NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS

Bing He; Yong Hong; Zhen Li; Bicheng Yang

doi:10.11948/20200086

2021 Volume 11 Issue 1

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Bing He, Yong Hong, Zhen Li, Bicheng Yang. NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 521-531. doi: 10.11948/20200086

Citation:

Bing He, Yong Hong, Zhen Li, Bicheng Yang. NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 521-531. doi: 10.11948/20200086

NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS

1.
Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China
2.
Department of Mathematics, Guangdong Baiyun University, Guangzhou, Guangdong 510450, China
3.
College of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou, Guangdong 510320, China

Corresponding author: Email address: lzhhymath@163.com(Z. Li)
Fund Project: The authors were supported by NNSF of China (No. 61772140), and Innovation Team Construction Project of Guangdong Province (2018KCXTD020)

Abstract

Abstract

The problem of equivalent parameters and the best constant factor for the existence of quasi-homogeneous half-discrete Hilbert type inequality $ \int_{R_ + ^m} {\sum\limits_{n = 1}^\infty G } \left( {{n^{{\lambda _1}}}/\left\| x \right\|_{m, \rho }^{{\lambda _2}}} \right){a_n}f(x){\rm{d}}x \le M{\left\| {\tilde a} \right\|_{p, \alpha }}{\left\| f \right\|_{q, \beta }} $ is discussed, and their applications in the study of operator boundedness and norm are also considered.
- Quasi homogeneous kernel /
- half-discrete Hilbert type inequality /
- equivalent condition /
- best constant factor /
- boundedness of operator
MSC: 26D15, 47A07

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References

[1]	T. Batbold, M. Krnić and P. Vuković, A unified approach to fractal Hilbert-type inequalities, J. Inequal. Appl., 2019. DOI: 10.1186/s13660-019-2076-9. CrossRef Google Scholar
[2]	O. F. Brevig, Sharp norm estimates for composition operators and Hilbert-type inequalities, B. Lond. Math. Soc., 2017, 49(6), 965-978. doi: 10.1112/blms.12092 CrossRef Google Scholar
[3]	A. Čižmešija, M. Krnić and J. Pečarić, General Hilbert-Type Inequalities with Non-conjugate Exponents, Math. Inequal. Appl., 2008, 11(2), 237-269. Google Scholar
[4]	G. M. Fichtingoloz, A course in differential and integral calculus, People's Education Press, Beijing, China, 1957, 404-423. Google Scholar
[5]	L. He, H. Liu and B. Yang, On a more accurate reverse Mulholland-type inequality with parameters, J. Inequal. Appl., 2019. DOI: 10.1186/s13660-019-2139-y. CrossRef Google Scholar
[6]	Y. Hong, On the norm of a singular multiple integral operator with homogeneous kernel and its application, Chinese Annals of Mathematics, 2011, 32(5), 599-606. doi: 10.1007/s10255-011-0044-3 CrossRef Google Scholar
[7]	Y. Hong, A Hilbert-Type Integral Inequality with Quasi-homoeneous Kernel and Several Functions, Acta Mathematics Sinica, Chinese Series, 2014, 57(5), 833-840. Google Scholar
[8]	Y. Hong, B. He and B. Yang, Necessary and sufficient conditions for the validity of Hilbert type integral inequalities with a class of quasi-homogeneous kernels and its application in operator theory, J. Math. Inequal., 2018, 12(3), 777-788. Google Scholar
[9]	Y. Hong, Q. Huang and B. Yang, et al., The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications, J. Inequal. Appl., 2017. DOI: 10.1186/s13660-017-1592-8. CrossRef Google Scholar
[10]	M. Krnić and P. Vuković, A class of Hilbert-type inequalities obtained via the improved Young inequality, Results math., 2017, 71, 185-196. doi: 10.1007/s00025-015-0506-7 CrossRef Google Scholar
[11]	J. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2004, 534-535. Google Scholar
[12]	R. Luo and B. Yang, Parameterized discrete Hilbert-type inequalities with intermediate variables, J. Inequal. Appl., 2019. DOI: 10.1186/s13660-019-2095-6. CrossRef Google Scholar
[13]	A. Osekowski, Inequalities for Hilbert operator and its extensions: The probabilistic approach, Ann. Probab., 2017, 45(1), 535-563. doi: 10.1214/15-AOP1026 CrossRef Google Scholar
[14]	M. T. Rassias, B. Yang and A. Raigorodskii, Two Kinds of the Reverse HardyType Integral Inequalities with the Equivalent Forms Related to the Extended Riemann Zeta Function, Appl. Anal. Discrete Math., 2018, 12, 273-296. doi: 10.2298/AADM180130011R CrossRef Google Scholar
[15]	S. H. Saker, A. A. El-Deeb and H. M. Rezk, et al., On Hilbert's inequality on time scales, Appl. Anal. Discr. Math., 2017, 11(2), 399-423. Google Scholar
[16]	B. Tserendorj, A. Vandanjav and L. E. Azar, A new discrete Hilbert-type inequality involving partial sums, J. Inequal. Appl., 2019. DOI: 10.1186/s13660- 019-2087-6. CrossRef Google Scholar
[17]	B. Yang, M. Huang and Y. Zhong, On a parametric Mulholland-type inequality and applications, Abstr. Appl. Anal., 2019. DOI: 10.1155/2019/8317029. CrossRef Google Scholar