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The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

  • Ahmet Turan Gürkanlı EMAIL logo , Öznur Kulak and Ayşe Sandıkçı
Published/Copyright: March 3, 2016
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 23 Issue 3

Abstract

Fix a nonzero window g∈𝒮⁢(ℝn), a weight function w on ℝ2⁢n and 1≤p,q≤∞. The weighted Lorentz type modulation space M⁢(p,q,w)⁢(ℝn) consists of all tempered distributions f∈𝒮′⁢(ℝn) such that the short time Fourier transform Vg⁢f is in the weighted Lorentz space L⁢(p,q,w⁢d⁢μ)⁢(ℝ2⁢n). The norm on M⁢(p,q,w)⁢(ℝn) is ∥f∥M⁢(p,q,w)=∥Vg⁢f∥p⁢q,w. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let 1<p1,p2<∞, 1≤q1,q2<∞, 1≤p3,q3≤∞, ω1,ω2 be polynomial weights and ω3 be a weight function on ℝ2⁢n. In the present paper, we define the bilinear multiplier operator from M⁢(p1,q1,ω1)⁢(ℝn)×M⁢(p2,q2,ω2)⁢(ℝn) to M⁢(p3,q3,ω3)⁢(ℝn) in the following way. Assume that m⁢(ξ,η) is a bounded function on ℝ2⁢n, and define

Bm⁢(f,g)⁢(x)=∫ℝn∫ℝnf^⁢(ξ)⁢g^⁢(η)⁢m⁢(ξ,η)⁢e2⁢π⁢i⁢〈ξ+η,x〉⁢𝑑ξ⁢𝑑η for all f,g∈𝒮⁢(ℝn).

The function m is said to be a bilinear multiplier on ℝn of type (p1,q1,ω1;p2,q2,ω2;p3,q3,ω3) if Bm is the bounded bilinear operator from M⁢(p1,q1,ω1)⁢(ℝn)×M⁢(p2,q2,ω2)⁢(ℝn) to M⁢(p3,q3,ω3)⁢(ℝn). We denote by BM⁢(p1,q1,ω1;p2,q2,ω2)⁢(ℝn) the space of all bilinear multipliers of type (p1,q1,ω1;p2,q2,ω2;p3,q3,ω3), and define ∥m∥(p1,q1,ω1;p2,q2,ω2;p3,q3,ω3)=∥Bm∥. We discuss the necessary and sufficient conditions for Bm to be bounded. We investigate the properties of this space and we give some examples.

Keywords: Modulation space; Lorentz space; bilinear multiplier
MSC 2010: 42B15; 42B35

References

[1] Blasco O., Notes on the spaces of bilinear multipliers, Rev. Un. Mat. Argentina 50 (2009), no. 2, 23–37. Search in Google Scholar

[2] Chung H.-M., Hunt R. A. and Kurtz D. S., The Hardy–Littlewood maximal function on L⁢(p,q) spaces with weights, Indiana Univ. Math. J. 31 (1982), no. 1, 109–120. 10.1512/iumj.1982.31.31012Search in Google Scholar

[3] Coifman R. R. and Meyer Y., Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean Harmonic Analysis, Lecture Notes in Math. 779, Springer, Berlin (1980), 104–122. 10.1007/BFb0087669Search in Google Scholar

[4] Duyar C. and Gürkanlı A. T., Multipliers and relative completion in weighted Lorentz spaces, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003), no. 4, 467–476. 10.1016/S0252-9602(17)30490-3Search in Google Scholar

[5] Feichtinger H. G., Modulation spaces on locally compact Abelian groups, technical report, University of Vienna, 1983. Search in Google Scholar

[6] Gilbert J. E. and Nahmot A. R., Boundedness of bilinear operators with nonsmooth symbols, Math. Res. Lett. 7 (2000), no. 5–6, 767–778. 10.4310/MRL.2000.v7.n6.a9Search in Google Scholar

[7] Gilbert J. E. and Nahmot A. R., Bilinear operators with non-smooth symbol. I, J. Fourier Anal. Appl. 7 (2001), no. 5, 435–467. 10.1007/BF02511220Search in Google Scholar

[8] Gröchenig K., Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0003-1Search in Google Scholar

[9] Gürkanlı A. T., Time frequency analysis and multipliers of the spaces M⁢(p,q)⁢(𝐑d) and S⁢(p,q)⁢(𝐑d), J. Math. Kyoto Univ. 46 (2006), no. 3, 595–616. 10.1215/kjm/1250281751Search in Google Scholar

[10] Hunt R. A. and Kurtz D. S., The Hardy–Littlewood maximal function on L⁢(p,1), Indiana Univ. Math. J. 32 (1983), no. 1, 155–158. 10.1512/iumj.1983.32.32012Search in Google Scholar

[11] Kulak Ö. and Gürkanlı A. T., Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces, J. Inequal. Appl. 2013 (2013), Article ID 259. 10.1186/1029-242X-2013-259Search in Google Scholar

[12] Lacey M. T., On the bilinear Hilbert transform, Doc. Math. 2 (1998), 647–656. 10.4171/dms/1-2/62Search in Google Scholar

[13] Lacey M. and Thiele C., Lp estimates on the bilinear Hilbert transform for 2<p<∞, Ann. of Math. (2) 146 (1997), no. 3, 693–724. 10.2307/2952458Search in Google Scholar

[14] O’Neil R., Convolution operators and L⁢(p,q) spaces, Duke Math. J. 30 (1963), 129–142. 10.1215/S0012-7094-63-03015-1Search in Google Scholar

[15] Sandıkçı A., Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces, Turkish J. Math. 38 (2014), no. 4, 728–745. 10.3906/mat-1311-43Search in Google Scholar

[16] Sandıkçı A. and Gürkanlı A. T., Gabor analysis of the spaces M⁢(p,q,w)⁢(ℝd) and S⁢(p,q,r,w,ω)⁢(ℝd), Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 1, 141–158. 10.1016/S0252-9602(11)60216-6Search in Google Scholar

Received: 2014-8-11
Accepted: 2015-1-9
Published Online: 2016-3-3
Published in Print: 2016-9-1

© 2016 by De Gruyter

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  2. Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
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  4. On the modification of the Szaśz–Durrmeyer operators
  5. A spectral representation of the linear multivelocity transport problem
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https://doi.org/10.1515/gmj-2016-0003
Keywords for this article
Modulation space; Lorentz space; bilinear multiplier

Articles in the same Issue

  1. Frontmatter
  2. Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
  3. Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data
  4. On the modification of the Szaśz–Durrmeyer operators
  5. A spectral representation of the linear multivelocity transport problem
  6. A Tauberian theorem for the product of Abel and Cesàro summability methods
  7. The spaces of bilinear multipliers of weighted Lorentz type modulation spaces
  8. Summations of Schlömilch series containing Struve function terms
  9. On nondifferentiable minimax semi-infinite programming problems in complex spaces
  10. Modified relativistic Laguerre polynomials. Monomiality and Lie algebraic methods
  11. On the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector
  12. The Hardy--Littlewood maximal operator and BLO1/log class of exponents
  13. Bicritical domination and double coalescence of graphs
  14. The asymptotic behavior of a counting process in the max-scheme. A discrete case
  15. Functions of bounded fractional differential variation – A new concept
  16. Sensitivity analysis of the optimal exercise boundary of the American put option
  17. Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction
  18. A coupled system of nonlinear differential equations involving m nonlinear terms
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