We present necessary and sufficient conditions on triples of weights (u, v, w) for the boundedness of the dyadic weighted square function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_w$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>w</mi> </msub> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2(v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We use this characterization to obtain necessary and sufficient conditions for the boundedness of the t-Haar multipliers from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2(v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of boundedness of the dyadic weighted square function.

Two-weight estimates for the square function and t-Haar multipliers

期刊论文
发表日期：

2025年10月31日

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Daewon Chung,
Jean Carlo Moraes,
María Cristina Pereyra,
Brett D. Wick

摘要

We present necessary and sufficient conditions on triples of weights (u, v, w) for the boundedness of the dyadic weighted square function $S_w$ $S_{w}$ from $L^2(u)$ $L^{2} (u)$ into $L^2(v)$ $L^{2} (v)$ . We use this characterization to obtain necessary and sufficient conditions for the boundedness of the t-Haar multipliers from $L^2(u)$ $L^{2} (u)$ into $L^2(v)$ $L^{2} (v)$ in terms of boundedness of the dyadic weighted square function.