| Summary: | In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> bounds of these Maximal operators under the assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>∈</mo><msup><mi>L</mi><mi>q</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and then we employ these bounds along with Yano’s extrapolation argument to obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> boundedness of the aforementioned integral operators under a weaker condition in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> belongs to either the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mi>q</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mn>2</mn><msup><mi>τ</mi><mo>′</mo></msup></mfrac><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> or to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msup><mrow><mo>(</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>L</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>/</mo><msup><mi>τ</mi><mo>′</mo></msup></mrow></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and improve many previously known results.
|
|---|