In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $\mathcal{ B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{ R})$ as $t\to 0$ provided $s\geq\frac{ 1}{ 4}$ -- more precisely -- on the one hand, by constructing a counterexample in $\mathbb{ R}$ we discover that the optimal convergence index $s_{ c,1}=\frac14$; on the other hand, we find that the Hausdorff dimension of the disconvergence set for $\mathcal{ B}_tf$ is \begin{ align*} \alpha_{ 1,\mathcal{ B}}(s)&=\begin{ cases} 1-2s&\ \ \text{ as}\ \ \frac{ 1}{ 4}\leq s\leq\frac{ 1}{ 2};\\ 1 &\ \ \text{ as}\ \ 0<s<\frac{ 1}{ 4}. \end{ cases} \end{ align*} Moreover, Theorem 1.3 presents a higher dimensional lift of Theorems 1.1- 1.2 under $f$ being radial.
Categories: math.CA