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Embeddings between weighted Copson and Cesàro function spaces. (English) Zbl 06819576
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 Cesàro 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 Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators
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