Embedding relations between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces

In this paper embedding relations between weighted complementary local Morrey-type spaces $^cLM_{p\theta,\omega}(\mathbb{R}^n,v)$ and weighted local Morrey-type spaces $LM_{p\theta,\omega}(\mathbb{R}^n,v)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality
$$\left( \int_0^\infty\left( \int_{B(0,t)} f(x)^{p_2}v_2(x)\,dx \right)^{\frac{q_2}{p_2}}u_2(t)\,dt \right)^{\frac1{q_2}} \leqslant c \left(\int_0^\infty\left(\int_{^cB(0,t)}f(x)^{p_1}v_1(x)\,dx\right)^{\frac{q_1}{p_1}}u_1(t)\,dt\right)^{\frac1{q_1}},\quad f\geqslant0$$
are obtained, where $p_1$, $p_2$, $q_1$, $q_2\in(0,\infty)$, $p_2\leqslant q_2$ and $u_1$, $u_2$ and $v_1$, $v_2$ are weights on $(0,\infty)$ and $\mathbb{R}^n$, respectively. The proof is based on the combination of the duality techniques with estimates of optimal constants of the embedding relations between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which allows to reduce the problem to using of the known Hardy-type inequalities.