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Embeddings between weighted Copson and Cesàro function spaces

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Abstract

In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality

$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$

where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 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.

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Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Amiran Gogatishvili

  2. Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, B. Vahabzade St. 9, Baku, AZ 1141, Azerbaijan

    Rza Mustafayev

  3. Department of Mathematics Faculty of Science and Arts, Kırıkkale University, 71450, Yahsihan, Kırıkkale, Turkey

    Rza Mustafayev & Tuğçe Ünver

Authors
  1. Amiran Gogatishvili
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  2. Rza Mustafayev
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  3. Tuğçe Ünver
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Correspondence to Amiran Gogatishvili.

Additional information

The research of A.Gogatishvili was partly supported by the grants GA13-14743S of the Grant Agency of the Czech Republic and RVO: 67985840, by Shota Rustaveli National Science Foundation grants no. 31/48 (Operators in some function spaces and their applications in Fourier Analysis) and no. DI/9/5-100/13 (Function spaces, weighted inequalities for integral operators and problems of summability of Fourier series). The research of all authors was partly supported by the joint project between the Czech Academy of Sciences and the Scientific and Technological Research Council of Turkey.

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Gogatishvili, A., Mustafayev, R. & Ünver, T. Embeddings between weighted Copson and Cesàro function spaces. Czech Math J 67, 1105–1132 (2017). https://doi.org/10.21136/CMJ.2017.0424-16

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