In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $mathcal{B}_tf$ converges pointwise to its initial data $fin H^s(mathbb{R})$ as $tto 0$ provided $sgeqfrac{1}{4}$ -- more precisely -- on the one hand, by constructing a counterexampl
e 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 sleqfrac{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.
We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted
norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmunds $L{(log L)^{{frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.
In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding em
bedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two upper-half planes.