Rocky Mountain Journal of Mathematics

On weak continuity of the Moser functional in Lorentz-Sobolev spaces

Robert Černý

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Let $B(R)\subset \mathbb{R}^n $, $n\in \mathbb{N} $, $n\geq 2$, be an open ball. By a result from~\cite {AdT}, the Moser functional with the borderline exponent from the Moser-Trudinger inequality fails to be sequentially weakly continuous on the set of radial functions from the unit ball in $W_0^{1,n}(B(R))$, only in the exceptional case of sequences acting like a~concentrating Moser sequence.

We extend this result into the Lorentz-Sobolev space $W_0^1L^{n,q}(B(R))$, with $q\in (1,n]$, equipped with the norm $$ ||\nabla u||_{n,q}:= ||t^{1/n-1/q}|\nabla u|^*(t)||_{L^q((0,|B(R)|))}. $$ We also consider the case of a nontrivial weak limit and the corresponding Moser functional with the borderline exponent from the concentration-compactness alternative.

Article information

Rocky Mountain J. Math., Volume 47, Number 3 (2017), 757-788.

First available in Project Euclid: 24 June 2017

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Primary: 26D10: Inequalities involving derivatives and differential and integral operators 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Sobolev spaces Lorentz-Sobolev spaces Moser-Trudinger in­equal­ity concentration-compactness principle sharp constants


Černý, Robert. On weak continuity of the Moser functional in Lorentz-Sobolev spaces. Rocky Mountain J. Math. 47 (2017), no. 3, 757--788. doi:10.1216/RMJ-2017-47-3-757.

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  • Tintarev K. Adimurthi, On compactness in the Trudinger-Moser inequality, Ann. Scuola Norm. 13 (2014), 399–416.
  • A. Alvino, V. Ferone and Q. Trombetti, Moser-type inequalities in Lorentz spaces, Potent. Anal. 5 (1996), 273–299.
  • R. Černý, Concentration-compactness principle for generalized Moser-Trudinger inequalities: Characterization of the non-compactness in the radial case, Math. Inequal. Appl. 17 (2014), 1245–1280.
  • ––––, Concentration-compactness principle for Moser-type inequalities in Lorentz-Sobolev spaces, Potent. Anal. 43 (2015), 97–126.
  • R. Černý, A. Cianchi and S. Hencl, Concentration-compactness principle for Moser-Trudinger inequalities: New results and proofs, Ann. Mat. Pura Appl. 192 (2013), 225–243.
  • J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
  • G.B. Folland, Real analysis: Modern techniques and their applications, John Wiley & Sons, Inc., New York, 1999.
  • I. Halperin, Uniform convexity in function spaces, Duke Math. J. 21 (1954), 195–204.
  • P.L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, I, Rev. Mat. Iber. 1 (1985), 145–201.
  • V. Maz'ya, Sobolev spaces, Springer-Verlag, Berlin, 1985.
  • J. Moser, A sharp form of an inequality by N.,Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.
  • G. Talenti, An inequality between $u^*$ and $\nabla u$, Int. Numer. Math. 103, Birkhäuser, Basel, 1992.
  • N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.