<p>Given a bilinear (or sub-bilinear) operator <i>B</i>, we prove restricted weighted weak type inequalities of the form <Equation ID="Equ16"> <EquationSource Format="TEX">$ ||B(f_1, f_2)||_{L^{p, \infty }(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi>B</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>w</mi> <mn>1</mn> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </msubsup> <msubsup> <mi>w</mi> <mn>2</mn> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≲</mo> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <mrow> <msup> <mi>L</mi> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <mrow> <msup> <mi>L</mi> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </Equation>whenever <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B(f_1, f_2)= (T_1f_1) (T_2 f_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in Baena-Miret et al. (Int Math Res Not IMRN 2023(24):21943–21975, 2023).</p>

Restricted weighted weak boundedness for product type operators

期刊论文
发表日期：

2026年3月2日

查看全文

María Jesús Carro,
Sheldy Ombrosi

摘要

Given a bilinear (or sub-bilinear) operator B, we prove restricted weighted weak type inequalities of the form $ ||B(f_1, f_2)||_{L^{p, \infty }(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $ $| | B (f_{1}, f_{2}) {| |}_{L^{p, \infty} (w_{1}^{p / p_{1}} w_{2}^{p / p_{2}})} ≲ | | f_{1} {| |}_{L^{p_{1}, 1} (w_{1})} | | f_{2} {| |}_{L^{p_{2}, 1} (w_{2})},$ whenever $B(f_1, f_2)= (T_1f_1) (T_2 f_2)$ $B (f_{1}, f_{2}) = (T_{1} f_{1}) (T_{2} f_{2})$ is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in Baena-Miret et al. (Int Math Res Not IMRN 2023(24):21943–21975, 2023).