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Correction to: A \({\texttt {p}}({\cdot })\)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

  • Thomas BieskeEmail author
  • Robert D. Freeman
Correction
  • 271 Downloads

1 Correction to: Anal.Math.Phys. (2018) 8:289–308  https://doi.org/10.1007/s13324-018-0235-7

In this short note, we remark on the main theorem of [1, Theorem 5.1]. First, the authors would like to thank the referee for pointing out that a more general result related to [1, Theorem 5.1], was proved for vector fields satisfying Hörmander’s condition by Li et al. [3, Theorem 1.10]. In [3, Theorem 3.1], the authors first show a more general result for boundedness of fractional integral operators in variable Lebesgue spaces on spaces of homogeneous type and combine this with a representation formula proved in [2] to establish a higher order Poincaré inequality for vector fields satisfying Hörmander’s condition on Boman chains (and therefore metric balls). Here they took the assumptions that \({\texttt {p}}({\cdot })\) is log-Hölder continuous and \(1< {\texttt {p}}^- \le {\texttt {p}}^+ < Q/m\), where Q is the homogenous dimension and m is the dimension of the variable exponent space (in our case \(m=1\)).

However, the main theorem [1, Theorem 5.1] does not follow directly from [3, Theorem 1.10]. The difference is in the assumption that \({\texttt {p}}({\cdot })\) is log-Hölder continuous in [3, Theorem 1.10] while [1, Theorem 5.1] requires that the variable exponent \({\texttt {p}}({\cdot })\) satisfies the jump condition. This allows for the case where \({\texttt {p}}^- > Q\) in the gauge balls discussed in Section 5 of [1]. Thus, the condition \(1< {\texttt {p}}^- \le {\texttt {p}}^+ < Q\) in [3, Theorem 1.10] is violated. Consequently, when \({\texttt {p}}^- > Q\), then the hypotheses of [3, Theorem 1.10] do not hold but those of [1, Theorem 5.1] do hold.

Furthermore, the main theorem [1, Theorem 5.1] does not follow from combining [3, Theorem 3.1] and the representation formula of functions with boundary value zero for vector fields established in [4]. Again, we have the possibility that \({\texttt {p}}^- > Q\) in gauge balls but this fails the conditions in [3, Theorem 3.1].

In both cases, it is obvious that if \({\texttt {p}}^+ \le \frac{Q{\texttt {p}}^-}{Q- {\texttt {p}}^-}\) in gauge balls, then we have the jump condition satisfied and this also agrees with the conditions in [3, Theorem 1.10] and thus [1, Theorem 5.1] does indeed follow directly from [3, Theorem 1.10]. Similarly, this is also true for [3, Theorem 3.1] and we could use representation formulas in [4] to easily obtain [1, Theorem 5.1].

Notes

References

  1. 1.
    Bieske, T., Freeman, R.D.: A \(p(x)\)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. Anal. Math. Phys. 8, 289–308 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Franchi, B., Lu, G., Wheeden, R.L.: Representation formulas and weighted Poincaré inequalities for Hörmander vector fields. Ann. Inst. Fourier (Grenoble) 45, 577–604 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, X., Lu, G.Z., Tang, H.L.: Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces. Acta Math. Sin. Engl. Ser. 31(7), 1067–1085 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
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    Lu, G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Rev. Mat. Iberoam. 8, 367–439 (1992)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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