Research Article
Year 2025,
Volume: 54 Issue: 2, 529 - 541, 28.04.2025
Abstract
Let $\mathbb{A}$ be the affine group, $\Phi_1, \Phi_2$ be Young functions. We study the Orlicz amalgam spaces $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$ defined on $\mathbb{A}$, where the local and global component spaces are the Orlicz spaces $L^{\Phi_1}(\mathbb{A})$ and $L^{\Phi_2}(\mathbb{A})$, respectively. We obtain an equivalent discrete norm on the amalgam space $W(L^{\Phi_1} (\mathbb{A}), L^{\Phi_2} (\mathbb{A}))$ using the constructions related to the affine group. Using the discrete norm we compute the dual space of $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$. We also prove that the Orlicz amalgam space is a left $L^1(\mathbb{A})$-module with respect to convolution under certain conditions. Finally, we investigate some inclusion relations between the Orlicz amalgam spaces.
Thanks
I would like to thank Prof. S. Öztop for critical reading of the manuscript and helpful suggestions on the subject.
References
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- [10] H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86, 307-340, 1989.
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- [12] C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames, J. Approx. Theory, 147, 28-46, 2007.
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- [16] S. Öztop and E. Samei, Twisted Orlicz algebras I, Studia Mathematica, 236, 271-296, 2017.
- [17] M.M. Rao, Extensions of the Hausdorff-Young theorem, Israel J. Math. 6, 133-149, 1967.
- [18] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
- [19] M.M. Rao and Z.D. Ren, Application of Orlicz Spaces, Marcel Dekker, New York, 2002.
- [20] M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in: Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, 267283, Birkhäuser/Springer, 2012.
- [21] N. Wiener, On the representation of functions by trigonometric integrals, Math. Z., 24, 575616, 1926.
- [22] N. Wiener, Tauberian theorems, Ann. of Math., 33, 1100, 1932.
- [23] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933.
Year 2025,
Volume: 54 Issue: 2, 529 - 541, 28.04.2025
Abstract
References
- [1] B. Ars and S. Öztop, Wiener amalgam spaces with respect to Orlicz spaces on the affine group, J. Pseudo. Differ. Oper. Appl. 14, 23, 2023.
- [2] A. Benedek and R. Panzone, The spaces $L^p$ with mixed norm, Duke Math. J. 28, 301-324, 1961.
- [3] J.P. Bertrandias, C. Datry and C. Dupuis, Unions et intersections despaces Lp invariantes par translation ou convolution, Ann. Inst. Fourier 28, 53-84, 1978.
- [4] R.C. Busby and H.A. Smith, Product-convolution operators and mixed-norm spaces, Trans. Amer. Math. Soc. 263, 309-341, 1981.
- [5] D.L. Cohn, Measure Theory, 2nd ed., Birkhäuser/Springer, New York, 2013.
- [6] E. Cordero and F. Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80, 105-116, 2009.
- [7] H.G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, Colloq. Math. Soc. 35, 509-524, János Bolyai North-Holland, Amsterdam, 1983.
- [8] H.G. Feichtinger, Banach spaces of distributions defined by decomposition methods, II, Math. Nachr. 132, 207237, 1987.
- [9] H.G. Feichtinger and P. Gröbner, Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123, 97-120, 1985.
- [10] H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86, 307-340, 1989.
- [11] C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and Their Applications, 183216, Allied Publishers, New Delhi, 2003.
- [12] C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames, J. Approx. Theory, 147, 28-46, 2007.
- [13] C. Heil and G. Kutyniok, Convolution and Wiener amalgam spaces on the affine group, in: Recent Advances in Computational Science, 209217, World Scientific, Singapore, 2008.
- [14] F. Holland, Harmonic analysis on amalgams of $L^p$ and $\ell^q$, J. London Math. Soc. 10, 295-305, 1975.
- [15] A. Osançlıol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99, 399-414, 2015.
- [16] S. Öztop and E. Samei, Twisted Orlicz algebras I, Studia Mathematica, 236, 271-296, 2017.
- [17] M.M. Rao, Extensions of the Hausdorff-Young theorem, Israel J. Math. 6, 133-149, 1967.
- [18] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
- [19] M.M. Rao and Z.D. Ren, Application of Orlicz Spaces, Marcel Dekker, New York, 2002.
- [20] M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in: Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, 267283, Birkhäuser/Springer, 2012.
- [21] N. Wiener, On the representation of functions by trigonometric integrals, Math. Z., 24, 575616, 1926.
- [22] N. Wiener, Tauberian theorems, Ann. of Math., 33, 1100, 1932.
- [23] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933.
There are 23 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Lie Groups, Harmonic and Fourier Analysis, Operator Algebras and Functional Analysis |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 27, 2024 |
| Publication Date | April 28, 2025 |
| Submission Date | March 29, 2024 |
| Acceptance Date | June 1, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 2 |