Weighted Hardy-type inequality in two dimensions

Abstract Weighted Hardy-type inequality in two dimensions has been characterized for pairs of weight functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>α</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\omega_{1}(\alpha_{1},\alpha_{2})} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>α</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\omega_{2}(\alpha_{1},\alpha_{2})} defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:msubsup> </m:math> {\mathbb{R}_{+}^{2}} , for which the sum of two-dimensional Hardy-type integral operators is bounded from a weighted Lebesgue space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:msub> <m:mi>ω</m:mi> <m:mn>1</m:mn> </m:msub> <m:mi>β</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L_{\omega_{1}}^{\beta}(\mathbb{R}_{+}^{2})} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:msub> <m:mi>ω</m:mi> <m:mn>2</m:mn> </m:msub> <m:mi>γ</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L_{\omega_{2}}^{\gamma}(\mathbb{R}_{+}^{2})} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>β</m:mi> <m:mo>&lt;</m:mo> <m:mi>γ</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {1&lt;\beta&lt;\gamma&lt;\infty} .