an:06819576
Zbl 06819576
Gogatishvili, Amiran; Mustafayev, Rza; ??nver, Tu????e
Embeddings between weighted Copson and Ces??ro function spaces
EN
Czech. Math. J. 67, No. 4, 1105-1132 (2017).
00372715
2017
j
46E30 26D10
Ces??ro and Copson function spaces; embedding; iterated Hardy inequalities
Summary: In this paper, characterizations of the embeddings between weighted Copson function spaces \(\text{Cop}_{p_1,q_1}(u_1,v_1)\) and weighted Ces??ro function spaces \(\text{Ces}_{p_2,q_2}(u_2,v_2)\) are given. In particular, two-sided estimates of the optimal constant \(c\) in the inequality
\[
\biggl(\int_0^{\infty} \biggl(\int_0^t f(\tau)^{p_2}v_2(\tau)\text{d}\tau\biggr)^{\! q_2/p_2}u_2(t)\text{d} t\biggr)^{\! 1/{q_2}}\leq c\biggl(\int_0^{\infty}\biggl(\int_t^{\infty}f(\tau)^{p_1}v_1(\tau)\text{d}\tau\biggr)^{\! q_1/p_1}u_1(t)\text{d}t\biggr)^{\! 1/q_1},
\]
where \(p_1,p_2,q_1,q_2 \in (0,\infty)\), \(p_2 \leq q_2\) and \(u_1\), \(u_2\), \(v_1\), \(v_2\) are weights on \((0,\infty)\), are obtained. The most innovative part consists of the fact that possibly different parameters \(p_1\) and \(p_2\) and possibly different inner weights \(v_1\) and \(v_2\) are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Ces??ro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.