Embeddings between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces

In this paper embeddings between weighted complementary local Morrey-type spaces ${\,^{^{\bf c}}\!}LM_{p\theta,\omega}({\mathbb R}^n,v)$ 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 \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_{B(0,t)} f(x)^{p_2}v_2(x)\,dx \bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_{{\,^{^{\bf c}}\!}B(0,t)} f(x)^{p_1} v_1(x)\,dx\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}} \end{equation*} are obtained, where $p_1,\,p_2,\,q_1,\,q_2 \in (0,\infty)$, $p_2 \le q_2$ 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 duality techniques with estimates of optimal constants of the embeddings between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of the iterated Hardy-type inequalities.