Mathematical Researches

fa مطالعه کرانداری عملگر لیتلوود-پالی ${B, lambda}_*^g$ وابسته به عملگر دیفرانسیل بسل On the Boundedness of Littlewood-Paley $ g^{*}_{B, lambda}$ operator associated with Bessel differential operator آنالیز Anal مقاله مستقل Original Manuscript عملگرهای لیتلوود-پلی، بدلیل کاربردهایشان در معادلات دیفرانسیل با مشتقات جزئی و دیگر شاخههای ریاضی، نقش مهمی در آنالیز فوریه دارند. از جمله عملگرهای لیتلوود-پلی، عملگر <m:omath><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>g</m:r></m:e><m:sub><m:r>λ</m:r></m:sub><m:sup><m:r>*</m:r></m:sup></m:ssubsup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:11.4pt; height:15pt" >  می‌باشد. به هنگام مطالعه کرانداری این عملگر در حالت کلاسیک، وابسته بودن کرانداری از نوع قوی این عملگر در فضای <m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>1<</m:r><m:r>p</m:r><m:r><∞</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>L</m:r></m:e><m:sup><m:r>p</m:r></m:sup></m:ssup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHoAAAAZCAIAAABxW61HAAAAAXNSR0IArs4c6QAAAAlwSFlzAAASdAAAEnQB3mYfeAAAAyBJREFUaEPtWT12ozAQlvcsdoo8TgAnsNOnTYfLpEmXG7iB0unSukqzcALnBH5bBO7inRmBECD0g2ycfYsqDBrN6JuZb0by4nw+s3lMhcCvqRTNehCBGe5J42CG+5Zwl3m6jaI0n9SGn6WsTKNom+bldayCUlmPIglZmGSF9Oo/fQQkwjDOxu2+SOKQAZBFkeEDPdYrsTbWaqiLDFaQhMaZ8W9J8djTh14WixQQvikSCFgSDmMSx0niaw1366Uc8ZWHJBF32NDNXPc1x6W12ACOc1rhyzcIaApXKeAmqX7ywOsYyWXgsxk7Sgswh1a52riWFgu81cjIaOOMBlmKbhOcpu99ICveQgI04FwTHJkk2M6ON+210NpAyJT8wIu1VZxn6aUi/cx4Y+j2KKcRg/20vxPcSqFODVUEvzJcETH7gIbtIslQwiVJzJ1DlKgH3E1LFVECZaKdjLDgmQd+U+b3EMXWO1dayglG7cTLwV2lNNgP5rsRR4cB6edAmRqpRU5vsq3CRHZqb44xDgepZDBW8JhTfp8YC+6WXp1mvtu8nAKIzuN+vXRcqvjzBfB+PJvFPLQoNhhn+3Wz6dV9E5VWUJS/D2D3/ao9Of98Z++bKFU27gj38i5g7PTt19ivX7MkOG1Wi2ibl25LobvDxwcBNqI/4P6xWgDK1gbz3ctX2H7FSK3LoCiR7Oay6z1mz3Egdm7L3apCbUHdVFPtK0SHuwVZYefVFE25g6jIUMvd7g0EXL7eujPpEaSpbsslnHcyFv0P36WiC0G38aHoTLSm9NGmAqyvWvyYY3CUrnZpl7fpiHnR4lHGGwTTYa6r0kaLW+3ms/V9YMcXdrGuP1ViQLTqhzIIjFvRnvewIYilwDM16oPapjxVAjAKljfHidWdiRFPjwmm1tZjaS9R8xFnzPLivnv5fCweDyu4fHVrK1xKuWIudSXdXspzTX9xuIR9OgTZUHsxXoH89wIi/sY+n6a873buvsZv1VIS77t37O0Dzg+WEi7TxqTE5WT4Fab5ZuVyGm+70gLUu3hnnuuFwPxfpRd8rsIz3K6Iec3/C7DbxVcBIH2IAAAAAElFTkSuQmCC" style="width:73.2pt; height:15pt" > ، به پارامتر <m:omath><m:r>λ</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAZCAIAAACHEijfAAAAAXNSR0IArs4c6QAAAAlwSFlzAAASdAAAEnQB3mYfeAAAAKhJREFUOE9j/P//PwNuwIRHDihFL+k7E9OtGYHAeuIdhIOALgeDbRPSJtwG0WkMVmAGGDDAGDD69gQrhrRtMB6Gy1XUdJD8iiF959YV5KBANRxkNAOS4Sh2QySRnYZk+J2J8QXH0oA+wGY3VHKmKgOy3VDdMElPtBgCS29PVwUZOxNdEigFDick50CD5fa2NFDgMKBKgkMV6HwrSAj/ZxwSqQVbmiWQUgFgOasDWv3t8QAAAABJRU5ErkJggg==" style="width:6pt; height:15pt" >  مشاهده می‌شود. ما عملگر <m:omath><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>g</m:r></m:e><m:sub><m:r>λ</m:r></m:sub><m:sup><m:r>*</m:r></m:sup></m:ssubsup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:11.4pt; height:15pt" >  وابسته به عملگر بسل را با <m:omath><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>g</m:r></m:e><m:sub><m:r>B</m:r><m:r>,</m:r><m:r>λ</m:r></m:sub><m:sup><m:r>*</m:r></m:sup></m:ssubsup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" >  نشان می‌دهیم و <m:omath><m:r>-</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>L</m:r></m:e><m:sub><m:r>p</m:r><m:r>,</m:r><m:r>v</m:r></m:sub></m:ssub></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:25.8pt; height:16.2pt" > کرانداری عملگر <m:omath><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>g</m:r></m:e><m:sub><m:r>B</m:r><m:r>,</m:r><m:r>λ</m:r></m:sub><m:sup><m:r>*</m:r></m:sup></m:ssubsup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:18.6pt; height:16.2pt" >  برای <m:omath><m:r>2≤</m:r><m:r>p</m:r><m:r>≤∞</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:50.4pt; height:15pt" >   به ازای <m:omath><m:r>λ</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>></m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:num><m:den><m:r>p</m:r></m:den></m:f><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>4</m:r><m:r>v</m:r></m:num><m:den><m:r>np</m:r></m:den></m:f><m:r>  </m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:53.4pt; height:22.2pt" >  و   نیز بی‌کرانی عملگر <m:omath><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>g</m:r></m:e><m:sub><m:r>B</m:r><m:r>,</m:r><m:r>λ</m:r></m:sub><m:sup><m:r>*</m:r></m:sup></m:ssubsup></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:18.6pt; height:16.2pt" > ، به‌ازای <m:omath><m:r>λ</m:r><m:r><</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>2</m:r></m:num><m:den><m:r>p</m:r></m:den></m:f><m:r>+</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>4</m:r><m:r>v</m:r></m:num><m:den><m:r>pn</m:r></m:den></m:f></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:48.6pt; height:22.2pt" >  را در فضای <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>L</m:r></m:e><m:sub><m:r>p</m:r><m:r>,</m:r><m:r>v</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>R</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>+</m:r></m:sub><m:sup><m:r>n</m:r></m:sup></m:ssubsup></m:e></m:d></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:41.4pt; height:16.2pt" >   به‌دست میآوریم. <p<infty$><p<infty< span=""><infty< span="">.</infty<></p<infty<></p<infty$> The study of classical Littlewood-Paley operators has an intrinsic interest for their essential role in harmonic analysis due to their applications in PDEs and other fields. One of the Littlewood-Paley operators is <m:ctrlpr></m:ctrlpr> <m:r>g</m:r> <m:r>λ</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r>  operator which its <m:ctrlpr></m:ctrlpr> <m:r>p</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>p</m:r>  strong boundedness depends on the parameter <m:r>λ</m:r> . For example, Fefferman showed strong boundedness of classical <m:ctrlpr></m:ctrlpr> <m:r>g</m:r> <m:r>λ</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r>  for <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1<</m:r><m:r>p</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr><∞</m:r>  in <m:ctrlpr></m:ctrlpr> <m:r>L</m:r> <m:r>p</m:r>   when <m:r>λ</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>></m:r><m:r>max</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1,</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r> <m:r>p</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>.</m:r>  In this work, We consider the Laplace-Bessel differential operator and correspondingly we define the relevant Littlewood-Paley operator <m:ctrlpr></m:ctrlpr> <m:r>g</m:r> <m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>λ</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r>  to investigate both <m:ctrlpr></m:ctrlpr> <m:r>L</m:r> <m:r>p</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>ν</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>-</m:r> boundedness of <m:ctrlpr></m:ctrlpr> <m:r>g</m:r> <m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>λ</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r>  for <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2≤</m:r><m:r>P</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr><∞</m:r>  and <m:r>λ</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>>1+</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r><m:r>v</m:r> <m:r>n</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r> and its unboundedness for <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>0<</m:r><m:r>λ</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr><</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r> <m:r>P</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4ν</m:r> <m:r>pn</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r>in</m:r> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r>L</m:r> <m:r>p</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>ν</m:r> <m:ctrlpr></m:ctrlpr> <m:ctrlpr></m:ctrlpr> <m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>R</m:r> <m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r> <m:r>n</m:r> <m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>. </m:r> <p<infty$ in=""></p<infty$> عملگرهای لیتلوود-پالی ${ lambda}_{*}^g$, کرانداری از نوع قوی, عملگر ${B, lambda}_{*}^g$ متناطر با عملگر بسل, بی‌کرانی عملگر ${B, lambda}_{*}^g$ . Littlewood-Paley operators $g^{*}_{lambda}$ , strong boundedness, Littlewood-Paley operators $g^{*}_{B, lambda}$ associated with Bessel differential operator, unboundedness of $g^{*}_{B, lambda}$ 1 17 http://mmr.khu.ac.ir/browse.php?a_code=A-10-326-2&slc_lang=fa&sid=1 Arash Ghorbanalizadeh آرش قربانعلی زاده gurbanalizade@gmail.com 10031947532846006509 10031947532846006509 Yes IASBS دانشگاه تحصیلات تکمیلی علوم پایه زنجان Monire Mikaeili nia منیره مکائیلی نیا moniremikaeili@iasbs.ac.ir 10031947532846006510 10031947532846006510 No IASBS دانشگاه تحصیلات تکمیلی علوم پایه زنجان