Geodesic
Embeddings between weighted Copson and Cesàro function spaces
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1105-1132
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In this paper, characterizations of the embeddings between weighted Copson function spaces ${\rm Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\rm Ces}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$ \def \frc #1#2{{#1/#2}} \begin{aligned}d \biggl ( \int _0^{\infty } \biggl ( \int _0^t f(\tau )^{p_2}v_2(\tau ) {\rm d}\tau \biggr )^{\frc {q_2}{p_2}} u_2(t) {\rm d} t\biggr )^{\frc {1}{q_2}}\\ \le c \biggl ( \int _0^{\infty } \biggl ( \int _t^{\infty } f(\tau )^{p_1} v_1(\tau ) {\rm d}\tau \biggr )^{\frc {q_1}{p_1}} u_1(t) {\rm d} t\biggr )^{\frc {1}{q_1}}, \end{aligned}d $$ where $p_1,p_2,q_1,q_2 \in (0,\infty )$, $p_2 \le q_2$ 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.
In this paper, characterizations of the embeddings between weighted Copson function spaces ${\rm Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\rm Ces}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$ \def \frc #1#2{{#1/#2}} \begin{aligned}d \biggl ( \int _0^{\infty } \biggl ( \int _0^t f(\tau )^{p_2}v_2(\tau ) {\rm d}\tau \biggr )^{\frc {q_2}{p_2}} u_2(t) {\rm d} t\biggr )^{\frc {1}{q_2}}\\ \le c \biggl ( \int _0^{\infty } \biggl ( \int _t^{\infty } f(\tau )^{p_1} v_1(\tau ) {\rm d}\tau \biggr )^{\frc {q_1}{p_1}} u_1(t) {\rm d} t\biggr )^{\frc {1}{q_1}}, \end{aligned}d $$ where $p_1,p_2,q_1,q_2 \in (0,\infty )$, $p_2 \le q_2$ 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.
DOI : 10.21136/CMJ.2017.0424-16
Classification : 26D10, 46E30
Keywords: Cesàro and Copson function spaces; embedding; iterated Hardy inequalities 
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Gogatishvili, Amiran; Mustafayev, Rza; Ünver, Tuğçe. Embeddings between weighted Copson and Cesàro function spaces. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1105-1132. doi: 10.21136/CMJ.2017.0424-16

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