@article{8636227,
  abstract     = {{We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy-Sobolev type inequalities. Most of the obtained estimates are new already for the case of R-n. For example, for any f is an element of C-0(infinity) (R-n \ { 0 }) we obtain the range of critical Hardy inequalities of the form 
sup(R>0)parallel to f - f(R)/vertical bar x vertical bar(n/p)logR/vertical bar x vertical bar parallel to(Lp(Rn)) <= p/p - 1 parallel to 1/vertical bar x vertical bar(n/p-1)del f parallel to(Lp(Rn)) ,1 < p < infinity, 
where f(R) = f (R x/vertical bar x vertical bar), with sharp constant P/p-1 , recovering the known cases of p = n and p = 2. Moreover, we also show a new type of a critical Hardy inequality of the form 
parallel to f/vertical bar x vertical bar parallel to(Ln(Rn)) <= n parallel to(log vertical bar x vertical bar)del f parallel to(Ln(Rn)), 
for all f is an element of C-0(infinity)(R-n \{ 0 }), where the constant n is sharp.}},
  author       = {{Ruzhansky, Michael and Suragan, Durvudkhan}},
  issn         = {{1239-629X}},
  journal      = {{ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA}},
  keywords     = {{Critical Hardy inequality,homogeneous Lie group,uncertainty principle,RELLICH}},
  language     = {{eng}},
  pages        = {{1159--1174}},
  title        = {{Critical Hardy inequalities}},
  url          = {{http://doi.org/10.5186/aasfm.2019.4467}},
  volume       = {{44}},
  year         = {{2019}},
}

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