Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces
Guanghui Lu, Miaomiao Wang, Shuangping Tao
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{"title":"Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces","authors":"Guanghui Lu, Miaomiao Wang, Shuangping Tao","doi":"10.1515/agms-2023-0101","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant=\"script\">X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left({\\mathcal{X}},d,\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{p}^{u}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>1\\le p\\lt \\infty </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mi mathvariant=\"script\">X</m:mi> <m:mo>×</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>u\\left(x,r):{\\mathcal{X}}\\times \\left(0,\\infty )\\to \\left(0,\\infty )</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> <jats:tex-math>{u}_{1},{u}_{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>u</m:mi> </m:math> <jats:tex-math>u</jats:tex-math> </jats:alternatives> </jats:inline-formula> belong to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">W</m:mi> </m:mrow> <m:mrow> <m:mi>τ</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{\\mathbb{W}}}_{\\tau }</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>τ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\tau \\in \\left(0,2)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove that the bilinear <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>θ</m:mi> </m:math> <jats:tex-math>\\theta </jats:tex-math> </jats:alternatives> </jats:inline-formula>-type generalized fractional integral <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_010.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mo stretchy=\"true\">˜</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{\\widetilde{T}}_{\\theta ,\\alpha }</jats:tex-math> </jats:alternatives> </jats:inline-formula> is bounded from the product of spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_011.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{{p}_{1}}^{{u}_{1}}\\left(\\mu )\\times {M}_{{p}_{2}}^{{u}_{2}}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> into spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_012.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{q}^{u}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_013.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>u</m:mi> </m:math> <jats:tex-math>{u}_{1}{u}_{2}=u</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_014.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\alpha \\in \\left(0,1)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_015.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>α</m:mi> </m:math> <jats:tex-math>\\frac{1}{q}=\\frac{1}{{p}_{1}}+\\frac{1}{{p}_{2}}-2\\alpha </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_016.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{p}_{1},{p}_{2}\\in \\left(1,\\frac{1}{\\alpha })</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and also show that the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_017.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mo stretchy=\"true\">˜</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{\\widetilde{T}}_{\\theta ,\\alpha }</jats:tex-math> </jats:alternatives> </jats:inline-formula> is bounded from the product of spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_018.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{{p}_{1}}^{{u}_{1}}\\left(\\mu )\\times {M}_{{p}_{2}}^{{u}_{2}}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> into spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_019.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{1}^{u}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_020.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>1</m:mn> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>α</m:mi> </m:math> <jats:tex-math>1=\\frac{1}{{p}_{1}}+\\frac{1}{{p}_{2}}-2\\alpha </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Meanwhile, we prove that the commutator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_021.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mo stretchy=\"true\">˜</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:math> <jats:tex-math>{\\widetilde{T}}_{\\theta ,\\alpha ,{b}_{1},{b}_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> formed by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_022.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mover accent=\"true\"> <m:mrow> <m:mi mathvariant=\"normal\">RBMO</m:mi> </m:mrow> <m:mrow> <m:mo stretchy=\"true\">˜</m:mo> </m:mrow> </m:mover> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{b}_{1},{b}_{2}\\in \\widetilde{{\\rm{RBMO}}}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_023.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mo stretchy=\"true\">˜</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{\\widetilde{T}}_{\\theta ,\\alpha }</jats:tex-math> </jats:alternatives> </jats:inline-formula> is bounded from the product of spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_024.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{{p}_{1}}^{{u}_{1}}\\left(\\mu )\\times {M}_{{p}_{2}}^{{u}_{2}}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> into spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_025.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{q}^{u}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and it is also bounded from the product of spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_026.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{{p}_{1}}^{{u}_{1}}\\left(\\mu )\\times {M}_{{p}_{2}}^{{u}_{2}}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula> into spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_027.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{M}_{1}^{u}\\left(\\mu )</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"9 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Geometry in Metric Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2023-0101","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let ( X , d , μ ) \left({\mathcal{X}},d,\mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space M p u ( μ ) {M}_{p}^{u}\left(\mu ) , where 1 ≤ p < ∞ 1\le p\lt \infty and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) u\left(x,r):{\mathcal{X}}\times \left(0,\infty )\to \left(0,\infty ) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions u 1 , u 2 {u}_{1},{u}_{2} , and u u belong to W τ {{\mathbb{W}}}_{\tau } with τ ∈ ( 0 , 2 ) \tau \in \left(0,2) , we prove that the bilinear θ \theta -type generalized fractional integral T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , where u 1 u 2 = u {u}_{1}{u}_{2}=u , α ∈ ( 0 , 1 ) \alpha \in \left(0,1) , and 1 q = 1 p 1 + 1 p 2 − 2 α \frac{1}{q}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha with p 1 , p 2 ∈ ( 1 , 1 α ) {p}_{1},{p}_{2}\in \left(1,\frac{1}{\alpha }) , and also show that the T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) , where 1 = 1 p 1 + 1 p 2 − 2 α 1=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha . Meanwhile, we prove that the commutator T ˜ θ , α , b 1 , b 2 {\widetilde{T}}_{\theta ,\alpha ,{b}_{1},{b}_{2}} formed by b 1 , b 2 ∈ RBMO ˜ ( μ ) {b}_{1},{b}_{2}\in \widetilde{{\rm{RBMO}}}\left(\mu ) and T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , and it is also bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) .
新非齐次广义Morrey空间上双线性θ型广义分数积分及其对易子的估计
令(X, d, μ) \left({\mathcal{X}},d;\mu )是满足几何加倍和上加倍条件的非齐次度量测度空间。在这种情况下,我们首先引入广义Morrey空间M p u (μ) {m}_{p}^{你}\left(\mu ),其中1≤p &lt;∞1\le p\lt \infty u (x, r): x ×(0,∞)→(0,∞)u\left(x,r):{\mathcal{X}}\times \left(0;\infty )\to \left(0;\infty )是勒贝格可测函数。进一步,假设可测函数u1, u2 {你}_{1},{你}_{2} , u u属于W τ {{\mathbb{W}}}_{\tau } τ∈(0,2) \tau \in \left(0,2),我们证明双线性的θ \theta 型广义分数积分T ~ θ, α {\widetilde{T}}_{\theta ,\alpha } 从空间mp1u1 (μ) × mp2u2 (μ)的乘积有界 {m}_{{p}_{1}}^{{你}_{1}}\left(\mu )\times {m}_{{p}_{2}}^{{你}_{2}}\left(\mu )化成空间M q u (μ) {m}_{q}^{你}\left(\mu ),其中u 1 u 2 = u {你}_{1}{你}_{2}=u, α∈(0,1) \alpha \in \left(0,1)和1q = 1p1 + 1p2−2 α \frac{1}{q}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha 与p1, p2∈(1,1 α) {p}_{1},{p}_{2}\in \left(1)\frac{1}{\alpha }),也表明了T ~ θ, α {\widetilde{T}}_{\theta ,\alpha } 从空间mp1u1 (μ) × mp2u2 (μ)的乘积有界 {m}_{{p}_{1}}^{{你}_{1}}\left(\mu )\times {m}_{{p}_{2}}^{{你}_{2}}\left(\mu )到空间M 1u (μ) {m}_{1}^{你}\left(\mu ),其中1= 1p1 + 1p2−2 α 1=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha . 同时,我们证明了换向子T ~ θ, α, b1, b2 {\widetilde{T}}_{\theta ,\alpha ,{b}_{1},{b}_{2}} 由b1, b2∈RBMO≈(μ) {b}_{1},{b}_{2}\in \widetilde{{\rm{RBMO}}}\left(\mu )和T≈θ, α {\widetilde{T}}_{\theta ,\alpha } 从空间mp1u1 (μ) × mp2u2 (μ)的乘积有界 {m}_{{p}_{1}}^{{你}_{1}}\left(\mu )\times {m}_{{p}_{2}}^{{你}_{2}}\left(\mu )化成空间M q u (μ) {m}_{q}^{你}\left(\mu ),并且它也有界于空间mp1u1 (μ) × mp2u2 (μ)的积 {m}_{{p}_{1}}^{{你}_{1}}\left(\mu )\times {m}_{{p}_{2}}^{{你}_{2}}\left(\mu )到空间M 1u (μ) {m}_{1}^{你}\left(\mu )。
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