<p>Multilinear Fourier multipliers with Sobolev regularity have been extensively studied, and their boundedness under various integrability assumptions is by now well understood. In this paper, we study the weighted <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ L^{p_1}(w_1)\times \cdots \times L^{p_N}(w_N) \rightarrow L^{p}(w) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mi>L</mi> <msub> <mi>p</mi> <mi>N</mi> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mi>N</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> boundedness of multilinear Fourier multipliers satisfying Sobolev regularity conditions measured in mixed-norm spaces. Our analysis focuses on the endpoint regime, which allows some <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p_j = \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>j</mi> </msub> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, thereby substantially extending and improving previous known results.</p>

Weighted Estimates for Multilinear Fourier Multipliers with Mixed Norms

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发表日期：

2026年5月5日

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Jiali Yu,
Moyan Qin,
Qingying Xue

摘要

Multilinear Fourier multipliers with Sobolev regularity have been extensively studied, and their boundedness under various integrability assumptions is by now well understood. In this paper, we study the weighted $ L^{p_1}(w_1)\times \cdots \times L^{p_N}(w_N) \rightarrow L^{p}(w) $ $L^{p_{1}} (w_{1}) \times \dots \times L^{p_{N}} (w_{N}) \to L^{p} (w)$ boundedness of multilinear Fourier multipliers satisfying Sobolev regularity conditions measured in mixed-norm spaces. Our analysis focuses on the endpoint regime, which allows some $ p_j = \infty $ $p_{j} = \infty$ , thereby substantially extending and improving previous known results.