{"paper":{"title":"On Astala's theorem for martingales and Fourier multipliers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.FA"],"primary_cat":"math.PR","authors_text":"Adam Osekowski, Rodrigo Banuelos","submitted_at":"2013-06-16T13:32:45Z","abstract_excerpt":"We exhibit a large class of symbols $m$ on $\\R^d$, $d\\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\\R^d$ with $E\\subseteq D$ and $|D|<\\infty$, then $$ \\int_{D\\setminus E} |T_{m}\\chi_E(x)|\\mbox{d}x\\leq \\begin{cases} |E|+|E|\\ln\\left(\\frac{|D|}{2|E|}\\right), & \\mbox{if}|E|<|D|/2, |D\\setminus E|+\\frac{1}{2}|D \\setminus E|\\ln \\left(\\frac{|E|}{|D\\setminus E|}\\right), & \\mbox{if}|E|\\geq |D|/2. \\end{cases}. $$ Here $|\\cdot|$ denotes the Lebesgue measure on $\\bR^d$. When $d=2$, these multipliers include the real a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3659","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"}