Strongly nonlinear Gagliardo-Nirenberg inequality in Orlicz spaces and Boyd indices
Summary: Given an \(N\)-function \(A\) and a continuous function \(h\) satisfying certain assumptions, we derive the inequality \[ \int_{\mathbb R} A(|f^{\prime}(x)|h(f(x)))\,dx\leq C_1\int_{\mathbb R} A\left( C_2 \root p\of {|{\mathcal M}f^{\prime\prime}(x){\mathcal T}_{h,p}(f,x)| }\cdot h(f(x)) \right)dx, \] with constants \(C_1,C_2\) independent of \(f\), where \(f\geq 0\) belongs locally to the Sobolev space \(W^{2,1}(\mathbb R)\), \(f^{\prime}\) has compact support, \(p > 1\) is smaller than the lower Boyd index of \(A\), \({\mathcal T}_{h,p}(\cdot)\) is a certain nonlinear transform depending on \(h\) but not on \(A\), and \({\mathcal M}\) denotes the Hardy-Littlewood maximal function. Moreover, we show that when \(h\equiv 1\), then \({\mathcal M}f^{\prime\prime}\) can be improved by \(f^{\prime\prime}\). This inequality generalizes a previous result by the third author and \textit{J. Peszek} [Asymptotic Anal. 77, No. 3--4, 169--196 (2012; Zbl 1256.26017); J. Fixed Point Theory Appl. 13, No. 1, 271--290 (2013; Zbl 1278.46030)], which was dealing with \(p=2\).
scientific article; zbMATH DE number 6708149