Functional analytical setting

Abstract

For any r ∈ (1, +∞), we denote by L^r (ℝ ⁿ⁺¹₊ , y^a) the wheigted¹ Lebesgue space, endowed with the norm

$$ {{\left\| U \right\|}_{{{{L}^{r}}\left( {\mathbb{R}_{+}^{{n+1}},{{y}^{a}}} \right)}}}:={{\left( {{{\int }_{{\mathbb{R}_{+}^{{n+1}}}}}{{y}^{a}}{{{\left| U \right|}}^{r}}dX} \right)}^{{{{1} \left/ {r} \right.}}}}. $$

The following result shows that Ḣ ^s _a (ℝ ⁿ⁺¹₊ , y^a) is continuously embedded in ${{L}^{{{{2}_{\gamma }}}}}\left( {\mathbb{R}_{+}^{{n+1}},{{y}^{a}}} \right).$