{"paper":{"title":"Square functions of fractional homogeneity and Wolff potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Laura Prat, Vasileios Chousionis, Xavier Tolsa","submitted_at":"2014-10-20T13:42:51Z","abstract_excerpt":"In this paper it is shown that for anymeasure $\\mu$ in $\\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\\displaystyle{\\iint_0^\\infty(\\frac{\\mu(B(x,r))}{r^s})^2\\,\\frac{dr}{r}d\\mu(x)}$ is comparable to\n  $$\\iint_0^\\infty(\\frac{\\mu(B(x,r))}{r^s} - \\frac{\\mu(B(x,2r))}{(2r)^s})^2\\,\\frac{dr}rd\\mu(x),$$ unlike in the case when $s$ is an integer. We also study the relation with the $L^2-$norm of $s$-Riesz transforms, $0<s<1$, and we provide a counterexample in the integer case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5272","kind":"arxiv","version":2},"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"}