We introduce the classes <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_{p,q} (\varphi , \psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {B}_{p,q} (\varphi , \psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of pairs (u, v) of weights u and v on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and prove two-weight norm estimates between two local generalized Morrey spaces for the multi-dimensional Hardy operator and its adjoint for weights in these classes. Such estimates were earlier known in the one-weight case and for radial weights. The proof is based on point-wise estimates for the Hardy operator and its adjoint via the norm of the domain space. A number of theorems on inclusion of pairs of radial weights into the classed <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B_{p,q} (\varphi , \psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {B}_{p,q} (\varphi , \psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is proved. Belonging of pairs of weights to the introduced classes proves to have a form of criterion for two-weight norm estimates in the case of classical Morrey spaces and power weights. We also apply the obtained results to two-weight study of multidimensional versions of Calderón and Stieltjes operators in Morrey spaces.

Two-weight norm inequalities for Hardy operators in generalized Morrey spaces

期刊论文
出版模式：

开放获取
发表日期：

2026年8月5日

查看全文

Natasha Samko

摘要

We introduce the classes $B_{p,q} (\varphi , \psi )$ $B_{p, q} (φ, ψ)$ and $\mathcal {B}_{p,q} (\varphi , \psi )$ $B_{p, q} (φ, ψ)$ of pairs (u, v) of weights u and v on $\mathbb {R}^n$ $R^{n}$ and prove two-weight norm estimates between two local generalized Morrey spaces for the multi-dimensional Hardy operator and its adjoint for weights in these classes. Such estimates were earlier known in the one-weight case and for radial weights. The proof is based on point-wise estimates for the Hardy operator and its adjoint via the norm of the domain space. A number of theorems on inclusion of pairs of radial weights into the classed $B_{p,q} (\varphi , \psi )$ $B_{p, q} (φ, ψ)$ and $\mathcal {B}_{p,q} (\varphi , \psi )$ $B_{p, q} (φ, ψ)$ is proved. Belonging of pairs of weights to the introduced classes proves to have a form of criterion for two-weight norm estimates in the case of classical Morrey spaces and power weights. We also apply the obtained results to two-weight study of multidimensional versions of Calderón and Stieltjes operators in Morrey spaces.