<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> be a proper open subset of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We provide sufficient conditions about local weights for the two-weight norm inequalities for local versions of the fractional maximal and the fractional integral operator acting on weighted Lebesgue spaces. The techniques applied, based on the use of Sparse operators, allow to get similar results to those known for the non-local versions, thus improving the known results for the local ones. As applications we obtain, in the first place, an a priori estimate for solutions of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta ^m U=f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>m</mi> </msup> <mi>U</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, acting on weighted Sobolev spaces involving the distance to the boundary and different local weights. In the context of Schödinger-type operators we also prove, as another application, the boundedness of the Riesz potential <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$I^\alpha _\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>I</mi> <mi>μ</mi> <mi>α</mi> </msubsup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a Radon measure on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

Sparse Approach to the Two-Weight Boundedness of Local Operators and Applications.

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发表日期：

2026年4月15日

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Mauricio Ramseyer,
Oscar Salinas,
Juan Sotto Ríos,
Marisa Toschi

摘要

Let $\Omega $ $Ω$ be a proper open subset of $\mathbb {R}^n$ $R^{n}$ . We provide sufficient conditions about local weights for the two-weight norm inequalities for local versions of the fractional maximal and the fractional integral operator acting on weighted Lebesgue spaces. The techniques applied, based on the use of Sparse operators, allow to get similar results to those known for the non-local versions, thus improving the known results for the local ones. As applications we obtain, in the first place, an a priori estimate for solutions of $\Delta ^m U=f$ $Δ^{m} U = f$ in $\Omega $ $Ω$ , acting on weighted Sobolev spaces involving the distance to the boundary and different local weights. In the context of Schödinger-type operators we also prove, as another application, the boundedness of the Riesz potential $I^\alpha _\mu $ $I_{μ}^{α}$ , where $\mu $ $μ$ is a Radon measure on $\mathbb {R}^n$ $R^{n}$ .