Sections of Functions and Sobolev-Type Inequalities
2014 (English)In: Proceedings of the Steklov Institute of Mathematics, ISSN 0081-5438, E-ISSN 1531-8605, Vol. 284, no 1, 192-203 p.Article in journal (Refereed) Published
We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy the Lipschitz condition of order 0 < alpha a parts per thousand currency sign 1. We prove that if for a function f the Lip alpha-norms of these sections belong to the Lorentz space L (p,1)(a"e) (p = 1/alpha), then f can be modified on a set of measure zero so as to become bounded and uniformly continuous on a"e(2). For alpha = 1 this gives an extension of Sobolev's theorem on continuity of functions of the space W (1) (2,2) (a"e(2)). We show that the exterior L (p,1)-norm cannot be replaced by a weaker Lorentz L (p,q) -norm with q > 1.
Place, publisher, year, edition, pages
Maik Nauka/Interperiodica, 2014. Vol. 284, no 1, 192-203 p.
IdentifiersURN: urn:nbn:se:kau:diva-41538DOI: 10.1134/S0081543814010131ISI: 000335559000012OAI: oai:DiVA.org:kau-41538DiVA: diva2:923121