<p>In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ p \in (1, \infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \gamma \in [0, 1] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> for which the quantities <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \Vert \Delta ^\gamma f\Vert _p $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>γ</mi> </msup> <msub> <mrow> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \Vert \nabla f\Vert _p $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> can be compared. In particular, we prove that in such metric spaces, the Riesz transform <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \nabla \Delta ^{-1/2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <msup> <mi mathvariant="normal">Δ</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is unbounded on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ L^p $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ p \in (2, \infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, thereby demonstrating a clear departure from the behavior observed in the Euclidean setting.</p>

In Spaces with a Slow Diffusion, the Riesz Transform is Unbounded on $L^p$, $p\in (2,\infty )$

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2026年1月4日

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Joseph Feneuil

摘要

In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters $ p \in (1, \infty ) $ $p \in (1, \infty)$ and $ \gamma \in [0, 1] $ $γ \in [0, 1]$ for which the quantities $ \Vert \Delta ^\gamma f\Vert _p $ $‖ Δ^{γ} {f ‖}_{p}$ and $ \Vert \nabla f\Vert _p $ ${‖ \nabla f ‖}_{p}$ can be compared. In particular, we prove that in such metric spaces, the Riesz transform $ \nabla \Delta ^{-1/2} $ $\nabla Δ^{- 1 / 2}$ is unbounded on $ L^p $ $L^{p}$ for all $ p \in (2, \infty ) $ $p \in (2, \infty)$ , thereby demonstrating a clear departure from the behavior observed in the Euclidean setting.

In Spaces with a Slow Diffusion, the Riesz Transform is Unbounded on \(L^p\), \(p\in (2,\infty )\)

摘要