<p>We establish <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{p_1}(\mathbb R^d) \times \cdots \times L^{p_n}(\mathbb R^d) \rightarrow L^{r}(\mathbb R^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> <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> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mi>r</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> bounds for the <i>n</i>-linear spherical averaging operators <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal A^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">A</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> in all dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, for indices <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1 \le p_1,\dots ,p_n \le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\tfrac{1}{p_1}+\cdots +\tfrac{1}{p_n}=\tfrac{1}{r}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> </mstyle> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>n</mi> </msub> </mfrac> </mstyle> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mstyle> </mrow> </math></EquationSource> </InlineEquation>. Our argument begins by showing that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal A^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">A</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> maps <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^1 \times \cdots \times L^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, our result recovers and extends the bilinear theory previously developed by Iosevich, Palsson, and Sovine. We also obtain analogous estimates for the lacunary maximal spherical averages in the largest possible open region of indices.</p>

Lebesgue Bounds for Multilinear Spherical and Lacunary Maximal Averages

期刊论文
发表日期：

2026年1月8日

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Xinyu Gao

摘要

We establish $L^{p_1}(\mathbb R^d) \times \cdots \times L^{p_n}(\mathbb R^d) \rightarrow L^{r}(\mathbb R^d)$ $L^{p_{1}} (R^{d}) \times \dots \times L^{p_{n}} (R^{d}) \to L^{r} (R^{d})$ bounds for the n-linear spherical averaging operators $\mathcal A^n$ $A^{n}$ in all dimensions $d \ge 2$ $d \geq 2$ , for indices $1 \le p_1,\dots ,p_n \le \infty $ $1 \leq p_{1}, \dots, p_{n} \leq \infty$ satisfying $\tfrac{1}{p_1}+\cdots +\tfrac{1}{p_n}=\tfrac{1}{r}$ $\frac{1}{p_{1}} + \dots + \frac{1}{p_{n}} = \frac{1}{r}$ . Our argument begins by showing that $\mathcal A^n$ $A^{n}$ maps $L^1 \times \cdots \times L^1$ $L^{1} \times \dots \times L^{1}$ to $L^1$ $L^{1}$ . For $n=2$ $n = 2$ , our result recovers and extends the bilinear theory previously developed by Iosevich, Palsson, and Sovine. We also obtain analogous estimates for the lacunary maximal spherical averages in the largest possible open region of indices.