On Rough Parametric Marcinkiewicz Integrals Along Certain Surfaces
In this paper, we study rough Marcinkiewicz integrals associated with surfaces defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Ψ</mi&g...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/8/1287 |
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| Summary: | In this paper, we study rough Marcinkiewicz integrals associated with surfaces defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Ψ</mi><mrow><mi mathvariant="script">P</mi><mo>,</mo><mi>ϕ</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mmultiscripts><mo>(</mo><none></none><none></none><mprescripts></mprescripts><none></none><mo stretchy="false">˜</mo></mmultiscripts><mi mathvariant="script">P</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>,</mo><mi>ϕ</mi><mo>(</mo><mfenced open="|" close="|"><mi>w</mi></mfenced><mo>)</mo><mo>)</mo></mrow><mo>:</mo><mi>w</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></mrow></semantics></math></inline-formula>}. We establish 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 these integrals when the kernel functions lie in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><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></mrow></mrow></semantics></math></inline-formula> space. Combining this result with Yano’s extrapolation technique, we further 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 under weaker kernel conditions—specifically, when the kernels belong to either the block 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><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</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></mrow></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msup><mrow><mo>(</mo><mo form="prefix">log</mo><mi>L</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and refine several previously known results on Marcinkiewicz integrals, offering broader applicability and sharper conclusions. |
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| ISSN: | 2227-7390 |