Trudinger inequalities without derivatives (Q2781395)

The limiting case of the Sobolev embedding theorem is \(W^{1,n}_{\text{loc}}(\mathbb{R}^n)\subset \exp L^{n'}(\mathbb{R}^n)\), where the latter space is the Orlicz space associated with the function \(\Phi(t)= \exp(t^{n'})- 1\) and \(n'\) is the conjugate index to \(n\). A more precise version is Trudinger's inequality on \(\mathbb{R}^n\): NEWLINE\[NEWLINE\|f- f_B\|_{\exp L^{n'}}\leq C\Biggl(\int_B |\nabla f|^n\Biggr)^{1/n}NEWLINE\]NEWLINE for any ball \(B\) in \(\mathbb{R}^n\). \textit{P. Hajłasz} and \textit{P. Koskela} [``Sobolev met Poincaré'', Mem. Am. Math. Soc. 688 (2000; Zbl 0954.46022)] considered a connected metric space \(X\) and functions \(f\in L^1_{\text{loc}}(X)\) which satisfy the Poincaré inequality: NEWLINE\[NEWLINE{1\over|B|} \int_B|f- f_B|\leq C\Biggl(\int_B|g|^2\Biggr)^{1/n}NEWLINE\]NEWLINE for any ball \(B\) and some fixed \(g\in L^1_{\text{loc}}(X)\). They used a self-improving argument to establish a Trudinger inequality: NEWLINE\[NEWLINE\|f- f_B\|_{\exp L^{n'}}\leq C\Biggl(\int_B|g|^n\Biggr)^{1/n}NEWLINE\]NEWLINE for any ball \(B\).NEWLINENEWLINENEWLINEThe present paper extends this idea. The authors take a space \((S,d,\mu)\) for homogeneous type, a doubling measure \(w\) absolutely continuous with respect to \(\mu\) and a class of nonnegative functionals \(a\) on the family of balls in \(S\). From a weak Poincaré inequality, they establish a Trudinger inequality of the form: NEWLINE\[NEWLINE\|f- f_B\|_{\exp L^{n'}(B, w)}\leq C\|f\|_a a(\kappa B)NEWLINE\]NEWLINE for any ball \(B\) and some fixed \(\kappa> 1\). The self-improving theorem of John-Nirenberg is shown to be a consequence of this result.