2023-03-22T18:56:29Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/696702022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116Inequalities associated with dilationsOzawa, TohruSasaki, Hironobuopen accessInequalitiesGenerator of semi-group of dilationsSobolev's inequality Poincar´e's inequalityHardy's inequality.410Some properties of distributions f satisfying x ¢ rf 2 Lp(Rn), 1 · p < 1, are studied. The operator x ¢ r is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x ¢r. Using the inequalities, we also show that if f 2 Lp loc(Rn), x ¢rf 2 Lp(Rn) and jxjn=pjf(x)j vanishes at infinity, then f belongs to Lp(Rn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2(Rn).Department of Mathematics, Hokkaido University2007engdepartmental bulletin paperVoRhttps://doi.org/10.14943/84011http://hdl.handle.net/2115/6967010.14943/84011Hokkaido University Preprint Series in Mathematics861112https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/69670/1/pre861.pdfapplication/pdf206.26 KB2007