New characterization of weighted inequalities involving superposition of Hardy integral operators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917698200403968 |
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| author | Gogatishvili, Amiran Ünver, Tuğçe |
| author_facet | Gogatishvili, Amiran Ünver, Tuğçe |
| contents | Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize the validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \bigg(\int_a^b f(x)^p v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions $f$ on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we provide some new scales of weight characterizations of this inequality in both discrete and continuous forms and we obtain previous characterizations as the special case of the parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New characterization of weighted inequalities involving superposition of Hardy integral operators Gogatishvili, Amiran Ünver, Tuğçe Analysis of PDEs Classical Analysis and ODEs Functional Analysis 26D10, 26D15 Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize the validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \bigg(\int_a^b f(x)^p v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions $f$ on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we provide some new scales of weight characterizations of this inequality in both discrete and continuous forms and we obtain previous characterizations as the special case of the parameter. |
| title | New characterization of weighted inequalities involving superposition of Hardy integral operators |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 26D10, 26D15 |
| url | https://arxiv.org/abs/2406.11298 |