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

Abstract

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

$\mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\rm e}^{{{\rm i}{t}(\sqrt{-\Delta})^a}} f(y)=f(x), \quad \mathop {\mathop {\lim }\limits_{y \in \Gamma \left( {x,t} \right)} }\limits_{\left( {y,t} \right) \to \left( {x,0} \right)} {\cal B}_{t}f(y)=f(x), \quad {\rm a.e.},$

whenever f ∈ H^s(ℝ).

Keywords

Fractional Schrödinger propagator / Boussinesq operator / Pointwise convergence / Tangential curves / Sobolev space / 42B25 / 35S10

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