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W2046582302 abstract "We study the weighted<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mi>p</mml:mi><mml:mo>→</mml:mo><mml:mi>q</mml:mi></mml:math>-boundedness of the multidimensional weighted Hardy-type operators<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math>with radial type weight<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mo stretchy=false>|</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>|</mml:mo><mml:mo stretchy=false>)</mml:mo></mml:math>, in the generalized complementary Morrey spaces<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:msubsup><mml:mrow><mml:mrow><mml:mi>ℒ</mml:mi></mml:mrow><mml:mrow><mml:mo>∁</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=false>{</mml:mo><mml:mn mathvariant=normal>0</mml:mn><mml:mo stretchy=false>}</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>ψ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>)</mml:mo></mml:math>defined by an almost increasing function<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>. We prove a theorem which provides conditions, in terms of some integral inequalities imposed on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:math>, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math>and the weight<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:math>are power functions. We also prove that the spaces<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:msubsup><mml:mrow><mml:mrow><mml:mi>ℒ</mml:mi></mml:mrow><mml:mrow><mml:mo>∁</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=false>{</mml:mo><mml:mn mathvariant=normal>0</mml:mn><mml:mo stretchy=false>}</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>ψ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=false>(</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>over bounded domains Ω are embedded between weighted Lebesgue space<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M12><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>with the weight<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M13><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math>and such a space with the weight<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M14><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math>, perturbed by a logarithmic factor. Both the embeddings are sharp." @default.
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W2046582302 title "Weighted Hardy Operators in Complementary Morrey Spaces" @default.
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W2046582302 doi "https://doi.org/10.1155/2012/283285" @default.
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