A Gagliardo-Nirenberg inequality for Orlicz and Lorentz spaces on \(\mathbb R^n_+\) (Q934177)

The authors develop methods of their preceding papers, e.g. [East J. Approx. 10, No.~3, 371--377 (2004; Zbl 1113.26014); Vietnam J. Math. 33, No.~2, 207--213 (2005; Zbl 1119.26019)], to extend the Gagliardo-Nirenberg inequality in the form \(\| D^\alpha f\|_X\leq C\| f\| ^{1-|\alpha|/\ell}_X\bigl(\sum_{\beta=\ell}\| D^\beta f\|_X\bigr)^{|\alpha|/\ell}\), where \(X\) stands for the Orlicz space \(L_\Phi(\mathbb R^n_+)\) or for the Lorentz space \(N_\Psi(\mathbb R^n_+)\). The space \(L_\Phi(\mathbb R^n_+)\) is equipped with the Orlicz norm \(\| f\|_{L_\Phi(\mathbb R^n_+)}=\sup\bigl|\int_{\mathbb R^n_+}f(x)g(x)\,dx\bigr|\), where \(\Phi:[0,\infty)\to[0,\infty)\) is a Young function and the supremum is taken over all functions \(g\) with \(\int_{\mathbb R^n_+}\overline{\Phi}(g(x))\,dx\leq1\), where \(\overline{\Phi}(t)\) is the conjugate Young function defined by \(\overline{\Phi}(t)=\sup_{s\geq0}(ts-\Phi(s))\). The Lorentz space \(N_\Psi(\mathbb R^n_+)\) is equipped with the norm \(\| f\|_{N_\Psi(\mathbb R^n_+)}=\int_0^\infty\Psi(\lambda_f(y))\,dy\), where \(\Psi:[0,\infty)\to[0,\infty)\) is a non-zero, non-decreasing concave function with \(\Psi(0+)=\Psi(0)=0\) and \(\lambda_f(y)=\text{meas}\{x\in G:| f(x)| >y\}\), \(y\geq0\). The method is based on a certain mollification argument.