Two-Weight, Weak-Type Norm Inequalities for Fractional Integral Operators and Commutators on Weighted Morrey and Amalgam Spaces
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Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>γ</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> be the fractional integral operator of order γ , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mstyle><mml:mrow><mml:msub><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:msup><mml:mrow><mml:mfenced open="|" close="|" separators="|"><mml:mrow><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>γ</mml:mi><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mtext>d</mml:mtext><mml:mi>y</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:math> and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mfenced open="[" close="]" separators="|"><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> be the linear commutator generated by a symbol function b and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mfenced open="[" close="]" separators="|"><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>b</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>b</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:math>. This paper is concerned with two-weight, weak-type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.
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