On Weighted Local Morrey-type and Complementary Local Morrey-Type Spaces

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Ces\`{a}ro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Ces\`{a}ro and Copson spaces, we provide complete characterizations for the embeddings $$\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)$$ and $$\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).$$ Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.