| Title:
|
A weighted inequality for the Hardy operator involving suprema (English) |
| Author:
|
Hofmanová, Pavla |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
317-326 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $u$ be a weight on $(0, \infty)$. Assume that $u$ is continuous on $(0, \infty)$. Let the operator $S_{u}$ be given at measurable non-negative function $\varphi$ on $(0, \infty)$ by $$ S_{u}\varphi (t)= \sup_{0< \tau\leq t}u(\tau)\varphi (\tau). $$ We characterize weights $v,w$ on $(0, \infty)$ for which there exists a positive constant $C$ such that the inequality $$ \left( \int_{0}^{\infty}[S_{u}\varphi (t)]^{q}w(t)\,dt\right)^{\frac 1q} \lesssim \left( \int_{0}^{\infty}[\varphi (t)]^{p}v(t)\,dt\right)^{\frac 1p} $$ holds for every $0<p, q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces. (English) |
| Keyword:
|
Hardy operators involving suprema |
| Keyword:
|
weighted inequalities |
| MSC:
|
26D15 |
| MSC:
|
47G10 |
| idZBL:
|
Zbl 06674882 |
| idMR:
|
MR3554512 |
| DOI:
|
10.14712/1213-7243.2015.167 |
| . |
| Date available:
|
2016-09-22T15:23:42Z |
| Last updated:
|
2018-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145836 |
| . |
| Reference:
|
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| . |
