Weighted Sobolev inequality in Musielak-Orlicz space (Q662051)

Let \(L_{p,q,\beta}(\mathbb{R}^n)\) be the Lebesgue space with continuous variable exponents \(p\), \(q\) satisfying the log-Hölder and log-log-Hölder condition, respectively, defined by means of the quasi-norm \[ \| f\|_{L_{p,q,\beta}(\mathbb{R}^n)}= \text{inf}\{\lambda> 0: \int(1+ |y|)^{\beta(y)}|f(y)/\lambda|^{p(y)}\cdot[\log(e+ |y|)^{\beta(y)}\cdot|f(y)/\lambda|]J^{q(y)}dy\leq 1\}. \] Let \(p^*(x)\) and \(q^*(x)\) denote the Sobolev conjugates of \(p(x)\) and \(q(x)\), respectively. Finally, let \[ I_\alpha f(x)= \int|x- y|^{\alpha(x)-n} f(y)\,dy \] be the Riesz potential of order \(\alpha\). The authors prove the estimates \(\| Mf\|_{L_{p^*,q^*,\beta}(\mathbb{R}^n)}\leq C\| f\|_{L_{p,q,\beta}(\mathbb{R}^n)}\), where \(\alpha^+- np_\infty<\beta< n(t- 1/p^-)\) and \(Mf(x)\) is the maximal function of \(f\). This easily implies the estimate \(\| I_\alpha f\|_{L_{p^*,p^*q/p, \beta}(\mathbb{R}^n)}\leq C\| f\|_{L_{p,q,\beta}(\mathbb{R}^n)}\).