Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent
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Keywords:
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Hölder Inequality , Herz-Morrey Spaces , Variable Exponent
Abstract
The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.
References
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Gurka, P., P. Harjulehto, and A. Nekvinda (2007). Bessel Potential Spaces with Variable Exponent. Mathematical Inequalities and Applications, 10(3); 661
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Masta, A. A., S. Fatimah, A. Arsisari, Y. Yunika Putra, and F. Apriani (2018). Sufficient and Necessary Conditions for Generalized Hölder’s Inequality in p-Summable Sequence Spaces. arXiv, arXiv:1812.06612; 1–5
Masta, A. A., H. Gunawan, and W. Setya Budhi (2017). An Inclusion Property of Orlicz-Morrey Spaces. In Journal of Physics: Conference Series, volume 893. IOP Publishing, page 012015
Matkowski, J. (2009). A Converse of the Hölder Inequality Theorem. Mathematical Inequalities & Applications, 12(1); 21–23
Mizuta, Y. (2016). Duality of Herz-Morrey Spaces of Variable Exponent. Filomat, 30(7); 1891–1898
Mizuta, Y. and T. Ohno (2015). Herz–Morrey Spaces of Variable Exponent, Riesz Potential Operator and Duality. Complex Variables and Elliptic Equations, 60(2); 211–240
Nakai, E. and Y. Sawano (2012). Hardy Spaces with Variable Exponents and Generalized Campanato Spaces. Journal of Functional Analysis, 262(9); 3665–3748
Nogayama, T. (2019). Boundedness of Commutators of Fractional Integral Operators on Mixed Morrey Spaces. Integral Transforms and Special Functions, 30(10); 790–816
Orlicz, W. (1931). Über Konjugierte Exponentenfolgen. Studia Mathematica, 3(1); 200–211
Shanzhen, L. and X. Lifang (2005). Boundedness of Rough Singular Integral Operatorson the Homogeneous Morrey-Herz Spaces. Hokkaido Mathematical Journal, 34(2); 299–314
Sultan, B., F. M. Azmi, M. Sultan, T. Mahmood, N. Mlaiki, and N. Souayah (2022). Boundedness of Fractional Integrals on Grand Weighted Herz–Morrey Spaces with Variable Exponent. Fractal and Fractional, 6(11); 660
Vasyunin, V. (2004). The Sharp Constant in the Reverse Hölder Inequality for Muckenhoupt Weights. St. Petersburg Mathematical Journal, 15(1); 49–79
Wajih, A. and H. Gunawan (2020). On Generalized Hölder’s Inequality in Weak Morrey Spaces. Journal of the Indonesian Mathematical Society, 26(01); 165–169
Wang, H. and F. Liao (2020). Boundedness of Singular Integral Operators on Herz-Morrey Spaces with Variable Exponent. Chinese Annals of Mathematics, Series B, 41(1); 99–116
Wang, H. and Y. Wu (2016). Anisotropic Herz-Morrey Spaces with Variable Exponents. Khayyam Journal of Mathematics, 2(2); 177–187
Xu, J. and X. Yang (2015). Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications. Journal of Function Spaces, 2015(Article ID 160635); 1–19
Alshanti, W. G., I. M. Batiha, A. Alshanty, A. Zraiqat, and I. H. Jebril (2023). Perturbed Trapezoid Like Inequalities. Science and Technology Indonesia, 8(2); 205–211
Aoyama, H. (2009). Lebesgue Spaces with Variable Exponent on a Probability Space. Hiroshima Mathematical Journal, 39(2); 207–216
Chen, G. S., H. M. Srivastava, P. Wang, and W. Wei (2014). Some Further Generalizations of Hölder’s Inequality and Related Results on Fractal Space. In Abstract and Applied Analysis, volume 2014. Hindawi, pages 802–832
Dung, K. H., C. D. L. Minh, and T. T. Nang (2023). Boundedness of Hardy-Cesàro Operators on Variable Exponent Morrey-Herz Spaces. Filomat, 37(4); 1001–1016
Fan, X. (2010). Variable Exponent Morrey and Campanato Spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(11); 4148–4161
Gurka, P., P. Harjulehto, and A. Nekvinda (2007). Bessel Potential Spaces with Variable Exponent. Mathematical Inequalities and Applications, 10(3); 661
Ifronika, M. Idris, A. A. Masta, and H. Gunawan (2017). Generalized Hölder’s Inequality on Morrey Spaces. arXiv, arXiv:1706.01659; 1–12
Ifronika, A. A. Masta, M. Nur, and H. Gunawan (2018). Generalized Hölder’s Inequality in Orlicz Spaces. arXiv, arXiv:1809.00788; 1–10
Izuki, M. (2010). Boundedness of Sublinear Operators on Herz Spaces with Variable Exponent and Application to Wavelet Characterization. Analysis Mathematica, 36(1); 33–50
Masta, A. A., S. Fatimah, A. Arsisari, Y. Yunika Putra, and F. Apriani (2018). Sufficient and Necessary Conditions for Generalized Hölder’s Inequality in p-Summable Sequence Spaces. arXiv, arXiv:1812.06612; 1–5
Masta, A. A., H. Gunawan, and W. Setya Budhi (2017). An Inclusion Property of Orlicz-Morrey Spaces. In Journal of Physics: Conference Series, volume 893. IOP Publishing, page 012015
Matkowski, J. (2009). A Converse of the Hölder Inequality Theorem. Mathematical Inequalities & Applications, 12(1); 21–23
Mizuta, Y. (2016). Duality of Herz-Morrey Spaces of Variable Exponent. Filomat, 30(7); 1891–1898
Mizuta, Y. and T. Ohno (2015). Herz–Morrey Spaces of Variable Exponent, Riesz Potential Operator and Duality. Complex Variables and Elliptic Equations, 60(2); 211–240
Nakai, E. and Y. Sawano (2012). Hardy Spaces with Variable Exponents and Generalized Campanato Spaces. Journal of Functional Analysis, 262(9); 3665–3748
Nogayama, T. (2019). Boundedness of Commutators of Fractional Integral Operators on Mixed Morrey Spaces. Integral Transforms and Special Functions, 30(10); 790–816
Orlicz, W. (1931). Über Konjugierte Exponentenfolgen. Studia Mathematica, 3(1); 200–211
Shanzhen, L. and X. Lifang (2005). Boundedness of Rough Singular Integral Operatorson the Homogeneous Morrey-Herz Spaces. Hokkaido Mathematical Journal, 34(2); 299–314
Sultan, B., F. M. Azmi, M. Sultan, T. Mahmood, N. Mlaiki, and N. Souayah (2022). Boundedness of Fractional Integrals on Grand Weighted Herz–Morrey Spaces with Variable Exponent. Fractal and Fractional, 6(11); 660
Vasyunin, V. (2004). The Sharp Constant in the Reverse Hölder Inequality for Muckenhoupt Weights. St. Petersburg Mathematical Journal, 15(1); 49–79
Wajih, A. and H. Gunawan (2020). On Generalized Hölder’s Inequality in Weak Morrey Spaces. Journal of the Indonesian Mathematical Society, 26(01); 165–169
Wang, H. and F. Liao (2020). Boundedness of Singular Integral Operators on Herz-Morrey Spaces with Variable Exponent. Chinese Annals of Mathematics, Series B, 41(1); 99–116
Wang, H. and Y. Wu (2016). Anisotropic Herz-Morrey Spaces with Variable Exponents. Khayyam Journal of Mathematics, 2(2); 177–187
Xu, J. and X. Yang (2015). Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications. Journal of Function Spaces, 2015(Article ID 160635); 1–19
Authors
Rahman, H., & Karim, C. (2024). Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent. Science and Technology Indonesia, 9(3), 637–641. https://doi.org/10.26554/sti.2024.9.3.637-641
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