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The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

  • Ahmet Turan Gürkanlı EMAIL logo , Öznur Kulak und Ayşe Sandıkçı
Veröffentlicht/Copyright: 3. März 2016
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Georgian Mathematical Journal
Aus der Zeitschrift Georgian Mathematical Journal Band 23 Heft 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

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

© 2016 by De Gruyter

Artikel in diesem Heft

  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
https://doi.org/10.1515/gmj-2016-0003
Schlagwörter für diesen Artikel
Modulation space; Lorentz space; bilinear multiplier

Artikel in diesem Heft

  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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