| Summary: | In this paper, we introduce mixed-norm amalgam spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>L</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover></msup><mo>,</mo><msup><mi>L</mi><mover accent="true"><mi>s</mi><mo>→</mo></mover></msup><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mrow><mo>(</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>γ</mi></msub><mo>]</mo></mrow></semantics></math></inline-formula> generated by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi><mi>M</mi><mi>O</mi><mo>(</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>I</mi><mi>γ</mi></msub></semantics></math></inline-formula> on mixed-norm amalgam spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>L</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover></msup><mo>,</mo><msup><mi>L</mi><mover accent="true"><mi>s</mi><mo>→</mo></mover></msup><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mrow><mo>(</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><msup><mi>L</mi><mover accent="true"><mi>p</mi><mo>→</mo></mover></msup><mo>,</mo><msup><mi>L</mi><mover accent="true"><mi>s</mi><mo>→</mo></mover></msup><mo stretchy="false">)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msubsup><mi>t</mi><mrow><mi>r</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.
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