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 \( cLM^p_2, θ_2_ω_2(R^n, v_2) \) and weighted local Morrey-type spaces \( LM^p_1, θ_1_ω_1(R^n, v_1) \) are characterized. In particular, two-sided estimates of the optimal constant \( c \) in the inequality \[ ( _0^∞ ( _B(0,t) f(x)^p_2 v_2(x) dx )^q_2p_2 u_2(t) dt )^1q_2 \] \[ ≤ c ( _0^∞ ( _cB(0,t) f(x)^p_1 v_1(x) dx )^q_1p_1 u_1(t) dt )^1q_1 , f ≥ 0 \] are obtained, where \( p_1, p_2, q_1, q_2 (0,∞) \), \( p_2 ≤ q_2 \) and \( u_1, u_2 \) and \( v_1, v_2 \) are weights on \( (0,∞) \) and \( R^n \), 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. , number=1, journal=Eurasian Mathematical Journal, author=Gogatishvili, Amiran and Mustafayev, Rza and Ünver, Tuğçe, year=2017, month=Mar., pages=34–49