%0 Journal Article
%T Weighted Estimates for Marcinkiewicz Integral Operator Associated with Schrödinger Operator and Its Commutator on Generalized Fractional Morrey Spaces
%A Yiyuan Wang
%J Applied Mathematics
%P 470-481
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.166026
%X The aim of this paper is to investigate the boundedness of a Marcinkiewicz integral operator <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x03BC;</mi>
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</math>

 associated with Schr&#246;dinger operator and its commutator on weighted generalized fractional Morrey spaces <math xmlns='http://www.w3.org/1998/Math/MathML'>
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  </msup>
  <mrow><mo>(</mo>
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    <msup>
     <mi>&#x211D;</mi>
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    <mo>,</mo><mi>&#x03C9;</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

. Under assumption that the function <math xmlns='http://www.w3.org/1998/Math/MathML'>
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 satisfies doubling conditions, the author proves that the <math xmlns='http://www.w3.org/1998/Math/MathML'>
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 is bounded on spaces <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>L</mi>
   <mrow>
    <mi>p</mi><mo>,</mo><mi>&#x03B7;</mi><mo>,</mo><mi>&#x03C6;</mi></mrow>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>&#x211D;</mi>
     <mi>n</mi>
    </msup>
    <mo>,</mo><mi>&#x03C9;</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

, where <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>0</mn><mo>&#x003C;</mo><mi>&#x03B7;</mi><mo>&#x003C;</mo><mfrac>
   <mi>n</mi>
   <mi>m</mi>
  </mfrac>
  </mrow>
</math>

, <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mfrac>
   <mi>n</mi>
   <mrow>
    <mi>m</mi><mi>&#x03B7;</mi></mrow>
  </mfrac>
  <mo>&#x003C;</mo><mi>p</mi><mo>&#x003C;</mo><mfrac>
   <mi>n</mi>
   <mi>&#x03B7;</mi>
  </mfrac>
  </mrow>
</math>

, and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C9;</mi><mo>&#x2208;</mo><msubsup>
   <mi>A</mi>
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    <mi>&#x03C1;</mi><mo>,</mo><mi>&#x03B8;</mi></mrow>
  </msubsup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>&#x211D;</mi>
     <mi>n</mi>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

. Furthermore, the boundedness of the commutator <math xmlns='http://www.w3.org/1998/Math/MathML'>
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   <mi>b</mi><mo>,</mo><msubsup>
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   </mrow> <mo>]</mo></mrow></mrow>
</math>

 formed by <math xmlns='http://www.w3.org/1998/Math/MathML'>
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%K Marcinkiewicz Integral
%K Schr&#246
%K dinger Operator
%K Generalize Fractional Morrey Space
%K Weighted Boundedness
%K Commutator
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=143492