Banach Journal of Mathematical Analysis

New function spaces related to Morrey spaces and the Fourier transform

Shohei Nakamura and Yoshihiro Sawano

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We introduce new function spaces to handle the Fourier transform on Morrey spaces and investigate fundamental properties of the spaces. As an application, we generalize the Stein–Tomas Strichartz estimate to our spaces. The geometric property of Morrey spaces and related function spaces will improve some well-known estimates.

Article information

Banach J. Math. Anal., Volume 12, Number 1 (2018), 1-30.

Received: 25 May 2016
Accepted: 23 October 2016
First available in Project Euclid: 16 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 46B10: Duality and reflexivity [See also 46A25] 42B35: Function spaces arising in harmonic analysis

Morrey spaces Fourier transform Schrödinger propagator Strichartz estimates


Nakamura, Shohei; Sawano, Yoshihiro. New function spaces related to Morrey spaces and the Fourier transform. Banach J. Math. Anal. 12 (2018), no. 1, 1--30. doi:10.1215/17358787-2017-0014.

Export citation


  • [1] D. R. Adams,A note on Riesz potentials, Duke Math. J.42(1975), no. 4, 765–778.
  • [2] P. Bégout and A. Vargas,Mass concentration Phenomena for the $L^{2}$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc.359(2007), no. 11, 5257–5282.
  • [3] O. Blasco, A. Ruiz, and L. Vega,Non-interpolation in Morrey–Campanate and block spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)28(1999), 31–40.
  • [4] F. Chiarenza and M. Frasca,Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl.7(1987), no. 3–4, 273–279.
  • [5] L. C. Evans,Partial Differential Equations, 2nd ed., Grad. Stud. Math.19, Amer. Math. Soc., Providence, 2010.
  • [6] S. Nakamura, T. Noi, and Y. Sawano,Generalized Morrey spaces and trace operator, Sci. China Math.59(2016), 281–336.
  • [7] S. Nakamura and Y. Sawano,The singular integral operator and its commutator on weighted Morrey spaces, Collect. Math.68(2017), no. 2, 145–174.
  • [8] P. Olsen,Fractional integration, Morrey spaces and a Schrödinger equation, Comm. Partial Differential Equations.20(1995), no. 11–12, 2005–2055.
  • [9] J. Peetre,On the theory of $\mathcal{L}_{p,\lambda}$ spaces, J. Funct. Anal.4(1969), 71–87.
  • [10] V. S. Rychkov,Littlewood–Paley theory and function spaces with $A_{p}^{\mathrm{loc}}$ weights, Math. Nachr.224(2001), 145–180.
  • [11] Y. Sawano, S. Sugano, and H. Tanaka, “Olsen’s inequality and its applications to Schrödinger equations” inHarmonic Analysis and Nonlinear Partial Differential Equations (Kyoto, 2010), RIMS Kôkyûroku BessatsuB26, RIMS, Kyoto, 2011, 51–80.
  • [12] E. M. Stein, “Oscillatory integrals in Fourier analysis” inBeijing Lectures in Harmonic Analysis, Ann. of Math. Stud.112, Princeton Univ. Press, Princeton, NJ, 1986.
  • [13] E. M. Stein and R. Shakarchi,Functional Analysis: Introduction to Further Topics in Analysis, Princeton Lect. Anal.4, Princeton Univ. Press, Princeton, NJ, 2011.
  • [14] P. A. Tomas,A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. (N. S.)81(1975), 477–478.
  • [15] H. Triebel,Theory of Function Spaces, Mongr. Math.78, Birkhäuser, Basel, 1983.
  • [16] C. T. Zorko,Morrey space, Proc. Amer. Math. Soc.98(1986), no. 4, 586–592.