Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
Let L=−△+VL=-\bigtriangleup +V be the Schrödinger operator on Rn{{\mathbb{R}}}^{n}, where V≠0V\ne 0 is a non-negative function satisfying the reverse Hölder class RHq1R{H}_{{q}_{1}} for some q1>n⁄2{q}_{1}\gt n/2. △\bigtriangleup is the Laplacian on Rn{{\mathbb{R}}}^{n}. Assume that bb is a membe...
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De Gruyter
2024-11-01
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| Online Access: | https://doi.org/10.1515/math-2024-0082 |
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| author | Celik Suleyman Guliyev Vagif S. Akbulut Ali |
| author_facet | Celik Suleyman Guliyev Vagif S. Akbulut Ali |
| author_sort | Celik Suleyman |
| collection | DOAJ |
| description | Let L=−△+VL=-\bigtriangleup +V be the Schrödinger operator on Rn{{\mathbb{R}}}^{n}, where V≠0V\ne 0 is a non-negative function satisfying the reverse Hölder class RHq1R{H}_{{q}_{1}} for some q1>n⁄2{q}_{1}\gt n/2. △\bigtriangleup is the Laplacian on Rn{{\mathbb{R}}}^{n}. Assume that bb is a member of the Campanato space Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with LL is ℐβL{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}. We study the boundedness of the commutators [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b∈Λνθ(ρ)b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces Mp→,φα,V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, vanishing generalized mixed Morrey spaces VMp→,φα,VV{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, and LMp→,φα,V,{x0}L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when bb belongs to Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) with θ>0\theta \gt 0, 0<ν<10\lt \nu \lt 1, and (φ1,φ2)\left({\varphi }_{1},{\varphi }_{2}) satisfying certain requirements are bounded from LMp→,φ1α,V,{x0}L{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V,\left\{{x}_{0}\right\}} to LMq→,φ2α,V,{x0}L{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V,\left\{{x}_{0}\right\}}; from Mp→,φ1α,V{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to Mq→,φ2α,V{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, and from VMp→,φ1α,VV{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to VMq→,φ2α,VV{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, ∑i=1n1⁄pi−∑i=1n1⁄qi=β+ν{\sum }_{i=1}^{n}1/{p}_{i}-{\sum }_{i=1}^{n}1/{q}_{i}=\beta +\nu . |
| format | Article |
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| institution | Kabale University |
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| publishDate | 2024-11-01 |
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| spelling | doaj-art-17100b9953e844fd8a3e01f19713a20f2024-12-02T12:04:24ZengDe GruyterOpen Mathematics2391-54552024-11-0122111513410.1515/math-2024-0082Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spacesCelik Suleyman0Guliyev Vagif S.1Akbulut Ali2Social Sciences Vocational School Department of Banking, Finance and Insurance, Firat University, 23119 Elazig, TurkeyDepartment of Mathematics, Faculty of Arts and Sciences, Kirsehir Ahi Evran University, 40100 Kirsehir, TurkeyDepartment of Mathematics, Faculty of Arts and Sciences, Kirsehir Ahi Evran University, 40100 Kirsehir, TurkeyLet L=−△+VL=-\bigtriangleup +V be the Schrödinger operator on Rn{{\mathbb{R}}}^{n}, where V≠0V\ne 0 is a non-negative function satisfying the reverse Hölder class RHq1R{H}_{{q}_{1}} for some q1>n⁄2{q}_{1}\gt n/2. △\bigtriangleup is the Laplacian on Rn{{\mathbb{R}}}^{n}. Assume that bb is a member of the Campanato space Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with LL is ℐβL{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}. We study the boundedness of the commutators [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b∈Λνθ(ρ)b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces Mp→,φα,V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, vanishing generalized mixed Morrey spaces VMp→,φα,VV{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, and LMp→,φα,V,{x0}L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when bb belongs to Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) with θ>0\theta \gt 0, 0<ν<10\lt \nu \lt 1, and (φ1,φ2)\left({\varphi }_{1},{\varphi }_{2}) satisfying certain requirements are bounded from LMp→,φ1α,V,{x0}L{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V,\left\{{x}_{0}\right\}} to LMq→,φ2α,V,{x0}L{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V,\left\{{x}_{0}\right\}}; from Mp→,φ1α,V{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to Mq→,φ2α,V{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, and from VMp→,φ1α,VV{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to VMq→,φ2α,VV{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, ∑i=1n1⁄pi−∑i=1n1⁄qi=β+ν{\sum }_{i=1}^{n}1/{p}_{i}-{\sum }_{i=1}^{n}1/{q}_{i}=\beta +\nu .https://doi.org/10.1515/math-2024-0082schrödinger operatorfractional integralcommutatorlipschitz functionlocal generalized mixed morrey space42b3535j1047h50 |
| spellingShingle | Celik Suleyman Guliyev Vagif S. Akbulut Ali Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces Open Mathematics schrödinger operator fractional integral commutator lipschitz function local generalized mixed morrey space 42b35 35j10 47h50 |
| title | Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces |
| title_full | Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces |
| title_fullStr | Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces |
| title_full_unstemmed | Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces |
| title_short | Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces |
| title_sort | commutator of fractional integral with lipschitz functions related to schrodinger operator on local generalized mixed morrey spaces |
| topic | schrödinger operator fractional integral commutator lipschitz function local generalized mixed morrey space 42b35 35j10 47h50 |
| url | https://doi.org/10.1515/math-2024-0082 |
| work_keys_str_mv | AT celiksuleyman commutatoroffractionalintegralwithlipschitzfunctionsrelatedtoschrodingeroperatoronlocalgeneralizedmixedmorreyspaces AT guliyevvagifs commutatoroffractionalintegralwithlipschitzfunctionsrelatedtoschrodingeroperatoronlocalgeneralizedmixedmorreyspaces AT akbulutali commutatoroffractionalintegralwithlipschitzfunctionsrelatedtoschrodingeroperatoronlocalgeneralizedmixedmorreyspaces |