{"title":"Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces","authors":"Munirah Aljuaid, M. A. Bakhit","doi":"10.1155/2024/5581805","DOIUrl":null,"url":null,"abstract":"This article examines the norms of composition operators from the weighted harmonic Bloch space <span><svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 23.549 16.0921\" width=\"23.549pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,-5.741)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,3.784)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,20.585,0)\"></path></g></svg><span></span><svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"25.678183800000003 -12.2582 22.001 16.0921\" width=\"22.001pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.728,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,30.226,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,40.098,0)\"></path></g></svg><span></span><svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"51.3101838 -12.2582 18.437 16.0921\" width=\"18.437pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,51.36,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,62.166,0)\"><use xlink:href=\"#g117-91\"></use></g></svg><span></span><svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"73.3791838 -12.2582 18.108 16.0921\" width=\"18.108pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,73.429,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,86.689,0)\"></path></g></svg></span> to the weighted harmonic Zygmund space <span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"-0.0498162 -13.4156 23.315 17.5066\" width=\"23.315pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,-6.899)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,4.041)\"><use xlink:href=\"#g50-73\"></use></g><g transform=\"matrix(.013,0,0,-0.013,20.351,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"25.4441838 -13.4156 22.001 17.5066\" width=\"22.001pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.494,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.992,0)\"><use xlink:href=\"#g113-49\"></use></g><g transform=\"matrix(.013,0,0,-0.013,39.864,0)\"><use xlink:href=\"#g117-91\"></use></g></svg><span></span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"51.0771838 -13.4156 18.817 17.5066\" width=\"18.817pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,51.127,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,62.313,0)\"><use xlink:href=\"#g117-91\"></use></g></svg><span></span><span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"73.5261838 -13.4156 18.105 17.5066\" width=\"18.105pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,73.576,0)\"><use xlink:href=\"#g113-30\"></use></g><g transform=\"matrix(.013,0,0,-0.013,86.836,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>.</span></span> The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between <svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 20.7306 16.0921\" width=\"20.7306pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-3\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,-5.741)\"><use xlink:href=\"#g50-233\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,3.784)\"><use xlink:href=\"#g50-73\"></use></g></svg> and <svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"-0.0498162 -13.4156 20.4958 17.5066\" width=\"20.4958pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-27\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,-6.899)\"><use xlink:href=\"#g50-224\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,4.041)\"><use xlink:href=\"#g50-73\"></use></g></svg> is bounded. Secondly, we will study the compactness case of the composition operator between <svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 20.7306 16.0921\" width=\"20.7306pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-3\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,-5.741)\"><use xlink:href=\"#g50-233\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,3.784)\"><use xlink:href=\"#g50-73\"></use></g></svg> and <span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"-0.0498162 -13.4156 20.4958 17.5066\" width=\"20.4958pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-27\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,-6.899)\"><use xlink:href=\"#g50-224\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,4.041)\"><use xlink:href=\"#g50-73\"></use></g></svg>.</span> Finally, we will estimate the essential norms of the composition operator between <svg height=\"16.0921pt\" style=\"vertical-align:-3.8339pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 20.7306 16.0921\" width=\"20.7306pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-3\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,-5.741)\"><use xlink:href=\"#g50-233\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.35,3.784)\"><use xlink:href=\"#g50-73\"></use></g></svg> and <span><svg height=\"17.5066pt\" style=\"vertical-align:-4.091pt\" version=\"1.1\" viewbox=\"-0.0498162 -13.4156 20.4958 17.5066\" width=\"20.4958pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-27\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,-6.899)\"><use xlink:href=\"#g50-224\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,12.116,4.041)\"><use xlink:href=\"#g50-73\"></use></g></svg>.</span>","PeriodicalId":15840,"journal":{"name":"Journal of Function Spaces","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Function Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/5581805","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This article examines the norms of composition operators from the weighted harmonic Bloch space to the weighted harmonic Zygmund space . The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between and is bounded. Secondly, we will study the compactness case of the composition operator between and . Finally, we will estimate the essential norms of the composition operator between and .
期刊介绍：
Journal of Function Spaces (formerly titled Journal of Function Spaces and Applications) publishes papers on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. As well as original research, Journal of Function Spaces also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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