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Capacities and embeddings via symmetrization and conductor inequalities

Author: Pilar Silvestre
Journal: Proc. Amer. Math. Soc. 142 (2014), 497-505
MSC (2000): Primary 46E30; Secondary 28A12
Published electronically: October 11, 2013
MathSciNet review: 3133991
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Abstract: Using estimates of rearrangements in terms of modulus of continuity, some isocapacitary inequalities are derived for Besov, Lipschitz or $ H^{\omega }_p$ capacities. These inequalities allow us to show that capacitary Lorentz spaces, based on Besov spaces, are between the homogeneous Besov spaces and the usual Lorentz spaces. Moreover, we extend a result of Adams-Xiao to other function spaces and we improve embeddings of Lipschitz and $ H^{\omega }_p$ spaces in Lorentz spaces.

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  • [AH] David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441 (97j:46024)
  • [AX] D. R. Adams and J. Xiao, Strong type estimates for homogeneous Besov capacities, Math. Ann. 325 (2003), no. 4, 695-709. MR 1974564 (2004d:31004),
  • [C] Joan Cerdà, Lorentz capacity spaces, Interpolation theory and applications, Contemp. Math., vol. 445, Amer. Math. Soc., Providence, RI, 2007, pp. 45-59. MR 2381885 (2009h:46056),
  • [CMS1] Joan Cerdà, Joaquim Martín, and Pilar Silvestre, Capacitary function spaces, Collect. Math. 62 (2011), no. 1, 95-118. MR 2772330 (2012c:46060),
  • [CMS] J. Cerdà, J. Martín, and P. Silvestre, Conductor Sobolev type estimates and isocapacitary inequalities, to appear in Indiana Univ. Math. J.
  • [Co] Şerban Costea, Scaling invariant Sobolev-Lorentz capacity on $ \mathbb{R}^n$, Indiana Univ. Math. J. 56 (2007), no. 6, 2641-2669. MR 2375696 (2008k:31011),
  • [Co1] Şerban Costea, Besov capacity and Hausdorff measures in metric measure spaces, Publ. Mat. 53 (2009), no. 1, 141-178. MR 2474119 (2009m:31016),
  • [CoMa] Serban Costea and Vladimir Mazya, Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 103-121. MR 2484623 (2010a:31003),
  • [H] Keijo Hildén, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math. 18 (1976), no. 3, 215-235. MR 0409773 (53 #13525)
  • [K2] Kolyada, V. On imbedding in classes $ \varphi (L)$, English transl. in Math. USSR Izv. 9 (1975), 395-413.
  • [K5] V. I. Kolyada, On the differential properties of the rearrangements of functions, Progress in approximation theory (Tampa, FL, 1990) Springer Ser. Comput. Math., vol. 19, Springer, New York, 1992, pp. 333-352. MR 1240790 (95j:26025),
  • [K4] V. I. Kolyada, Mixed norms and Sobolev type inequalities, Approximation and probability, Banach Center Publ., vol. 72, Polish Acad. Sci. Inst. Math., Warsaw, 2006, pp. 141-160. MR 2325743 (2008c:46044),
  • [K1] Viktor I. Kolyada, On embedding theorems, NAFSA 8--Nonlinear analysis, function spaces and applications. Vol. 8, Czech. Acad. Sci., Prague, 2007, pp. 34-94. MR 2657117 (2012d:46073)
  • [L] Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916 (2010m:46049)
  • [MM3] Joaquim Martín and Mario Milman, Symmetrization inequalities and Sobolev embeddings, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2335-2347. MR 2213707 (2007c:46033),
  • [MM1] Joaquim Martín and Mario Milman, Pointwise symmetrization inequalities for Sobolev functions and applications, Adv. Math. 225 (2010), no. 1, 121-199. MR 2669351 (2011m:46050),
  • [M85] Vladimir G. Mazja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985 (87g:46056)
  • [M05] Vladimir Mazya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings, J. Funct. Anal. 224 (2005), no. 2, 408-430. MR 2146047 (2006m:31009),
  • [M06] Vladimir Mazya, Conductor inequalities and criteria for Sobolev type two-weight imbeddings, J. Comput. Appl. Math. 194 (2006), no. 1, 94-114. MR 2230972 (2007a:46035),
  • [M11] Vladimir Mazya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530 (2012a:46056)
  • [M] Mario Milman, On interpolation of entropy and block spaces, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 531-540. MR 1315462 (96b:46107),
  • [S] P. Silvestre, Capacitary function spaces and applications, PhD thesis, TDX, Barcelona Univ., 2012.
  • [T] Giorgio Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994), Prometheus, Prague, 1994, pp. 177-230. MR 1322313 (96a:46062)
  • [U] P. L. Ul'yanov, The embedding of certain function spaces $ H^\omega _p$, English transl. in Math. USSR Izv. 2 (1968), 601-637.

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Additional Information

Pilar Silvestre
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona, Spain

Keywords: Rearrangement estimates, function spaces, modulus of continuity, isocapacitary inequalities, embeddings
Received by editor(s): December 8, 2011
Received by editor(s) in revised form: March 5, 2012, and March 15, 2012
Published electronically: October 11, 2013
Additional Notes: This work was partially supported by Grant MTM2010-14946 and by a grant from the Ferran Sunyer i Balaguer Foundation
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2013 American Mathematical Society

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