<p>In this paper, we establish Trudinger–Moser inequalities for fractional Sobolev–Slobodeckij spaces in both bounded domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> of the Heisenberg group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {H}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and the entire <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {H}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. To prove the Trudinger–Moser inequality in bounded domains, we utilize the characterization of Sobolev–Slobodeckij spaces in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {H}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> through interpolation, using Peetre’s K–method. In extending the local inequality to the global setting, we develop a new technique to circumvent the use of symmetrization. Finally, we establish the concentration–compactness principle for these inequalities. This Lions type result significantly improves the Trudinger–Moser inequality along particular sequences and it has important applications in the study of semilinear sub–elliptic PDEs with exponential nonlinearities, providing a crucial tool to address the lack of compactness. Our results are presented for the Heisenberg group, but the same proofs still work on general Carnot groups.</p>

Trudinger–Moser inequalities for fractional Sobolev–Slobodeckij spaces in the Heisenberg group and the concentration–compactness principle

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发表日期：

2025年9月16日

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Guozhen Lu,
Letizia Temperini

摘要

In this paper, we establish Trudinger–Moser inequalities for fractional Sobolev–Slobodeckij spaces in both bounded domains $\Omega $ $Ω$ of the Heisenberg group ${\mathbb {H}}^n$ $H^{n}$ and the entire $\mathbb {H}^n$ $H^{n}$ . To prove the Trudinger–Moser inequality in bounded domains, we utilize the characterization of Sobolev–Slobodeckij spaces in $\mathbb {H}^n$ $H^{n}$ through interpolation, using Peetre’s K–method. In extending the local inequality to the global setting, we develop a new technique to circumvent the use of symmetrization. Finally, we establish the concentration–compactness principle for these inequalities. This Lions type result significantly improves the Trudinger–Moser inequality along particular sequences and it has important applications in the study of semilinear sub–elliptic PDEs with exponential nonlinearities, providing a crucial tool to address the lack of compactness. Our results are presented for the Heisenberg group, but the same proofs still work on general Carnot groups.