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Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel\u2019s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel\u2019s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel\u2019s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schr\u00f6dinger-type operators with generalized kinetic energies.\n<\/jats:p>","DOI":"10.1007\/s00222-022-01144-7","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T12:10:45Z","timestamp":1662379845000},"page":"111-167","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Cwikel\u2019s bound reloaded"],"prefix":"10.1007","volume":"231","author":[{"given":"Dirk","family":"Hundertmark","sequence":"first","affiliation":[]},{"given":"Peer","family":"Kunstmann","sequence":"additional","affiliation":[]},{"given":"Tobias","family":"Ried","sequence":"additional","affiliation":[]},{"given":"Semjon","family":"Vugalter","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,5]]},"reference":[{"key":"1144_CR1","doi-asserted-by":"publisher","first-page":"195","DOI":"10.4310\/MRL.2000.v7.n2.a5","volume":"7","author":"R Benguria","year":"2000","unstructured":"Benguria, R., Loss, M.: A simple proof of a theorem of Laptev and Weidl. 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