<p>We give a certain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{\infty }(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-estimate for the heat semigroup <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{e^{t\Delta }\}_{t \ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msup> <mi>e</mi> <mrow> <mi>t</mi> <mi mathvariant="normal">Δ</mi> </mrow> </msup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> that is closely related to the fact <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1(\mathbb {R}^2) \not \subset L^{\infty }(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>⊄</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallouët inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t \rightarrow 0^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> is indeed optimal.</p>

An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding

期刊论文
发表日期：

2026年2月25日

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Yi C. Huang,
Tohru Ozawa,
Chenmin Sun,
Taiki Takeuchi

摘要

We give a certain $L^{\infty }(\mathbb {R}^2)$ $L^{\infty} (R^{2})$ -estimate for the heat semigroup $\{e^{t\Delta }\}_{t \ge 0}$ ${e^{t Δ}}_{t \geq 0}$ that is closely related to the fact $H^1(\mathbb {R}^2) \not \subset L^{\infty }(\mathbb {R}^2)$ $H^{1} (R^{2}) ⊄ L^{\infty} (R^{2})$ , i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallouët inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as $t \rightarrow 0^+$ $t \to 0^{+}$ is indeed optimal.