Assume that Δh is the hyperbolic Laplacian in the unit ball <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb{B}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> </math></EquationSource> </InlineEquation> and assume that Φn is the unique radial solution of Poisson equation Δh log Φn = − 4(n − 1)2 satisfying the condition Φn(0) = 1 and Φn(ζ) = 0 for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\zeta\in\partial\mathbb{B}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>ζ</mi> <mo>∈</mo> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> </math></EquationSource> </InlineEquation>. We explicitly solve the question of maximizing <Equation ID="Equ1"> <EquationSource Format="TEX">${R_n}({f,\Omega}) = {{\int_\Omega {{{\vert {f(x)} \vert}^2}\Phi _n^\alpha ({\vert x \vert})d\tau (x)}} \over {\Vert f \Vert_{{\rm{B}}_\alpha ^2}^2}},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>R</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mfrac> <mrow> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <mrow> <mrow> <msup> <mrow> <mo fence="false" stretchy="false">|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mi>n</mi> <mi>α</mi> </msubsup> <mo stretchy="false">(</mo> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo fence="false" stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">∥</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">∥</mo> <mrow> <msubsup> <mrow> <mrow> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mi>α</mi> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> </math></EquationSource> </Equation> over all f ∈ B<Stack> α 2 </Stack> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega\subset\mathbb{B}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> </math></EquationSource> </InlineEquation> with τ(Ω) = s, where dτ denotes the invariant measure on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb{B}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Vert f \Vert_{B_\alpha ^2}^2 = \int_{\mathbb{B}} {{{\vert {f(x)} \vert}^2}\Phi _n^\alpha ({\vert x \vert})d\tau} (x)< \infty$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo>∥</mo> <mi>f</mi> <msubsup> <mo>∥</mo> <mrow> <msubsup> <mi>B</mi> <mi>α</mi> <mn>2</mn> </msubsup> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mo>∫</mo> <mrow> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> </mrow> </msub> <mrow> <mrow> <msup> <mrow> <mo fence="false" stretchy="false">|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mi>n</mi> <mi>α</mi> </msubsup> <mo stretchy="false">(</mo> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo fence="false" stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>τ</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> </InlineEquation>.This result extends the main result of Tilli and the second author [20] to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with additional technical challenges arising from the definition of the weights Φn through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-subharmonic functions and one for the Wavelet transform is only available in dimension one.

A Faber–Krahn type inequality for log-subharmonic functions in the hyperbolic ball

期刊论文
发表日期：

2025年6月29日

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David Kalaj,
João P. G. Ramos

摘要

Assume that Δ_h is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ $B$ and assume that Φ_n is the unique radial solution of Poisson equation Δ_h log Φ_n = − 4(n − 1)² satisfying the condition Φ_n(0) = 1 and Φ_n(ζ) = 0 for $\zeta\in\partial\mathbb{B}$ $ζ \in \partial B$ . We explicitly solve the question of maximizing ${R_n}({f,\Omega}) = {{\int_\Omega {{{\vert {f(x)} \vert}^2}\Phi _n^\alpha ({\vert x \vert})d\tau (x)}} \over {\Vert f \Vert_{{\rm{B}}_\alpha ^2}^2}},$ $R_{n} (f, Ω) = \frac{\int_{Ω} {| f (x) |}^{2} Φ_{n}^{α} (| x |) d τ (x)}{∥ f ∥_{B_{α}^{2}}^{2}},$ over all f ∈ B _α ² and $\Omega\subset\mathbb{B}$ $Ω \subset B$ with τ(Ω) = s, where dτ denotes the invariant measure on $\mathbb{B}$ $B$ , and $\Vert f \Vert_{B_\alpha ^2}^2 = \int_{\mathbb{B}} {{{\vert {f(x)} \vert}^2}\Phi _n^\alpha ({\vert x \vert})d\tau} (x)< \infty$ $∥ f ∥_{B_{α}^{2}}^{2} = \int_{B} {| f (x) |}^{2} Φ_{n}^{α} (| x |) d τ (x) < \infty$ .

This result extends the main result of Tilli and the second author [20] to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with additional technical challenges arising from the definition of the weights Φ_n through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-subharmonic functions and one for the Wavelet transform is only available in dimension one.