While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the \n\n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n L_p<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Mat\u00e9rn and surface spline RBFs). In this way it extends the boundary-free construction recently presented by Fuselier, Hangelbroek and Narcowich [Localized bases for kernel spaces on the unit sphere<\/italic>, SIAM J. Numer. Anal. 51 (2013), no. 5, 2358-2562].<\/p>","DOI":"10.1090\/mcom\/3256","type":"journal-article","created":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T13:19:39Z","timestamp":1485177579000},"page":"1949-1989","source":"Crossref","is-referenced-by-count":11,"title":["An inverse theorem for compact Lipschitz regions in \u211d^{\ud835\udd55} using localized kernel bases"],"prefix":"10.1090","volume":"87","author":[{"given":"T.","family":"Hangelbroek","sequence":"first","affiliation":[]},{"given":"F.","family":"Narcowich","sequence":"additional","affiliation":[]},{"given":"C.","family":"Rieger","sequence":"additional","affiliation":[]},{"given":"J.","family":"Ward","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2017,10,17]]},"reference":[{"key":"1","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2008","ISBN":"http:\/\/id.crossref.org\/isbn\/9780387759333","edition":"3"},{"key":"2","doi-asserted-by":"crossref","unstructured":"Carsten Carstensen and Birgit Faermann, Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind, Engineering analysis with boundary elements 25 (2001), no. 7, 497\u2013509.","DOI":"10.1016\/S0955-7997(01)00012-1"},{"issue":"247","key":"3","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1090\/S0025-5718-03-01583-7","article-title":"Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method","volume":"73","author":"Dahmen, W.","year":"2004","journal-title":"Math. 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