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 Morreytype spaces (LMp theta,omega)-L-c(R-n, v) and weighted local Morrey-type spaces L LMp theta,omega(R-n, v) are characterized. In particular, two-sided estimates of the optimal constant c in the inequality (integral(infinity)(0) (integral(B(0,t)) f (x)(p2) v(2)(x)dx)(q2/p2) u(2)(t) dt)(1/q2) <= c (integral(infinity)(0)(integral(cB(0,t)) f (x)(p1) v(1)(x) dx)(q1/p1) u(1)(t) dt)(1/q1), f >= 0 are obtained, where p(1), p(2), q(1), q(2) is an element of (0, infinity), p(2) <= q(2) and u(1), u(2) and v(1), v(2) are weights on (0,infinity) 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.