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When\n \n \n $$p> 4$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n ><\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure\n \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s10884-026-10509-y","type":"journal-article","created":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T07:42:45Z","timestamp":1777102965000},"page":"455-471","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schr\u00f6dinger Equation"],"prefix":"10.1007","volume":"38","author":[{"given":"Guopeng","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiawei","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leonardo","family":"Tolomeo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,25]]},"reference":[{"key":"10509_CR1","doi-asserted-by":"crossref","unstructured":"Bakry, D., \u00c9mery, M.: Diffusions hypercontractives, S\u00e9minaire de probabilit\u00e9s, XIX, 1983\/84, 177\u2013206. 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