{"paper":{"title":"Logarithmic Bloch space and its predual","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Miroslav Pavlovi\\'c","submitted_at":"2011-04-24T13:24:04Z","abstract_excerpt":"We consider the space $\\bk^1_{\\log^\\alpha}$, of analytic functions on the unit disk $\\D,$ defined by the requirement $\\int_\\D|f'(z)|\\phi(|z|)\\,dA(z)<\\infty,$ where $\\phi(r)=\\log^\\alpha(1/(1-r))$ and show that it is a predual of the \"$\\log^\\alpha$-Bloch\" space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\\sum_{n=0}^\\infty a_nz^n$ with $a_n\\downarrow 0$ is in $\\bk^1_{\\log^\\alpha}$ iff $\\sum_{n=0}^\\infty \\log^\\alpha(n+2)/(n+1)<\\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\\bk"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4629","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}