weyl型算子的加权估计及其紧性

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-02-27 DOI:10.1002/mma.10834

Akbota Abylayeva, Alisher Otegen

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{\\left(\\int_a&#x0005E;b{\\left&#x0007C;f(x)\\right&#x0007C;}&#x0005E;pw(x) dx\\right)}&#x0005E;{\\frac{1}{p}}&lt;\\infty . $$</annotation>\n </semantics></math>\n </div>\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 9","pages":"9676-9683"},"PeriodicalIF":2.1000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weighted Estimates of the Weyl-Type Operator and Its Compactness\",\"authors\":\"Akbota Abylayeva, Alisher Otegen\",\"doi\":\"10.1002/mma.10834\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n In this paper, the results of the necessity and sufficiency conditions of the fact that the operator \\n<math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation>$$ T $$</annotation>\\n </semantics></math> for the case \\n<math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>≤</mo>\\n <mi>q</mi>\\n </mrow>\\n <annotation>$$ \\\\boldsymbol{p}\\\\boldsymbol{\\\\le}\\\\boldsymbol{q} $$</annotation>\\n </semantics></math> is bounded and compact from the weighted Lebesgue space \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>,</mo>\\n <mi>w</mi>\\n </mrow>\\n </msub>\\n <mo>=</mo>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>,</mo>\\n <mi>w</mi>\\n </mrow>\\n </msub>\\n <mo>(</mo>\\n <mi>I</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$$ {L}_{p,w}&#x0003D;{L}_{p,w}(I) $$</annotation>\\n </semantics></math> to weighted Lebesgue space \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>q</mi>\\n <mo>,</mo>\\n <mi>v</mi>\\n </mrow>\\n </msub>\\n <mo>=</mo>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>q</mi>\\n <mo>,</mo>\\n <mi>v</mi>\\n </mrow>\\n </msub>\\n <mo>(</mo>\\n <mi>I</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$$ {L}_{q,v}&#x0003D;{L}_{q,v}(I) $$</annotation>\\n </semantics></math> are obtained, where \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>,</mo>\\n <mi>w</mi>\\n </mrow>\\n </msub>\\n <mo>=</mo>\\n <msub>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </msub>\\n <mo>(</mo>\\n <mi>w</mi>\\n <mo>,</mo>\\n <mi>I</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$$ {L}_{p,w}&#x0003D;{L}_p\\\\left(w,I\\\\right) $$</annotation>\\n </semantics></math> is the set of all measurable functions \\n<math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n </mrow>\\n <annotation>$$ f $$</annotation>\\n </semantics></math> in the interval \\n<math>\\n <semantics>\\n <mrow>\\n <mi>I</mi>\\n <mo>=</mo>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n <mo>)</mo>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mn>0</mn>\\n <mo>≤</mo>\\n <mi>a</mi>\\n <mo><</mo>\\n <mi>b</mi>\\n <mo>≤</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$$ I&#x0003D;\\\\left(a,b\\\\right),\\\\kern0.3em 0\\\\le a&lt;b\\\\le \\\\infty $$</annotation>\\n </semantics></math>, which the norm is finite: \\n\\n <div>\\n <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <mi>f</mi>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>,</mo>\\n <mi>w</mi>\\n </mrow>\\n </msub>\\n <mo>:</mo>\\n <mo>=</mo>\\n <msup>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <munderover>\\n <mrow>\\n <mo>∫</mo>\\n </mrow>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n </munderover>\\n <msup>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </msup>\\n <mi>w</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n <mi>d</mi>\\n <mi>x</mi>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <mrow>\\n <mfrac>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </mfrac>\\n </mrow>\\n </msup>\\n <mo><</mo>\\n <mi>∞</mi>\\n <mo>.</mo>\\n </mrow>\\n <annotation>$$ {&#x0007C;|f&#x0007C;|}_{p,w}:&#x0003D; 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引用次数: 0

摘要

在本文中，在加权勒贝格空间中，得到了p≤q $$ $ \boldsymbol{p}\boldsymbol{\le}\boldsymbol{q} $$是有界紧致的算子T $$ $ T $$的充要条件的结果L p w = L p，w (I) $$ {L}_{p,w}={L}_{p,w}(I) $$到加权勒贝格空间lq，v = lq,v (I) $$ {L}_{q,v}={L}_{q,v}(I) $$，其中L p， w = L p (w，I) $$ {L}_{p,w}={L}_p\left(w,I\right) $$是区间I = (a, b)内所有可测函数f $$ f $$ $的集合，0≤a &lt；b≤∞$$ I=\left(a,b\right),\kern0； 3em 0\le a<b\le \ inty $$，其范数是有限的：F p，w :=∫abF (x)pw (x) dx1 p &lt；∞ .$ $ {, # x0007C; |的# x0007C; |} _ {p w}:, # x0003D;{\离开(\ int_a& # x0005E; b {\ left& # x0007C; f (x) \ right& # x0007C;}, # x0005E; pw (x) dx \右)},# x0005E;{\压裂{1}{p}}, lt; \ infty。$ $

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Weighted Estimates of the Weyl-Type Operator and Its Compactness

In this paper, the results of the necessity and sufficiency conditions of the fact that the operator $T$ for the case $p \leq q$ is bounded and compact from the weighted Lebesgue space $L_{p, w} = L_{p, w} (I)$ to weighted Lebesgue space $L_{q, v} = L_{q, v} (I)$ are obtained, where $L_{p, w} = L_{p} (w, I)$ is the set of all measurable functions $f$ in the interval $I = (a, b), 0 \leq a < b \leq \infty$ , which the norm is finite:

{(f)}_{p, w} : = {(\int_{a}^{b} {(f (x))}^{p} w (x) d x)}^{\frac{1}{p}} < \infty .

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.