{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:OBP5RXT37RNYZQ7UUFPNN57DEN","short_pith_number":"pith:OBP5RXT3","schema_version":"1.0","canonical_sha256":"705fd8de7bfc5b8cc3f4a15ed6f7e323415e2aa152e915b4c8e4fbec14ea7bb3","source":{"kind":"arxiv","id":"1601.00938","version":1},"attestation_state":"computed","paper":{"title":"Sharp inequalities for one-sided Muckenhoupt weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ioannis Parissis, Olli Saari, Paul A. Hagelstein","submitted_at":"2016-01-05T18:58:15Z","abstract_excerpt":"Let $A_\\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\\mathsf M^+:L^p(w)\\to L^{p,\\infty}(w)$ for some $p>1$, where $\\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\\in A_\\infty ^+$ if and only if there exist numerical constants $\\gamma\\in(0,1)$ and $c>0$ such that $$ w(\\{x \\in \\mathbb{R} : \\, \\mathsf M ^+\\mathbf 1_E (x)>\\gamma\\})\\leq c w(E) $$ for all measurable sets $E\\subset \\mathbb R$. Furthermore, letting $$ \\mathsf C_w ^+(\\alpha):= \\sup_{01$, where $\\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\\in A_\\infty ^+$ if and only if there exist numerical constants $\\gamma\\in(0,1)$ and $c>0$ such that $$ w(\\{x \\in \\mathbb{R} : \\, \\mathsf M ^+\\mathbf 1_E (x)>\\gamma\\})\\leq c w(E) $$ for all measurable sets $E\\subset \\mathbb R$. Furthermore, letting $$ \\mathsf C_w ^+(\\alpha):= \\sup_{0