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

Abstract

In this paper embedding relations between weighted complementary local Morrey-type spaces \( cLM^{p_2, \theta_2}_{\omega_2}(\mathbb{R}^n, v_2) \) and weighted local Morrey-type spaces \( LM^{p_1, \theta_1}_{\omega_1}(\mathbb{R}^n, v_1) \) are characterized. In particular, two-sided estimates of the optimal constant \( c \) in the inequality

\[ \left( \int_0^{\infty} \left( \int_{B(0,t)} f(x)^{p_2} v_2(x) \, dx \right)^{\frac{q_2}{p_2}} u_2(t) \, dt \right)^{\frac{1}{q_2}} \]

\[ \leq c \left( \int_0^{\infty} \left( \int_{cB(0,t)} f(x)^{p_1} v_1(x) \, dx \right)^{\frac{q_1}{p_1}} u_1(t) \, dt \right)^{\frac{1}{q_1}}, \quad f \geq 0 \]

are obtained, where \( p_1, p_2, q_1, q_2 \in (0,\infty) \), \( p_2 \leq q_2 \) 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.