In this paper, the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in ℝ × ℝ given by {(y, t): y ∈ Γ(x, t)} for each t ∈ [0, 1], where Γ(x, t) = {γ(x, t, θ): θ ∈ Θ} for a given compact set Θ in ℝ of the fractional Schrödinger propagator and Boussinesq operator. They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11401_2025_31_Article_Equa.gif" Format="GIF" Height="50" Rendition="HTML" Resolution="72" Type="Linedraw" Width="448" /> </MediaObject> <EquationSource Format="TEX">$\mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\rm e}^{{{\rm i}{t}(\sqrt{-\Delta})^a}} f(y)=f(x), \quad \mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\cal B}_{t}f(y)=f(x), \quad {\rm a.e.},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mrow class="MJX-TeXAtom-OP"> <munder> <mrow class="MJX-TeXAtom-OP"> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>y</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </munder> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </munder> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> <mrow> <mi>t</mi> </mrow> <mo stretchy="false">(</mo> <msqrt> <mo>−</mo> <mi mathvariant="normal">Δ</mi> </msqrt> <msup> <mo stretchy="false">)</mo> <mi>a</mi> </msup> </mrow> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mrow class="MJX-TeXAtom-OP"> <munder> <mrow class="MJX-TeXAtom-OP"> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>y</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </munder> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </munder> <msub> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mrow> <mi mathvariant="normal">a</mi> <mo>.</mo> <mi mathvariant="normal">e</mi> <mo>.</mo> </mrow> <mo>,</mo> </math></EquationSource> </Equation> whenever f ∈ Hs(ℝ).

A Note on the Convergence Along Tangential Curve Associated with Fractional Schrödinger Propagator and Boussinesq Operator

期刊论文
发表日期：

2025年7月28日

查看全文

Dan Li,
Junfeng Li

摘要

In this paper, the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in ℝ × ℝ given by {(y, t): y ∈ Γ(x, t)} for each t ∈ [0, 1], where Γ(x, t) = {γ(x, t, θ): θ ∈ Θ} for a given compact set Θ in ℝ of the fractional Schrödinger propagator and Boussinesq operator. They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which $\mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\rm e}^{{{\rm i}{t}(\sqrt{-\Delta})^a}} f(y)=f(x), \quad \mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\cal B}_{t}f(y)=f(x), \quad {\rm a.e.},$ $\underset{(y, t) \to (x, 0)}{lim_{y \in Γ (x, t)}} e^{i t (\sqrt{- Δ})^{a}} f (y) = f (x), \underset{(y, t) \to (x, 0)}{lim_{y \in Γ (x, t)}} B_{t} f (y) = f (x), a . e .,$ whenever f ∈ H^s(ℝ).