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

IF 0.9 3区 数学 Q2 MATHEMATICS
Guanghui Lu, Miaomiao Wang, Shuangping Tao
{"title":"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","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>&lt;</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":"{\"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>&lt;</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}","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}
引用次数: 0

摘要

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 )。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces
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 ) .
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来源期刊
Analysis and Geometry in Metric Spaces
Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology
CiteScore
1.80
自引率
0.00%
发文量
8
审稿时长
16 weeks
期刊介绍: Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.
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