{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T18:24:10Z","timestamp":1749320650585,"version":"3.37.3"},"reference-count":0,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["SFRH\/BPD\/34258\/2006"],"award-info":[{"award-number":["SFRH\/BPD\/34258\/2006"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Function Spaces"],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider non\u2010standard H\u00f6lder spaces <jats:italic>H<\/jats:italic><jats:sup><jats:italic>\u03bb<\/jats:italic>(\u22c5)<\/jats:sup>(<jats:italic>X<\/jats:italic>) of functions <jats:italic>f<\/jats:italic> on a metric measure space (<jats:italic>X, d, \u03bc<\/jats:italic>), whose H\u00f6lder exponent <jats:italic>\u03bb<\/jats:italic>(<jats:italic>x<\/jats:italic>) is variable, depending on <jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>. We establish theorems on mapping properties of potential operators of variable order <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>), from such a variable exponent H\u00f6lder space with the exponent  <jats:italic>\u03bb<\/jats:italic>(<jats:italic>x<\/jats:italic>) to another one with a \u201cbetter\u201d exponent  <jats:italic>\u03bb<\/jats:italic>(<jats:italic>x<\/jats:italic>) + <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>), and similar mapping properties of hypersingular integrals of variable order <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>) from such a space into the space with the \u201cworse\u201d exponent <jats:italic>\u03bb<\/jats:italic>(<jats:italic>x<\/jats:italic>) \u2212 <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>) in the case <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>) &lt; <jats:italic>\u03bb<\/jats:italic>(<jats:italic>x<\/jats:italic>). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces <jats:italic>H<\/jats:italic><jats:sup><jats:italic>\u03bb<\/jats:italic>(\u22c5)<\/jats:sup>(<jats:italic>X<\/jats:italic>), but also the generalized H\u00f6lder spaces <jats:italic>H<\/jats:italic><jats:sup><jats:italic>w<\/jats:italic>(\u22c5,\u22c5)<\/jats:sup>(<jats:italic>X<\/jats:italic>) of functions whose continuity modulus is dominated by a given function <jats:italic>w<\/jats:italic>(<jats:italic>x, h<\/jats:italic>), <jats:italic>x<\/jats:italic> \u2208 <jats:italic>X, h<\/jats:italic> &gt; 0. We admit variable complex valued orders <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>), where <jats:italic>\u211c<\/jats:italic><jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized H\u00f6lder spaces with the weight <jats:italic>\u03b1<\/jats:italic>(<jats:italic>x<\/jats:italic>).<\/jats:p>","DOI":"10.1155\/2010\/659456","type":"journal-article","created":{"date-parts":[[2011,12,26]],"date-time":"2011-12-26T11:36:41Z","timestamp":1324899401000},"page":"215-244","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["Fractional integrals  and hypersingular integrals in variable order H\u00f6lder spaces  on homogeneous spaces"],"prefix":"10.1155","volume":"8","author":[{"given":"Natasha","family":"Samko","sequence":"first","affiliation":[]},{"given":"Stefan","family":"Samko","sequence":"additional","affiliation":[]},{"given":"Boris","family":"Vakulov","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,1]]},"container-title":["Journal of Function Spaces"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2010\/659456.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2010\/659456","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T20:41:45Z","timestamp":1723063305000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2010\/659456"}},"subtitle":[],"editor":[{"given":"Vakhtang","family":"Kokilashvili","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":0,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1155\/2010\/659456"],"URL":"https:\/\/doi.org\/10.1155\/2010\/659456","archive":["Portico"],"relation":{},"ISSN":["2314-8896","2314-8888"],"issn-type":[{"type":"print","value":"2314-8896"},{"type":"electronic","value":"2314-8888"}],"subject":[],"published":{"date-parts":[[2010,1]]},"assertion":[{"value":"2009-02-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2010-01-01","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}