<p>We study boundedness properties for the bilinear Bochner-Riesz operator at the critical index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha = m - \tfrac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </math></EquationSource> </InlineEquation>. Starting from the weighted <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2 \times L^2 \rightarrow L^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> estimate previously established by Jotsaroop, Shrivastava, and Shuin, we develop a technique that combines quantitative bilinear Rubio de Francia extrapolation with a suitable bilinear version of Yano’s extrapolation theorem. This method yields a range of new weighted endpoint estimates. Our results cover all open endpoints for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {B}^{m - \frac{1}{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation>, and include both one-weight and two-weight inequalities.</p>

The Bilinear Bochner-Riesz Operator at the Critical Index

期刊论文
发表日期：

2025年12月12日

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María J. Carro,
Teresa Luque,
Laura Sánchez-Pascuala

摘要

We study boundedness properties for the bilinear Bochner-Riesz operator at the critical index $\alpha = m - \tfrac{1}{2}$ $α = m - \frac{1}{2}$ . Starting from the weighted $L^2 \times L^2 \rightarrow L^1$ $L^{2} \times L^{2} \to L^{1}$ estimate previously established by Jotsaroop, Shrivastava, and Shuin, we develop a technique that combines quantitative bilinear Rubio de Francia extrapolation with a suitable bilinear version of Yano’s extrapolation theorem. This method yields a range of new weighted endpoint estimates. Our results cover all open endpoints for $\mathcal {B}^{m - \frac{1}{2}}$ $B^{m - \frac{1}{2}}$ , and include both one-weight and two-weight inequalities.