Estimates of maximal functions measuring local smoothness
Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(In) (I≡[0,1]. Set, where the supremum is taken over all cubes containing the pointx. Forη=tα (0<α≤1) this definition was given by A.Calderón.In the paper we prove estimates of the maximal functions, along wit...
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| Published in | Analysis mathematica (Budapest) Vol. 25; no. 1; pp. 277 - 300 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Heidelberg
Springer Nature B.V
1999
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| Abstract | Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(In) (I≡[0,1]. Set, where the supremum is taken over all cubes containing the pointx. Forη=tα (0<α≤1) this definition was given by A.Calderón.In the paper we prove estimates of the maximal functions, along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if, then. Furthermore, we obtain estimates of in terms of theLp-modulus of continuity off. We find sharp conditions for to belong toLp(In) and the Orlicz classϕ(L), too.Пустьη-неубываюшая функция на (0,1] такая, чтоη(t)/t убывает иη(+0)=0. Пустьf ∈L(In) (I≡[0,1]). Положим где верхняя грань берется по всем кубам, содержашим точкуx. Дляη=tα (0<α<1) Это определение было дано А. Кальдероном.В статье иэучаются оценки максимальных функций, а также некоторые теоремы вложения. Докаэываются неравенства типа Соболева: если, то Далее, получены оценки в терминахLp-модуля непрерывностиf. Найдены точные условия для принадлежности пространствамLq(In) и классамϕ(L). |
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| AbstractList | Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(In) (I≡[0,1]. Set, where the supremum is taken over all cubes containing the pointx. Forη=tα (0<α≤1) this definition was given by A.Calderón.In the paper we prove estimates of the maximal functions, along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if, then. Furthermore, we obtain estimates of in terms of theLp-modulus of continuity off. We find sharp conditions for to belong toLp(In) and the Orlicz classϕ(L), too.Пустьη-неубываюшая функция на (0,1] такая, чтоη(t)/t убывает иη(+0)=0. Пустьf ∈L(In) (I≡[0,1]). Положим где верхняя грань берется по всем кубам, содержашим точкуx. Дляη=tα (0<α<1) Это определение было дано А. Кальдероном.В статье иэучаются оценки максимальных функций, а также некоторые теоремы вложения. Докаэываются неравенства типа Соболева: если, то Далее, получены оценки в терминахLp-модуля непрерывностиf. Найдены точные условия для принадлежности пространствамLq(In) и классамϕ(L). |
| Author | Коляда, В. И. Kolyada, V. I. |
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| Cites_doi | 10.4064/sm-44-6-563-582 10.4064/sm-62-1-75-92 10.1007/978-3-642-65711-5 10.1090/memo/0293 |
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| Copyright | Akadémiai Kiadó 1999. |
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| References | S. M. Nikol’skii (BF02908442_CR9) 1975 V. I. Kolyada (BF02908442_CR6) 1987; 293 V. I. Kolyada (BF02908442_CR8) 1988; 181 BF02908442_CR4 V. I. Kolyada (BF02908442_CR7) 1988; 1936 A. P. Calderón (BF02908442_CR2) 1972; 44 A. P. Calderón (BF02908442_CR3) 1978; 62 P. L. Ul’yanov (BF02908442_CR12) 1970; 81 V. I. Kolyada (BF02908442_CR5) 1975; 39 E. M. Stein (BF02908442_CR11) 1970 C. Bennett (BF02908442_CR1) 1988 K. I. Oskolkov (BF02908442_CR10) 1977; 103 |
| References_xml | – volume: 44 start-page: 167 year: 1972 ident: BF02908442_CR2 publication-title: Studia Math. doi: 10.4064/sm-44-6-563-582 contributor: fullname: A. P. Calderón – volume: 62 start-page: 75 year: 1978 ident: BF02908442_CR3 publication-title: Studia Math. doi: 10.4064/sm-62-1-75-92 contributor: fullname: A. P. Calderón – volume: 293 start-page: 534 year: 1987 ident: BF02908442_CR6 publication-title: Dokl. Akad. Nauk SSSR contributor: fullname: V. I. Kolyada – volume: 1936 start-page: 3 year: 1988 ident: BF02908442_CR7 publication-title: Mat. Sbornik contributor: fullname: V. I. Kolyada – volume: 181 start-page: 117 year: 1988 ident: BF02908442_CR8 publication-title: Trudy Mat. Inst. Steklov contributor: fullname: V. I. Kolyada – volume-title: Approximation of functions of several variables and embedding theorems year: 1975 ident: BF02908442_CR9 doi: 10.1007/978-3-642-65711-5 contributor: fullname: S. M. Nikol’skii – volume-title: Interpolation of operators year: 1988 ident: BF02908442_CR1 contributor: fullname: C. Bennett – volume-title: Singular integrals and differentiability property of functions year: 1970 ident: BF02908442_CR11 contributor: fullname: E. M. Stein – volume: 103 start-page: 563 year: 1977 ident: BF02908442_CR10 publication-title: Mat. Sbornik contributor: fullname: K. I. Oskolkov – ident: BF02908442_CR4 doi: 10.1090/memo/0293 – volume: 39 start-page: 418 year: 1975 ident: BF02908442_CR5 publication-title: Izv. Akad. Nauk SSSR, Ser. Mat. contributor: fullname: V. I. Kolyada – volume: 81 start-page: 104 year: 1970 ident: BF02908442_CR12 publication-title: Mat. Sbornik contributor: fullname: P. L. Ul’yanov |
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| Snippet | Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(In) (I≡[0,1]. Set, where the supremum is taken over all cubes... |
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| Title | Estimates of maximal functions measuring local smoothness |
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