theory and Applications |
| [36] | Stein, EM; Weiss, G., Introduction to Fourier Analysis on Euclidean spaces, 1971, Princeton: Princeton University Press, Princeton · Zbl 0232.42007 |
| [37] | Toft, J.; Üster, R.; Nabizadeh, E.; Öztop, S., Continuity and Bargmann mapping properties of quasi-Banach Orlicz modulation spaces, Forum Math., 34, 5, 1205-1232, 2022 · Zbl 1505.46029 |
| [38] | Üster, R., Multipliers for the weighted Orlicz spaces of a locally compact abelian group, Results Math., 76, 4, 183, 2021 · Zbl 1481.43001 · doi:10.1007/s00025-021-01493-4 |
| [39] | Villarroya, P., Bilinear multipliers on Lorentz spaces, Czechoslovac Math. J., 58, 133, 1045-1057, 2008 · Zbl 1174.42011 · doi:10.1007/s10587-008-0067-y |
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