{"corpus_id":116994708,"paper_sha":"5a58bbe8bc88422bd0d2e325f1daf301fe0123a9","doi":"10.1093/IMRN/RNW323","arxiv_id":"1506.04883","pmid":null,"pmcid":null,"mag_id":2963297860,"dblp_id":null,"acl_id":null,"title":"Spectral multipliers, Bochner-Riesz means and uniform Sobolev inequalities for elliptic operators","year":2015,"publication_date":"2015-06-16","venue":"","journal":{"name":"arXiv: Analysis of PDEs","pages":null,"volume":""},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":63,"citation_count":25,"influential_citation_count":0,"is_open_access":true,"arxiv_categories":["math.AP"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"http://arxiv.org/pdf/1506.04883","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/5a58bbe8bc88422bd0d2e325f1daf301fe0123a9","s2_open_access_license":null,"s2_open_access_status":"GREEN","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\\in {\\mathbb C}\\backslash [0, \\infty)$, which give an extension of the second order results of Kenig-Ruiz-Sogge \\cite{KRS}. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schr\\\"odinger operators $P(D)+V$ with small integrable potentials $V$. Finally we deduce spectral multiplier estimates for all these operators, including sharp Bochner-Riesz summability results.","claims":[{"public_id":"cl_0654d7017315a35674ea445302d46134","status":"active","text":"Lp-Lq bounds for spectral multipliers and Bochner-Riesz means with negative index are established in the general setting of abstract self-adjoint operators.","confidence":0.93,"contributors":[{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["extraction"],"url":"https://sah.borca.ai/u/322360f1c1"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_0654d7017315a35674ea445302d46134"},{"public_id":"cl_c33b897fa28cf327adf41e81faaf7f79","status":"active","text":"Perturbation techniques yield uniform Sobolev estimates for Schrödinger operators P(D)+V with small integrable potentials V.","confidence":0.93,"contributors":[{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["extraction"],"url":"https://sah.borca.ai/u/322360f1c1"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_c33b897fa28cf327adf41e81faaf7f79"},{"public_id":"cl_8055b545e969c3022472d4715201554c","status":"active","text":"Sharp Bochner-Riesz summability results and spectral multiplier estimates are deduced for higher order elliptic operators and Schrödinger operators with small potentials.","confidence":0.9,"contributors":[{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale 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