Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
Let be a stratified Lie group, and let be a basis of the left-invariant vector fields of degree one on and be the sub-Laplacian of . Given that , this article studies the commutators of order and establishes their uniform two-weight boundedness from to for any and via space, assuming that and . Base...
Saved in:
| Published in | Advances in nonlinear analysis Vol. 14; no. 1; pp. 1207 - 1230 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
De Gruyter
04.07.2025
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | Let
be a stratified Lie group, and let
be a basis of the left-invariant vector fields of degree one on
and
be the sub-Laplacian of
. Given that
, this article studies the commutators
of order
and establishes their uniform two-weight boundedness from
to
for any
and
via
space, assuming that
and
. Based on this, we also give the characterization of
space with respect to the uniform weighted compactness of
for
. As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on
can be recovered as |
|---|---|
| AbstractList | Let G {\mathcal{G}} be a stratified Lie group, and let { X j } 1 ≤ j ≤ n 1 {\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector fields of degree one on G {\mathcal{G}} and Δ = − ∑ j = 1 n 1 X j 2 \Delta =-{\sum }_{j=1}^{{n}_{1}}{X}_{j}^{2} be the sub-Laplacian of G {\mathcal{G}} . Given that 0 ≤ α < Q 0\le \alpha \lt {\mathbb{Q}} , this article studies the commutators [ b , X j Δ − 1 + α 2 ] m , j = 1 , … , n 1 {\left[b,{X}_{j}{\Delta }^{-\frac{1+\alpha }{2}}]}^{m},\hspace{0.33em}j=1,\ldots ,{n}_{1} of order m ∈ N m\in {\mathbb{N}} and establishes their uniform two-weight boundedness from L p ( μ p ) {L}^{p}\left({\mu }^{p}) to L q ( w q ) {L}^{q}\left({w}^{q}) for any 0 < α < Q 0\lt \alpha \lt {\mathbb{Q}} and 1 q = 1 p − α Q \frac{1}{q}=\frac{1}{p}-\frac{\alpha }{{\mathbb{Q}}} via BMO ν 1 ⁄ m ( G ) {{\rm{BMO}}}_{{\nu }^{1/m}}\left({\mathcal{G}}) space, assuming that μ , w ∈ A p , q \mu ,w\in {A}_{p,q} and ν = μ w \nu =\frac{\mu }{w} . Based on this, we also give the characterization of VMO ( G ) {\rm{VMO}}\left({\mathcal{G}}) space with respect to the uniform weighted compactness of [ b , X j Δ − 1 + α 2 ] \left[b,{X}_{j}{\Delta }^{-\tfrac{1+\alpha }{2}}] for 0 ≤ α < Q 0\le \alpha \lt {\mathbb{Q}} . As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on G {\mathcal{G}} can be recovered as α → 0 \alpha \to 0 . Let be a stratified Lie group, and let be a basis of the left-invariant vector fields of degree one on and be the sub-Laplacian of . Given that , this article studies the commutators of order and establishes their uniform two-weight boundedness from to for any and via space, assuming that and . Based on this, we also give the characterization of space with respect to the uniform weighted compactness of for . As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on can be recovered as Let G{\mathcal{G}} be a stratified Lie group, and let {Xj}1≤j≤n1{\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector fields of degree one on G{\mathcal{G}} and Δ=−∑j=1n1Xj2\Delta =-{\sum }_{j=1}^{{n}_{1}}{X}_{j}^{2} be the sub-Laplacian of G{\mathcal{G}}. Given that 0≤α<Q0\le \alpha \lt {\mathbb{Q}}, this article studies the commutators [b,XjΔ−1+α2]m,j=1,…,n1{\left[b,{X}_{j}{\Delta }^{-\frac{1+\alpha }{2}}]}^{m},\hspace{0.33em}j=1,\ldots ,{n}_{1} of order m∈Nm\in {\mathbb{N}} and establishes their uniform two-weight boundedness from Lp(μp){L}^{p}\left({\mu }^{p}) to Lq(wq){L}^{q}\left({w}^{q}) for any 0<α<Q0\lt \alpha \lt {\mathbb{Q}} and 1q=1p−αQ\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{{\mathbb{Q}}} via BMOν1⁄m(G){{\rm{BMO}}}_{{\nu }^{1/m}}\left({\mathcal{G}}) space, assuming that μ,w∈Ap,q\mu ,w\in {A}_{p,q} and ν=μw\nu =\frac{\mu }{w}. Based on this, we also give the characterization of VMO(G){\rm{VMO}}\left({\mathcal{G}}) space with respect to the uniform weighted compactness of [b,XjΔ−1+α2]\left[b,{X}_{j}{\Delta }^{-\tfrac{1+\alpha }{2}}] for 0≤α<Q0\le \alpha \lt {\mathbb{Q}}. As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on G{\mathcal{G}} can be recovered as α→0\alpha \to 0. |
| Author | Chen, Yanping Han, Xueting |
| Author_xml | – sequence: 1 givenname: Xueting surname: Han fullname: Han, Xueting email: xueting.han@xs.ustb.edu.cn organization: School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, China – sequence: 2 givenname: Yanping surname: Chen fullname: Chen, Yanping email: chenyanping@mail.neu.edu.cn organization: Department of Mathematics, Northeastern University, Shenyang, 110004, Liaoning, China |
| BookMark | eNp1UU1rGzEUFCGFJI7Pve4f2EYfq5XVWwhJGjAESg25Ca305MrYkpG0NM6vr9YOIZe8y5sZ3gw85gqdhxgAoe8E_yCc8BtduW4pprzFWNIzdEmJJK3k-OX8E75A85w3uM6CEyHwJUqr4F1Mu2aIY7BgA-Tc6GAbE3d7bcqR14Om_IVJ241Fl0qjq1cNvBYI2ccw8d8e8ltTkg75mFjVXFnxzoNtlh6adYrjPl-jb05vM8zf9wytHu7_3P1ql8-PT3e3y9YwzkvruKHQWykGI3tJuO0ZXgwWeiGcMJQaYTvXGW0tY2bQnanqIIUQzHDeDx2boadTro16o_bJ73Q6qKi9OgoxrZVOxZstqIVw1GmDNTDZUUsG1knZEVcxY71d1KybU5ZJMecE7iOPYDU1oI4NqKkBNTVQHT9Pjn96WyBZWKfxUIHaxDGF-vdXTtIR9h_vwpJw |
| Cites_doi | 10.1016/j.jfa.2010.02.004 10.1016/j.aim.2020.107001 10.1016/j.jde.2023.10.027 10.1017/CBO9780511662485 10.1512/iumj.2020.69.7959 10.1016/j.nonrwa.2019.04.004 10.1007/BF02387385 10.1007/s11118-008-9114-4 10.1007/978-0-387-09432-8 10.1007/978-3-540-71897-0 10.1007/978-0-387-09434-2 10.1016/j.jfa.2016.05.002 10.1016/j.jfa.2012.09.013 10.1016/j.jfa.2019.05.008 10.1007/s00205-018-1320-7 10.1016/j.matpur.2018.06.012 10.1007/s00025-021-01578-0 10.1007/s00013-012-0453-4 10.1515/9780691222455 10.1016/j.jfa.2020.108887 10.1016/j.jmaa.2021.125869 10.1007/BF02386204 10.4064/cm126-1-1 10.1016/j.jfa.2021.109196 10.1016/j.jde.2022.11.035 10.1515/agms-2020-0116 10.1007/s10013-016-0210-2 10.1007/s12220-019-00308-x 10.1142/9789812770561 |
| ContentType | Journal Article |
| DBID | AAYXX CITATION DOA |
| DOI | 10.1515/anona-2025-0092 |
| DatabaseName | CrossRef DOAJ Open Access Full Text |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering |
| EISSN | 2191-950X |
| EndPage | 1230 |
| ExternalDocumentID | oai_doaj_org_article_87f2fac0ae3942d1b349941f42d336d8 10_1515_anona_2025_0092 10_1515_anona_2025_0092141 |
| GroupedDBID | 0R~ 0~D 4.4 AAFPC AAFWJ AAQCX AASOL AASQH ABAOT ABAQN ABFKT ABIQR ABSOE ABUVI ABXMZ ACGFS ACXLN ACZBO ADGQD ADGYE ADJVZ ADOZN AEJTT AENEX AEQDQ AEXIE AFBAA AFBDD AFCXV AFPKN AFQUK AHGSO AIERV AJATJ AKXKS ALMA_UNASSIGNED_HOLDINGS AMVHM BAKPI BBCWN CFGNV EBS GROUPED_DOAJ HZ~ IY9 J9A O9- OK1 QD8 SA. AAYXX CITATION |
| ID | FETCH-LOGICAL-c355t-f5c2e6d97bc96915d6308bde677f7c22c7d4f4cadd33cba4cf7cb97773c556b43 |
| IEDL.DBID | DOA |
| ISSN | 2191-950X |
| IngestDate | Wed Aug 27 01:26:46 EDT 2025 Thu Jul 10 08:37:53 EDT 2025 Thu Jul 10 10:28:00 EDT 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Language | English |
| License | This work is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c355t-f5c2e6d97bc96915d6308bde677f7c22c7d4f4cadd33cba4cf7cb97773c556b43 |
| OpenAccessLink | https://doaj.org/article/87f2fac0ae3942d1b349941f42d336d8 |
| PageCount | 31 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_87f2fac0ae3942d1b349941f42d336d8 crossref_primary_10_1515_anona_2025_0092 walterdegruyter_journals_10_1515_anona_2025_0092141 |
| PublicationCentury | 2000 |
| PublicationDate | 2025-07-04 |
| PublicationDateYYYYMMDD | 2025-07-04 |
| PublicationDate_xml | – month: 07 year: 2025 text: 2025-07-04 day: 04 |
| PublicationDecade | 2020 |
| PublicationTitle | Advances in nonlinear analysis |
| PublicationYear | 2025 |
| Publisher | De Gruyter |
| Publisher_xml | – name: De Gruyter |
| References | 2025070412063746281_j_anona-2025-0092_ref_008 2025070412063746281_j_anona-2025-0092_ref_009 2025070412063746281_j_anona-2025-0092_ref_022 2025070412063746281_j_anona-2025-0092_ref_001 2025070412063746281_j_anona-2025-0092_ref_023 2025070412063746281_j_anona-2025-0092_ref_002 2025070412063746281_j_anona-2025-0092_ref_024 2025070412063746281_j_anona-2025-0092_ref_003 2025070412063746281_j_anona-2025-0092_ref_025 2025070412063746281_j_anona-2025-0092_ref_004 2025070412063746281_j_anona-2025-0092_ref_026 2025070412063746281_j_anona-2025-0092_ref_005 2025070412063746281_j_anona-2025-0092_ref_027 2025070412063746281_j_anona-2025-0092_ref_006 2025070412063746281_j_anona-2025-0092_ref_028 2025070412063746281_j_anona-2025-0092_ref_007 2025070412063746281_j_anona-2025-0092_ref_029 2025070412063746281_j_anona-2025-0092_ref_020 2025070412063746281_j_anona-2025-0092_ref_021 2025070412063746281_j_anona-2025-0092_ref_019 2025070412063746281_j_anona-2025-0092_ref_011 2025070412063746281_j_anona-2025-0092_ref_012 2025070412063746281_j_anona-2025-0092_ref_013 2025070412063746281_j_anona-2025-0092_ref_014 2025070412063746281_j_anona-2025-0092_ref_015 2025070412063746281_j_anona-2025-0092_ref_016 2025070412063746281_j_anona-2025-0092_ref_017 2025070412063746281_j_anona-2025-0092_ref_018 2025070412063746281_j_anona-2025-0092_ref_030 2025070412063746281_j_anona-2025-0092_ref_031 2025070412063746281_j_anona-2025-0092_ref_010 2025070412063746281_j_anona-2025-0092_ref_032 |
| References_xml | – ident: 2025070412063746281_j_anona-2025-0092_ref_021 doi: 10.1016/j.jfa.2010.02.004 – ident: 2025070412063746281_j_anona-2025-0092_ref_028 – ident: 2025070412063746281_j_anona-2025-0092_ref_003 doi: 10.1016/j.aim.2020.107001 – ident: 2025070412063746281_j_anona-2025-0092_ref_008 doi: 10.1016/j.jde.2023.10.027 – ident: 2025070412063746281_j_anona-2025-0092_ref_029 doi: 10.1017/CBO9780511662485 – ident: 2025070412063746281_j_anona-2025-0092_ref_001 doi: 10.1512/iumj.2020.69.7959 – ident: 2025070412063746281_j_anona-2025-0092_ref_030 doi: 10.1016/j.nonrwa.2019.04.004 – ident: 2025070412063746281_j_anona-2025-0092_ref_026 doi: 10.1007/BF02387385 – ident: 2025070412063746281_j_anona-2025-0092_ref_005 doi: 10.1007/s11118-008-9114-4 – ident: 2025070412063746281_j_anona-2025-0092_ref_013 doi: 10.1007/978-0-387-09432-8 – ident: 2025070412063746281_j_anona-2025-0092_ref_002 doi: 10.1007/978-3-540-71897-0 – ident: 2025070412063746281_j_anona-2025-0092_ref_014 doi: 10.1007/978-0-387-09434-2 – ident: 2025070412063746281_j_anona-2025-0092_ref_019 doi: 10.1016/j.jfa.2016.05.002 – ident: 2025070412063746281_j_anona-2025-0092_ref_018 doi: 10.1016/j.jfa.2012.09.013 – ident: 2025070412063746281_j_anona-2025-0092_ref_025 – ident: 2025070412063746281_j_anona-2025-0092_ref_004 doi: 10.1016/j.jfa.2019.05.008 – ident: 2025070412063746281_j_anona-2025-0092_ref_032 doi: 10.1007/s00205-018-1320-7 – ident: 2025070412063746281_j_anona-2025-0092_ref_010 doi: 10.1016/j.matpur.2018.06.012 – ident: 2025070412063746281_j_anona-2025-0092_ref_015 doi: 10.1007/s00025-021-01578-0 – ident: 2025070412063746281_j_anona-2025-0092_ref_024 doi: 10.1007/s00013-012-0453-4 – ident: 2025070412063746281_j_anona-2025-0092_ref_012 doi: 10.1515/9780691222455 – ident: 2025070412063746281_j_anona-2025-0092_ref_031 doi: 10.1016/j.jfa.2020.108887 – ident: 2025070412063746281_j_anona-2025-0092_ref_020 doi: 10.1016/j.jmaa.2021.125869 – ident: 2025070412063746281_j_anona-2025-0092_ref_011 doi: 10.1007/BF02386204 – ident: 2025070412063746281_j_anona-2025-0092_ref_017 doi: 10.4064/cm126-1-1 – ident: 2025070412063746281_j_anona-2025-0092_ref_006 doi: 10.1016/j.jfa.2021.109196 – ident: 2025070412063746281_j_anona-2025-0092_ref_007 doi: 10.1016/j.jde.2022.11.035 – ident: 2025070412063746281_j_anona-2025-0092_ref_016 doi: 10.1515/agms-2020-0116 – ident: 2025070412063746281_j_anona-2025-0092_ref_027 doi: 10.1007/s10013-016-0210-2 – ident: 2025070412063746281_j_anona-2025-0092_ref_009 doi: 10.1007/s12220-019-00308-x – ident: 2025070412063746281_j_anona-2025-0092_ref_023 doi: 10.1142/9789812770561 – ident: 2025070412063746281_j_anona-2025-0092_ref_022 |
| SSID | ssj0000851770 |
| Score | 2.3213696 |
| Snippet | Let
be a stratified Lie group, and let
be a basis of the left-invariant vector fields of degree one on
and
be the sub-Laplacian of
. Given that
, this article... Let G {\mathcal{G}} be a stratified Lie group, and let { X j } 1 ≤ j ≤ n 1 {\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector... Let G{\mathcal{G}} be a stratified Lie group, and let {Xj}1≤j≤n1{\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector fields of... |
| SourceID | doaj crossref walterdegruyter |
| SourceType | Open Website Index Database Publisher |
| StartPage | 1207 |
| SubjectTerms | 22E30 42B20 42B35 43A15 BMO space Riesz transform stratified Lie groups VMO space |
| Title | Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups |
| URI | https://www.degruyter.com/doi/10.1515/anona-2025-0092 https://doaj.org/article/87f2fac0ae3942d1b349941f42d336d8 |
| Volume | 14 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1JSwMxFA7Skz0UV6wbOXjwEjoz2WaOKpYi6kEs9BayioJV2imiv96XzFSqIF68TcJjZvhekve9LF8QOim0TzpyRGQ-EIj4FSk10wT6ZKis1FLbeN755laMxuxqwicrV33FPWGNPHAD3KCUoQjaZvDOihUuNxQ4OssDPFMqXDrmCzFvJZl6anZf5VJmrZYPxOyBhmxaQ5soOIk6Q9_CUFLr76LeW1qhdv5htnivlyuiKdAMN1CvZYj4rPmzTbTmp1uou6IbuI1mwBQj2cQmXorkXRyusJ46nHaU2zqVwQADu4t1z4s65tb4JYAVTvPecZIslu8gU_7A9ZK-YqhtlHQDcFN8_ehxOvcx30Hj4eX9xYi0lycQCxSiJoHbwgtXSWMrUeXcCZqVxnkhZZC2KKx0LDALwxul1mhmodYAGZTUci4Mo7uoA5D5PYRLYyiXlS21pUxYY8pozcvCc-GCKPvodImlem00MlTMLQB2lWBXEXYVYe-j84j1l1kUt04V4HLVulz95fI-oj88pdqON__tuznL9__j0wdovWlCkmTsEHXq2cIfASupzXFqgJ-f1uNv |
| linkProvider | Directory of Open Access Journals |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwELZKe4AeKp6i5eUDBy7WJvErOZaKssC2SNBKvVn22C5ItFtts0Lw65lxsquC4MItHo2VaDz2fDOxPzP2svGp8MgJU6UsMOJ3ovXKC5yTuQPrrQc673x0bKan6v2ZPttgB6uzMLStMqbzxfJHPzCkTuIcllQoW3MNYASeeMyNPY5wowWxBk2-9BffbrEtg_AfXXtrf_r288d1qYVQhbXVSOzzl-6_xaRC3b_Ndr6X39Xrb7kRdQ7vsp0RLvL9YXzvsY10eZ9t3yARfMAWCBsJefJANySlSGsX95eRl-3l0Jc2KnCEeiS7WPaUaPN5Ri1eiuBUMaP2J0ybf_J-hWU5Sgda3YxAlc--Jl4OgVw_ZKeHb04OpmK8SUEA4oleZA1NMrGzATrT1ToaWbUhJmNtttA0YKPKCnCtkxKCV4DSgMjQStDaBCUfsU00WXrMeBuC1LaD1oNUBkJoSVu3TdImZtPuslcrW7qrgTDDUaKBZnfF7I7M7sjsu-w12XqtRkzXRTBfnLtx4rjW5iZ7qNCnOtXEOkjM0VSd8VlKE_F98o-RcuMsvP7Xe2tV7_1Xrxfs9vTkaOZm744_PGF3Bv-xolJP2Wa_WKZniE_68Hz0v18kQudZ |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwELZgKyF6qHiKUh4-cOBibRK_kmNbWBYoBQGVerPssV1xoFtts0L01zPjpKsFwYVbPBrL0fgx34ztz4y9aHwqPHLCVCkL9PidaL3yAudk7sB664HuO384NvMT9e5Un27chaFjlTGdLVc_-4EhdRoXsKJE2ZprAD3w1GNs7LGHGy2INWh6EfNNtmVMJ_WEbe3P33z5uM60EKiwthp5ff5S-zeXVJj7t9nOj7Jbvf6VDaczu8N2RrTI94fuvctupPN7bHuDQ_A-WyJqJODJAz2QlCItXdyfR15Ol0NfyqjAEemR7PuqpzibLzJq8ZIDp4QZlT9j1HzF-2soy1E6sOpmxKn86Fvi5Q7I5QN2Mnv99XAuxocUBCCc6EXW0CQTOxugM12to5FVG2Iy1mYLTQM2qqwAlzopIXgFKA0IDK0ErU1Q8iGboMnSI8bbEKS2HbQepDIQQkvaum2SNjGbdpe9vLaluxj4MhzFGWh2V8zuyOyOzL7LDsjWazUiui6CxfLMjfPGtTY32UOFQ6pTTayDxBBN1Rm_pTQR25N_9JQbJ-Hlv9qtVf34v2o9Z7c-vZq5o7fH7_fY7WH4WFGpJ2zSL1fpKaKTPjwbh98v0YLmfw |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Uniform+boundedness+and+compactness+for+the+commutator+of+an+extension+of+Riesz+transform+on+stratified+Lie+groups&rft.jtitle=Advances+in+nonlinear+analysis&rft.au=Han%2C+Xueting&rft.au=Chen%2C+Yanping&rft.date=2025-07-04&rft.issn=2191-950X&rft.eissn=2191-950X&rft.volume=14&rft.issue=1&rft_id=info:doi/10.1515%2Fanona-2025-0092&rft.externalDBID=n%2Fa&rft.externalDocID=10_1515_anona_2025_0092 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2191-950X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2191-950X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2191-950X&client=summon |