ON SOME INEQUALITIES FOR EXPONENTIALLY WEIGHTED FRACTIONAL HARDY OPERATORS WITH ∆-INTEGRAL CALCULUS

Yıl 2024, Cilt: 10 Sayı: 1, 1 – 13, 30.06.2024

Lütfi Akın

https://doi.org/10.51477/mejs.1451041

Öz

Dynamic equations, inequalities, and operators are the indispensable cornerstones of harmonic analysis and time-scale calculus. Undoubtedly, one of the most important of these operators and inequalities is the Hardy operator and inequality. Because especially when we say variable exponent Lebesgue space, the first thing that comes to our mind is the Hardy operator. We know that the topics in question have many applications in different scientific fields. In this paper, some inequalities will be proved for variable exponentially weighted Hardy operators with ∆-integral calculus.

Anahtar Kelimeler

Hardy inequality, Variable exponent, Weight function, Time scale

Etik Beyan

The authors declare that this document does not require ethics committee approval or any special permission.

Teşekkür

Thanks

Kaynakça

• Orlicz, W. “Uber konjugierte Exponentenfolgen,” Stud. Math., 3, 200-212, 1931.
• Kovacik, O., Rakosnik,J. “On spaces L^(p(x))and W^(k,p(x)),” Czechoslovak Math. J., 41, no. 4, 592-618, 1991.
• Akın, L. “A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces.” Anal. Theory Appl., Vol. 36, No. 1, pp. 60-68, 2020.
• Akin, L. “A Characterization of Approximation of Hardy Operators in VLS”, Celal Bayar University Journal of Science, Volume 14, Issue 3, pp:333-336, 2018.
• Akın, L., Zeren, Y. “On innovations of the multivariable fractional Hardy-type inequalities on time scales”. Sigma J Eng Nat Sci ;41(2):415−422, 2023.
• Bandaliev, R.A. “On Hardy-type inequalities in weighted variable exponent spaces L^p (x) for 0<p<1,” Eurasian Math.J., 4, no. 4, 5-16, 2013.
• Mamedov, F.I., Zeren, Y., Akin, L. “Compactification of weighted Hardy operator in variable exponent Lebesgue spaces,” Asian Journal of Mathematics and Computer Research, 17(1): 38-47, 2017.
• Azzouz, N., Halim, B., Senouci, A. “An inequality for the weighted Hardy operator for 0<p<1”, Eurasian Math. J., 4, no. 3, 60-65, 2013.
• Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, Berlin. 1748, 2000.
• Bandaliev, R.A. “On an inequality in Lebesgue space with mixed norm and with variable summability exponent,” Mat. Zametki, 84, no. 3, 323-333 (in Russian). English translation: Math. Notes, 84,no. 3, 303-313, 2008.
• Samko, S.G. “Differentiation and integration of variable order and the spaces L^p (x),” Proc. Inter. Conf. "Operator theory for complex and hypercomplex analysis", Mexico, 1994, Contemp. Math., 212, 203-219, 1998.
• Burenkov, V.I. “Function spaces. Main integral inequalities related to L^p-space,” Peoples' Friendship University of Russia, Moscow, 96pp, 1989.
• Burenkov, V.I. “On the exact constant in the Hardy inequality with 0<p<1 for monotone functions,” Trudy Matem. Inst. Steklov 194, 58-62 (in Russian), 1992.; English translation in proc. Steklov Inst. Math., 194, no. 4, 59-63, 1993.
• Senouci, A., Tararykova, T. “Hardy-type inequality for 0<p<1,” Evraziiskii Matematicheskii Zhurnal, pp.112-116, 2007.
• Bendaoud, S.A., Senouci, A. “Inequalities for weighted Hardy operators in weighted variable exponent Lebesgue space with 0<p(x)<1,” Eurasian Math. J., Volume 9, Number 1, 30–39, 2018.
• Akın, L. “On some results of weighted Hölder type inequality on time scales,” Middle East Journal of Science. 6(1), 15-22 2020.
• Akın, L. “On innovations of n-dimensional integral-type inequality on time scales.” Adv. Differ. Equ. 148 (2021), 2021.
• Akın, L. “On the Fractional Maximal Delta Integral Type Inequalities on Time Scales,” Fractal Fract. 4(2), 1-10, 2020.
• Agarwal, R.P., Bohner, M., Saker, S.H. “Dynamic Littlewood-type inequalities”. Proc. Am. Math. Soc. 143(2), 667–677, 2015.
• Oguntuase, J.A., Persson, L.E. “Time scales Hardy-type inequalities via superquadracity.” Ann. Funct. Anal. 5(2), 61–73, 2014.
• Rehak, P. “Hardy inequality on time scales and its application to half-linear dynamic equations.” J. Inequal. Appl. 5, 495–507, 2005.
• Saker, S.H. “Hardy–Leindler type inequalities on time scales.” Appl. Math. Inf. Sci. 8(6), 2975–2981, 2014.
• Saker, S.H., O’Regan, D. “Extensions of dynamic inequalities of Hardy’s type on time scales.” Math. Slovaca 65(5), 993–1012, 2015.
• Saker, S.H., O’Regan, D. “Hardy and Littlewood inequalities on time scales.” Bull. Malays. Math. Sci. Soc. 39(2), 527–543, 2016.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Some dynamic inequalities of Hardy’s type on time scales.” Math. Inequal. Appl.17, 1183–1199, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales.” Math. Nachr. 287(5–6), 686–698, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Dynamic inequalities of Hardy and Copson types on time scales.” Analysis 34,391–402, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Littlewood and Bennett inequalities on time scales.” Mediterr. J. Math. 12, 605–619, 2015.
• Hilger, S. Ein Maßkettenkalkül mit Anwendung auf Zentrmsmannigfaltingkeiten, Ph.D. Thesis, Univarsi. Würzburg, 1988.
• Saker, S.H., Rezk, H.M., Krni´c, M. “More accurate dynamic Hardy-type inequalities obtained via superquadraticity.” Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2691–2713, 2019.
• Saker, S.H., Saied, A.I., Krni´c, M. “Some new weighted dynamic inequalities for monotone functions involving kernels.” Mediterr. J. Math. 17(2), 1–18, 2020.
• Saker, S.H., Saied, A.I., Krni´c, M. “Some new dynamic Hardy-type inequalities with kernels involving monotone functions.” Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 1–16, 2020.

Yıl 2024, Cilt: 10 Sayı: 1, 1 – 13, 30.06.2024

Lütfi Akın

https://doi.org/10.51477/mejs.1451041

Öz

Kaynakça

• Orlicz, W. “Uber konjugierte Exponentenfolgen,” Stud. Math., 3, 200-212, 1931.
• Kovacik, O., Rakosnik,J. “On spaces L^(p(x))and W^(k,p(x)),” Czechoslovak Math. J., 41, no. 4, 592-618, 1991.
• Akın, L. “A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces.” Anal. Theory Appl., Vol. 36, No. 1, pp. 60-68, 2020.
• Akin, L. “A Characterization of Approximation of Hardy Operators in VLS”, Celal Bayar University Journal of Science, Volume 14, Issue 3, pp:333-336, 2018.
• Akın, L., Zeren, Y. “On innovations of the multivariable fractional Hardy-type inequalities on time scales”. Sigma J Eng Nat Sci ;41(2):415−422, 2023.
• Bandaliev, R.A. “On Hardy-type inequalities in weighted variable exponent spaces L^p (x) for 0<p<1,” Eurasian Math.J., 4, no. 4, 5-16, 2013.
• Mamedov, F.I., Zeren, Y., Akin, L. “Compactification of weighted Hardy operator in variable exponent Lebesgue spaces,” Asian Journal of Mathematics and Computer Research, 17(1): 38-47, 2017.
• Azzouz, N., Halim, B., Senouci, A. “An inequality for the weighted Hardy operator for 0<p<1”, Eurasian Math. J., 4, no. 3, 60-65, 2013.
• Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, Berlin. 1748, 2000.
• Bandaliev, R.A. “On an inequality in Lebesgue space with mixed norm and with variable summability exponent,” Mat. Zametki, 84, no. 3, 323-333 (in Russian). English translation: Math. Notes, 84,no. 3, 303-313, 2008.
• Samko, S.G. “Differentiation and integration of variable order and the spaces L^p (x),” Proc. Inter. Conf. "Operator theory for complex and hypercomplex analysis", Mexico, 1994, Contemp. Math., 212, 203-219, 1998.
• Burenkov, V.I. “Function spaces. Main integral inequalities related to L^p-space,” Peoples' Friendship University of Russia, Moscow, 96pp, 1989.
• Burenkov, V.I. “On the exact constant in the Hardy inequality with 0<p<1 for monotone functions,” Trudy Matem. Inst. Steklov 194, 58-62 (in Russian), 1992.; English translation in proc. Steklov Inst. Math., 194, no. 4, 59-63, 1993.
• Senouci, A., Tararykova, T. “Hardy-type inequality for 0<p<1,” Evraziiskii Matematicheskii Zhurnal, pp.112-116, 2007.
• Bendaoud, S.A., Senouci, A. “Inequalities for weighted Hardy operators in weighted variable exponent Lebesgue space with 0<p(x)<1,” Eurasian Math. J., Volume 9, Number 1, 30–39, 2018.
• Akın, L. “On some results of weighted Hölder type inequality on time scales,” Middle East Journal of Science. 6(1), 15-22 2020.
• Akın, L. “On innovations of n-dimensional integral-type inequality on time scales.” Adv. Differ. Equ. 148 (2021), 2021.
• Akın, L. “On the Fractional Maximal Delta Integral Type Inequalities on Time Scales,” Fractal Fract. 4(2), 1-10, 2020.
• Agarwal, R.P., Bohner, M., Saker, S.H. “Dynamic Littlewood-type inequalities”. Proc. Am. Math. Soc. 143(2), 667–677, 2015.
• Oguntuase, J.A., Persson, L.E. “Time scales Hardy-type inequalities via superquadracity.” Ann. Funct. Anal. 5(2), 61–73, 2014.
• Rehak, P. “Hardy inequality on time scales and its application to half-linear dynamic equations.” J. Inequal. Appl. 5, 495–507, 2005.
• Saker, S.H. “Hardy–Leindler type inequalities on time scales.” Appl. Math. Inf. Sci. 8(6), 2975–2981, 2014.
• Saker, S.H., O’Regan, D. “Extensions of dynamic inequalities of Hardy’s type on time scales.” Math. Slovaca 65(5), 993–1012, 2015.
• Saker, S.H., O’Regan, D. “Hardy and Littlewood inequalities on time scales.” Bull. Malays. Math. Sci. Soc. 39(2), 527–543, 2016.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Some dynamic inequalities of Hardy’s type on time scales.” Math. Inequal. Appl.17, 1183–1199, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales.” Math. Nachr. 287(5–6), 686–698, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Dynamic inequalities of Hardy and Copson types on time scales.” Analysis 34,391–402, 2014.
• Saker, S.H., O’Regan, D., Agarwal, R.P. “Littlewood and Bennett inequalities on time scales.” Mediterr. J. Math. 12, 605–619, 2015.
• Hilger, S. Ein Maßkettenkalkül mit Anwendung auf Zentrmsmannigfaltingkeiten, Ph.D. Thesis, Univarsi. Würzburg, 1988.
• Saker, S.H., Rezk, H.M., Krni´c, M. “More accurate dynamic Hardy-type inequalities obtained via superquadraticity.” Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2691–2713, 2019.
• Saker, S.H., Saied, A.I., Krni´c, M. “Some new weighted dynamic inequalities for monotone functions involving kernels.” Mediterr. J. Math. 17(2), 1–18, 2020.
• Saker, S.H., Saied, A.I., Krni´c, M. “Some new dynamic Hardy-type inequalities with kernels involving monotone functions.” Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 1–16, 2020.

Toplam 32 adet kaynakça vardır.

Ayrıntılar

Kaynak Göster

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