Functions operating on modulation spaces and nonlinear dispersive equations. (English) Zbl 1348.42020
Summary: The aim of this paper is twofold. We show that if a complex function \(F\) on \(\mathbb{C}\) operates in the modulation spaces \(M^{p, 1}(\mathbb{R}^n)\) by composition, then \(F\) is real analytic on \(\mathbb{R}^2 \approx \mathbb{C}\). This answers negatively the open question posed in [M. Ruzhansky et al., in: Evolution equations of hyperbolic and Schrödinger type. Asymptotics, estimates and nonlinearities. Basel: Springer. Progress in Mathematics 301, 267–283 (2012; Zbl 1250.35006)] regarding the general power type nonlinearity of the form \(|u|^\alpha u\). We also characterise the functions that operate in the modulation space \(M^{1, 1}(\mathbb{R}^n)\).
The local well-posedness of the NLS, NLW and NLKG equations for the “real entire” nonlinearities are also studied in some weighted modulation spaces \(M_s^{p, q}(\mathbb{R}^n)\).
The local well-posedness of the NLS, NLW and NLKG equations for the “real entire” nonlinearities are also studied in some weighted modulation spaces \(M_s^{p, q}(\mathbb{R}^n)\).
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
| 42B37 | Harmonic analysis and PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
Keywords:
modulation spaces; nonlinear operation; dispersive equations; local well-posedness; NLS equationsCitations:
Zbl 1250.35006References:
| [1] | Bényi, Á.; Okoudjou, Kasso A., Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41, 3, 549-558 (2009) · Zbl 1173.35115 |
| [2] | Bényi, Á.; Gröchenig, K.; Okoudjou, K. A.; Rogers, L. G., Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246, 2, 366-384 (2007) · Zbl 1120.42010 |
| [3] | Bényi, Á.; Oh, T., Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math., 228, 5, 2943-2981 (2011) · Zbl 1229.42021 |
| [4] | Cazenave, T., Semilinear Schrödinger Equations, Courant Lect. Notes Math. (2003) · Zbl 1055.35003 |
| [5] | Cordero, E.; Nicola, F., Remarks on Fourier multipliers and applications to wave equations, J. Math. Anal. Appl., 353, 583-591 (2009) · Zbl 1164.35052 |
| [6] | Feichtinger, H. G., On a new Segal algebra, Monatsh. Math., 92, 4, 269-289 (1981) · Zbl 0461.43003 |
| [7] | Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal., 86, 307-340 (1989) · Zbl 0691.46011 |
| [8] | Feichtinger, H. G.; Gröchenig, K., Banach Spaces related to integrable group representations and their atomic decompositions. Part II, Monatsh. Math., 108, 129-148 (1989) · Zbl 0713.43004 |
| [9] | Folland, G. B., Real Analysis, Modern Techniques and Their Applications (1999), Wiley-Interscience Publ.: Wiley-Interscience Publ. New York · Zbl 0924.28001 |
| [10] | Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations. 1. The Cauchy problem, general case, J. Funct. Anal., 32, 1-32 (1979) · Zbl 0396.35028 |
| [11] | Gröbner, P., Banachräume Glatter Funktionen und Zerlegungsmethoden (1992), University of Vienna, Doctoral thesis |
| [12] | Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0966.42020 |
| [13] | Helson, H.; Kahane, J.-P.; Katznelson, Y.; Rudin, W., The functions which operate on Fourier transforms, Acta Math., 102, 135-157 (1959) · Zbl 0091.10902 |
| [14] | Hörmander, L., Estimates for translation invariant operators in \(L^p\) spaces, Acta Math., 104, 93-140 (1960) · Zbl 0093.11402 |
| [15] | Keel, M.; Tao, T., End point Strichartz estimates, Amer. J. Math., 120, 955-980 (1998) · Zbl 0922.35028 |
| [16] | Kobayashi, M., Modulation spaces \(M^{p, q}\) for \(0 < p, q \leq \infty \), J. Funct. Spaces Appl., 4, 3, 329-341 (2006) · Zbl 1133.46308 |
| [17] | Lebedev, V.; Olevskiĭ, A., \(C^1\) changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers, Geom. Funct. Anal., 4, 2, 213-235 (1994) · Zbl 0798.42004 |
| [18] | Lévy, P., Sur la convergence absolue des séries de Fourier, Compos. Math., 1, 1-14 (1934) · JFM 60.0227.01 |
| [19] | Okoudjou, K., Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., 132, 6, 1639-1647 (2004) · Zbl 1044.46030 |
| [20] | Rudin, W., Fourier Analysis on Groups (1990), Wiley Classics Library Edition · Zbl 0107.09603 |
| [21] | Ruzhansky, M.; Sugimoto, M.; Wang, B., Evolution equations of hyperbolic and Schrödinger type, (Progr. Math., vol. 301 (2012), Birkhäuser/Springer Basel AG: Birkhäuser/Springer Basel AG Basel), 267-283 · Zbl 1256.42038 |
| [22] | Ruzhansky, M.; Sugimoto, M.; Toft, J.; Tomita, N., Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr., 284, 16, 2078-2092 (2011) · Zbl 1228.35279 |
| [23] | Ryosuke, H.; Tsutsumi, M., On existence of global solutions of Schrödinger equations with subcritical nonlinearity for \(\hat{L^p} \)-initial data, Proc. Amer. Math. Soc., 140, 11, 3905-3920 (2012) · Zbl 1283.35126 |
| [24] | Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 3, 705-714 (1977) · Zbl 0372.35001 |
| [25] | Sugimoto, M., The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248, 1, 79-106 (2007) · Zbl 1124.42018 |
| [26] | Sugimoto, M.; Tomita, N.; Wang, B., Remarks on nonlinear operations on modulation spaces, Integral Transforms Spec. Funct., 22, 4-5, 351-358 (2011) · Zbl 1221.44007 |
| [27] | Tao, T., Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106 (2006), AMS · Zbl 1106.35001 |
| [28] | Toft, J., Continuity properties for modulation spaces with applications to pseudo-differential operators, J. Funct. Anal., 207, 399-429 (2004) · Zbl 1083.35148 |
| [29] | Triebel, H., Modulation spaces on the Euclidean \(n\)-space, Z. Anal. Anwend., 2, 5, 443-457 (1983) · Zbl 0521.46026 |
| [30] | Vargas, A.; Vega, L., Global well-posedness for 1D non-linear Schrödinger equation for data with an infinite \(L^2\) norm, J. Math. Pures Appl. (9), 80, 10, 1029-1044 (2001) · Zbl 1027.35134 |
| [31] | Wang, B. X.; Zhao, L.; Guo, B., Isometric decomposition operators, function spaces \(E_{p, q}^\lambda\) and applications to nonlinear evolution equations, J. Funct. Anal., 233, 1, 1-39 (2006) · Zbl 1099.46023 |
| [32] | Wang, B. X.; Zhaohui, H.; Chengchun, H.; Zihua, G., Harmonic Analysis Method for Nonlinear Evolution Equations. I (2011), World Scientific Publishing Co. Pte. Lt. · Zbl 1254.35002 |
| [33] | Wang, B. X.; Huang, C. Y., Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239, 213-250 (2007) · Zbl 1219.35289 |
| [34] | Wang, B. X.; Hudzik, H., The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232, 36-73 (2007) · Zbl 1121.35132 |
| [35] | Wiener, N., Tauberian theorems, Ann. of Math., 33, 1-100 (1932) · JFM 58.0226.02 |
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.
