arXiv subjects
Publications and source records attributed to Shahaboddin Shaabani.
In this note, we investigate the sharpness of existing bounds for various types of bi-parameter paraproducts acting between product Hardy spaces in the dyadic setting. We show that these bounds are sharp in most cases but fail to be so in one particular instance.
For a tempered distribution $g$, and $0 < p, q, r < \infty$ with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, we show that the operator norm of a Fourier paraproduct $Π_g$, of the form \[ Π_{g}(f) := \sum_{j \in \mathbb{Z}} (φ_{2^{-j}} * f) \cdot Δ_jg, \] from $H^p(\mathbb{R}^n)$ to $\dot{H}^q(\mathbb{R}^n)$ is comparable to $\|g\|_{\dot{H}^r(\mathbb{R}^n)}$. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.
It is shown that for $0<p,q,r<\infty$, with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, the operator norm of the dyadic paraproduct of the form \[ π_g(f) := \sum_{R \in \mathcal{D}\otimes\mathcal{D}} g_R \left\langle f \right\rangle_{R} h_R, \] from the bi-parameter dyadic Hardy space $H_d^p(\mathbb{R}\otimes\mathbb{R})$ to $\dot{H}_d^q(\mathbb{R}\otimes\mathbb{R})$ is comparable to $\|g\|_{\dot{H}_d^r(\mathbb{R}\otimes\mathbb{R})}$. We also prove that for all $0 < p < \infty$, there holds \[ \|g\|_{BMO_d(\mathbb{R}\otimes\mathbb{R})} \simeq \|π_g\|_{H_d^p(\mathbb{R}\otimes\mathbb{R}) \to \dot{H}_d^p(\mathbb{R}\otimes\mathbb{R})}. \] Similar results are obtained for bi-parameter Fourier paraproducts of the same form.
For a symmetric convex body $K\subset\mathbb{R}^n$ and $1\le p<\infty$, we define the space $S^p(K)$ to be the tent generalization of $\text{JN}_p(\mathbb{R}^n)$, i.e., the space of all continuous functions $f$ on the upper-half space $\mathbb{R}_+^{n+1}$ such that \[ \|f\|_{S^p(K)} := \big( \sup_{\mathcal{C}} \sum_{x+tK \in \mathcal{C}} |f(x,t)|^p \big)^{\frac{1}{p}} < \infty, \] where, in the above, the supremum is taken over all finite disjoint collections of homothetic copies of $K$. It is then shown that the dual of $S^1_0(K)$, the closure of the space of continuous functions with compact support in $S^1(K)$, consists of all Radon measures on $\mathbb{R}_+^{n+1}$ with uniformly bounded total variation on cones with base $K$ and vertex in $\mathbb{R}^n$. In addition, a similar scale of spaces is defined in the dyadic setting, and for $1\le p<\infty$, a complete characterization of their duals is given. We apply our results to study $\text{JN}_p$ spaces.
Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the literature. Comparing the resulting function spaces to known function spaces such as $\dot{W}^{1,p}(\rn)$, $\JNp$, $\Lp$ and weak-$\Lp$ gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.
We show that a proper open subset $Ω\subset \mathbb{R}^n$ is an extension domain for $H^p$ ($0<p\le1$), if and only if it satisfies a certain geometric condition. When $n(\frac{1}{p}-1)\in \mathbb{N}$ this condition is equivalent to the global Markov condition for $Ω^c$, for $p=1$ it is stronger, and when $n(\frac{1}{p}-1)\notin \mathbb{N}\cup \{0\}$ every proper open subset is an extension domain for $H^p$. It is shown that in each case a linear extension operator exists. We apply our results to study some complemented subspaces of $BMO(\mathbb{R}^n)$.
We prove that the Hardy-Littlewood maximal operator is discontinuous on $\bmorn$ and maps $\vmorn$ to itself. A counterexample to boundedness of the strong and directional maximal operators on $\bmorn$ is given, and properties of slices of $\bmorn$ functions are discussed.