Article ID: MTJPAM-D-23-00009

Title: Multilinear multipliers of function spaces with wavelet transform in Lorentz spaces


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00009; Volume 6 / Issue 3 / Year 2024, Pages 34-47

Document Type: Research Paper

Author(s): ‪İrem Adıyaman a , Öznur Kulak b

aDepartment of Mathematics, Institute of Sciences, Amasya University, Amasya, Turkey

bDepartment of Mathematics, Faculty of Arts and Sciences, Amasya University, Amasya, Turkey

Received: 26 May 2023, Accepted: 7 October 2023, Published: 25 October 2023

Corresponding Author: Öznur Kulak (Email address: oznur.kulak@amasya.edu.tr)

Full Text: PDF

Abstract

Let 1 ≤ pi, qi, ri < ∞, si ∈ ℝ+  (i = 1, …, d + 1) and wi, vi (i = 1, …, d + 1) be weight functions on ℝ. Let Lsi(W)wi, vipi,qi, ri(ℝ) (i = 1, …, d + 1) be weighted normed spaces of functions whose wavelet transforms are in Lorentz space. A bounded function m(ξ1, …, ξd) defined on ℝd is said to be a multilinear multiplier on ℝ of type L(W)(pi, qi, ri,wi, vi, si), if the multilinear operator Bm associated with the m

    B_{m}(f_{1},\dots,f_{d})(x)=\underset{\mathbb{R}^{d}}{\int }\widehat{f_{1}}(\xi _{1})\dots\widehat{f_{d}}(\xi _{d})m(\xi_{1},\dots,\xi_{d})e^{2\pi i\left\langle \xi_{1}+\cdots+\xi_{d},x\right\rangle}d\xi_{1}\dots d\xi_{d}

defines a bounded multilinear operator from

    L_{s_{1}}(W)_{w_{1},v_{1}}^{p_{1},q_{1},r_{1}}(\mathbb{R})\times \dots\times L_{s_{d}}(W)_{w_{d},v_{d}}^{p_{d},q_{d},r_{d}}(\mathbb{R})\text{\ \ into \ }L_{s_{d+1}}(W)_{w_{d+1},v_{d+1}}^{p_{d+1},q_{d+1},r_{d+1}}(\mathbb{R}).

Also BM[L(W)(pi,qi,ri,wi,vi,si)] denotes the space of all multilinear multipliers of type L(W)(pi, qi, ri, wi, vi, si). In this work, we discuss the behaviour of the multilinear multipliers under the translation and modulation operators. Moreover, we give methods of construction examples of multilinear multipliers.

Keywords: Multilinear multiplier, Lorentz space, wavelet transform

References:
  1. A. P. Blozinski, On a convolution theorem for L(p,q) spaces, Trans. Amer. Math. Soc. 164, 255–165, 1972.
  2. H. Çakir and Ö. Kulak, Bilinear multipliers theory on some function spaces, Montes Taurus J. Pure Appl. Math. 5 (3), 95–104, 2023.
  3. K. Gröchenig, Foundations of time-frequency analysis, Birkhauser, Boston, 2001.
  4. A. T. Gürkanli, Ö. Kulak and A. Sandıkcı, The spaces of bilinear multipliers of weighted Lorentz type modulation spaces, Georgian Math. J. 23 (3), 351–362, 2016.
  5. R. A. Hunt, On L(p,q) spaces, Extrait de L’Enseignement Mathematique T., XII, fasc. 4, 249–276, 1966.
  6. R. A. Hunt and D. S. Kurtz, The Hardy-Littlewood maximal function on L(p,1), Indiana Univ. Math. J. 32 (1), 155–158, 1983.
  7. Ö. Kulak, Boundedness of the bilinear Littlewood-Paley square function on variable Lorentz spaces, AIP Conf. Proc. 1863 (1), 2017; Article ID: 300022.
  8. Ö. Kulak, Bilinear multipliers of function spaces with wavelet transform in Lωp(ℝn), Proc. Jangjeon Math. Soc. 20 (1), 81–94, 2017.
  9. Ö. Kulak, The bilinear Hardy-Littlewood maximal function and Littlewood-Paley square function on weighted variable exponent Wiener Amalgam space, Cankaya University Journal of Science and Engineering 1 6(1), 63–80, 2019.
  10. Ö. Kulak, On the multilinear fractional transforms, Symmetry 13 (5), 2021; Article ID: 740.
  11. Ö. Kulak, Compact embedding and inclusion theorems for weighted function spaces with wavelet transform, Lectures of Pure Mathematics on Algebra Analysis and Geometry, Artikel Academy, 2022.
  12. Ö. Kulak and A. T. Gürkanli, On function spaces with wavelet transform in Lωp(ℝd×ℝ+), Hacet. J. Math. Stat. 40 (2), 163–177, 2011.
  13. Ö. Kulak and A. T. Gürkanli, Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces, J. Inequal. Appl. 2013, 2013; Article ID: 259.
  14. Ö. Kulak and A. T. Gürkanli, Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces, J. Inequal. Appl. 2014, 2014; Article ID: 476.
  15. Ö. Kulak and A. T. Gürkanli, Bilinear multipliers of weighted Lorentz spaces and variable exponent Lorentz spaces, Mathematics and Statistics 5 (1), 5–18, 2017.
  16. Ö. Kulak and A. T. Gürkanli, Bilinear multipliers of small Lebesgue spaces, Turkish J. Math. 45 (5), 1959–1984, 2017.
  17. H. Reiter, Classical harmonic analysis and locally compact group, Oxford Universty Pres, Oxford, 1968.