Citation: Guanghui Lu, Li Rui. $ \theta $-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces[J]. AIMS Mathematics, 2021, 6(9): 9619-9632. doi: 10.3934/math.2021560
[1] | Y. Cao, J. Zhou, Morrey spaces for nonhomogenerous metric measure spaces, Abstr. Appl. Anal., 2013 (2013), 1–8. |
[2] | C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L^{p}$ spaces, Rev. Mat. Iberoamer., 23 (2007), 743–770. |
[3] | L. Ephremidze, V. Kokilashvili, S. Samko, Fractional, maximal and singular operators in variable exponent Lorentz spaces, Fract. Calc. Appl. Anal., 11 (2008), 407–420. |
[4] | X. Fan, Variable exponent Morrey and Campanato spaces, Nonlinear Anal., 72 (2010), 4148–4161. doi: 10.1016/j.na.2010.01.047 |
[5] | X. Fu, D. Yang, W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math., 18 (2014), 509–557. |
[6] | G. Hu, H. Lin, D. Yang, Marcinkiewicz integrals with non-doubling measures, Integral Equations Operator Theory, 58 (2007), 205–238. doi: 10.1007/s00020-007-1481-5 |
[7] | V. Kokilashvili, A. Meskhi, Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure, Complex Var. Elliptic Equ., 55 (2010), 923–936. doi: 10.1080/17476930903276068 |
[8] | V. Kokilashvili, A. Meskhi, Maximal and Calderón-Zygmund operators in weighted grand variable exponent Lebesgue space, Trans. A. Razmadze Math. Inst., 173 (2019), 127–131. |
[9] | O. Kováčik, J. Rákosník, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Math., 41 (1991), 592–618. doi: 10.21136/CMJ.1991.102493 |
[10] | G. Lu, Parameter Marcinkiewicz integral on non-homogeneous Morrey space with variable exponent, Politehn. Univ. Bucharest Sci. Bull. Ser. A, 83 (2021), 89–98. |
[11] | G. Lu, Commutators of bilinear pseudo-differential operators on local Hardy spaces with variable exponents, Bull. Braz. Math. Soc., 51 (2020), 975–1000. doi: 10.1007/s00574-019-00184-7 |
[12] | G. Mingione, V. R$\breve{a}$dulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197. doi: 10.1016/j.jmaa.2021.125197 |
[13] | E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665–3748. doi: 10.1016/j.jfa.2012.01.004 |
[14] | A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R})$, Math. Inequal. Appl., 7 (2004), 255–265. |
[15] | W. Orlicz, Über konjugierte exponentenfolgen, Studia Math., 3 (1931), 200–212. doi: 10.4064/sm-3-1-200-211 |
[16] | V. Radulescu, D. Repovs, Partial differential equations with variable exponents: Variational methods and qualitative analysis, Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. |
[17] | M. Ragusa, A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), 710–728. |
[18] | Y. Sawano, H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1535–1544. doi: 10.1007/s10114-005-0660-z |
[19] | H. Shen, Y. Li, X. Shao, A GPIU method for fractional diffusion equations, Adv. Difference Equ., 2020 (2020), 1–17. doi: 10.1186/s13662-019-2438-0 |
[20] | X. Tolsa, BMO, $H^{1}$, and Calderón-Zygmund operators for non-doubling measures, Math. Ann., 319 (2001), 89–149. doi: 10.1007/PL00004432 |
[21] | L. Wang, L. Shu, Multilinear commutators of singular integral operators in variable exponent Herz-type spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), 1413–1432. doi: 10.1007/s40840-017-0554-0 |
[22] | L. Wang, L. Shu, Higher order commutators of fractional integrals on Morrey type spaces with variable exponents, Math. Nachr., 291 (2018), 1437–1449. doi: 10.1002/mana.201600438 |
[23] | L. Wang, S. Tao, Parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponent, Turkish J. Math., 40 (2016), 122–145. doi: 10.3906/mat-1412-52 |