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Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces

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19 pages
Our aim in this paper is to give Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem. MSC: 31B15, 46E35, 26A33. Our aim in this paper is to give Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem. MSC: 31B15, 46E35, 26A33.
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Sawano and Shimomura Journal of Inequalities and Applications 2013, 2013 :12 http://www.journalonequalitiesandapplications.com/content/2013/1/12
R E S E A R C H Open Access Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value over non-doubling measure spaces Yoshihiro Sawano 1,2* and Tetsu Shimomura 3 * Correspondence: ysawano@tmu.ac.jp Abstract 1 Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan Our aim in this paper is to give Sobolev’s inequality for Riesz potentials of functions in 2 Current address: Department of generalized Morrey spaces with variable exponent attaining the value 1 over Mathematics and Information non-doubling measure spaces. The main result is oriented to the outrange of the Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, well-known Adams theorem. Hachioji-shi, Tokyo 192-0397, Japan MSC: Primary 31B15; secondary 46E35; 26A33 Full list of author information is available at the end of the article Keywords: Riesz potentials; Sobolev’s inequality; Morrey spaces of variable exponents; non-doubling measure
1 Introduction The boundedness of fractional integral operators on Morrey spaces is known as the Adams theorem. Recently, many endpoint results have been obtained for this theorem, and in this paper we extend them to generalized Morrey spaces with variable exponent attaining the value  over non-doubling measure spaces. In  Morrey observed that a weaker regularity sufficed in order that the solutions in elliptic differential equations were smooth [ ]. This observation grew up to be a useful tool for partial differential equations in general. Nowadays, his technique turned out to be a wide theory of function spaces called Morrey spaces; see also [ ]. The (original) Morrey space L p , λ ( R N ) with  p < and  < λ N is a normed space whose norm is given by f L p , λ sup x R N , r > ( r N λ B ( x , r ) | f ( y ) | p dy ) p for f L l p oc ( R N ). We are oriented to building up a theory of a metric measure space ( X , d , μ ) for which the notion of dimension is not equipped with μ , where d is a distance function and μ is a Borel measure. For example, we encounter the situation where more than one dimension comes into play. . In R consider the set X Y Z , where Y ≡ { ( x , , ) : x R } and Z ≡ { ( x , y , z ) : z = x + y } . Denote by H s the s -dimensional Hausdorff measure in R for s . Consider μ H | Y + H | Z . Then μ has two different dimensions. . The above example is a little artificial. Consider the Cantor dust, which is given by E j = E j , where E j is given recursively by E [, ] , © 2013 Sawano and Shimomura; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and re-production in any medium, provided the original work is properly cited.