Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w
The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces E and in particular on Lorentz spaces Γp,w={f:∫(f**)pw<∞} for any 0<p<∞ and a nonnegative locally integrable weight function w, where f** is...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/682960 |
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| author | Maciej Ciesielski Anna Kamińska |
| author_facet | Maciej Ciesielski Anna Kamińska |
| author_sort | Maciej Ciesielski |
| collection | DOAJ |
| description | The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces E and in particular on Lorentz spaces Γp,w={f:∫(f**)pw<∞} for any 0<p<∞ and a nonnegative locally integrable weight function w, where f** is a maximal function of the decreasing rearrangement f* for any measurable function f on (0,α), with 0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of f∈E, where E is an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximants fϵ of f∈Γp,w or f∈Γp-1,w, 1<p<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function f∈Γp-1,w, 1<p<∞. |
| format | Article |
| id | doaj-art-49fd8f9e3f914c31bcfeed12ae84b433 |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-49fd8f9e3f914c31bcfeed12ae84b4332025-02-03T05:50:48ZengWileyJournal of Function Spaces and Applications0972-68021758-49652012-01-01201210.1155/2012/682960682960Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,wMaciej Ciesielski0Anna Kamińska1Institute of Mathematics, Poznań University of Technology, Piotrowo 3a, 60-965 Poznań, PolandDepartment of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USAThe paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces E and in particular on Lorentz spaces Γp,w={f:∫(f**)pw<∞} for any 0<p<∞ and a nonnegative locally integrable weight function w, where f** is a maximal function of the decreasing rearrangement f* for any measurable function f on (0,α), with 0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of f∈E, where E is an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximants fϵ of f∈Γp,w or f∈Γp-1,w, 1<p<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function f∈Γp-1,w, 1<p<∞.http://dx.doi.org/10.1155/2012/682960 |
| spellingShingle | Maciej Ciesielski Anna Kamińska Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w Journal of Function Spaces and Applications |
| title | Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w |
| title_full | Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w |
| title_fullStr | Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w |
| title_full_unstemmed | Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w |
| title_short | Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w |
| title_sort | lebesgue s differentiation theorems in r i quasi banach spaces and lorentz spaces γp w |
| url | http://dx.doi.org/10.1155/2012/682960 |
| work_keys_str_mv | AT maciejciesielski lebesguesdifferentiationtheoremsinriquasibanachspacesandlorentzspacesgpw AT annakaminska lebesguesdifferentiationtheoremsinriquasibanachspacesandlorentzspacesgpw |