We show that for all homogeneous polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ f_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> of degree m, in d variables, and each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$j = 1, \dots , d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>, we have <Equation ID="Equ13"> <EquationSource Format="TEX">$\begin{aligned} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb {S} ^{d-1}\right) } \ge \frac{\pi ^{2}}{4\left( m+ 2 d + 1 \right) ^{2}} \left\langle f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb {S}^{d-1}\right) }. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mfenced close="〉" open="〈"> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>f</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> </mfenced> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mfenced close=")" open="("> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </msub> <mo>≥</mo> <mfrac> <msup> <mi>π</mi> <mn>2</mn> </msup> <mrow> <mn>4</mn> <msup> <mfenced close=")" open="("> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mfenced> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mfenced close="〉" open="〈"> <msub> <mi>f</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> </mfenced> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mfenced close=")" open="("> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </msub> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces.

The Dirichlet Problem with Entire Data for Non-Hyperbolic Quadratic Hypersurfaces

期刊论文
发表日期：

2026年1月14日

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J. M. Aldaz,
H. Render

摘要

We show that for all homogeneous polynomials $ f_{m}$ $f_{m}$ of degree m, in d variables, and each $j = 1, \dots , d$ $j = 1, \dots, d$ , we have $\begin{aligned} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb {S} ^{d-1}\right) } \ge \frac{\pi ^{2}}{4\left( m+ 2 d + 1 \right) ^{2}} \left\langle f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb {S}^{d-1}\right) }. \end{aligned}$ $\begin{matrix} {〈x_{j}^{2} f_{m}, f_{m}〉}_{L^{2} (S^{d - 1})} \geq \frac{π^{2}}{4 {(m + 2 d + 1)}^{2}} {〈f_{m}, f_{m}〉}_{L^{2} (S^{d - 1})} . \end{matrix}$ This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces.