Banach August 2015

banach@mathdept.okstate.edu

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Abstract of a paper by Amiran Gogatishvili, Rza Mustafayev, and Tugce Unver
by alspach＠pelczynski.math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Embeddings between weighted Copson and Ces\`{a}ro function spaces" by Amiran Gogatishvili, Rza Mustafayev, and Tugce Unver. Abstract: In this paper embeddings between weighted Copson function spaces ${\operatorname{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Ces\`{a}ro function spaces ${\operatorname{Ces}}_{p_2,q_2}(u_2,v_2)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^t f(\tau)^{p_2}v_2(\tau)\,d\tau\bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_t^{\infty} f(\tau)^{p_1} v_1(\tau)\,d\tau\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}}, \end{equation*} 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 duality techniques with estimates of optimal constants of the embeddings between weighted Ces\`{a}ro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of the iterated Hardy-type inequalities. Archive classification: math.FA Mathematics Subject Classification: Primary 46E30, Secondary 26D10 Remarks: 25 pages Submitted from: rzamustafayev(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.07866 or http://arXiv.org/abs/1507.07866

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Banach August 2015