Two-Weight, Weak-Type Norm Inequalities for a Class of Sublinear Operators 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:msub><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfenced></mml:math> be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:mfenced></mml:math> be the commutators generated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mtext>BMO</mml:mtext><mml:mfenced open="(" close=")"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math> functions and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math>. This paper is concerned with two-weight, weak-type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criteria for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our main results, we can obtain the weak-type norm inequalities for several integral operators as well as the corresponding commutators in the framework of weighted Morrey and amalgam spaces.
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