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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1306.3659","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-16T13:32:45Z","cross_cats_sorted":["math.CV","math.FA"],"title_canon_sha256":"73f441e4284d6e7a472de9371ce7b461953661cd1f60f84ef74af1f59395d30e","abstract_canon_sha256":"62ce3c82cf086db9ea1f89b272e799090357e6c3541501da83d51d9102e884f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:46.875733Z","signature_b64":"Mb4eUn6rlN9TLyawpU/75WJscxLvUQwI3gnwZaBgFDV0c1ahJSKf32ccgZOBmGXay4z3qMQZMaFSLkR4gR7CAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a298f0836301a41da209d16c579c2880acdbef702dd8b694e8fdf6d58692cf6b","last_reissued_at":"2026-05-18T03:20:46.874846Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:46.874846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\bR^d$. 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