Narrow Results

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces H^p((0,∞),dm_λ) associated with △_λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dm_λ(x) ≡ x^2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

Let $\lambda \in (0,\infty )$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${ℝ}_{+}:=(0,\infty )$. In this paper, the authors show that $b\in \mathrm{\text{BMO}}({ℝ}_{+},d{m}_{\lambda })$ (or $\mathrm{\text{CMO}}({ℝ}_{+},d{m}_{\lambda })$, respectively) if and only if the Riesz transform commutator $[b,R_{{△}_{\lambda }}]$ is bounded (or compact, respectively) on Morrey spaces $L^{p,\kappa }({ℝ}_{+},d{m}_{\lambda })$, where $d{m}_{\lambda }(x):=x^{2\lambda }dx$, $p\in (1,\infty )$ and $\kappa \in (0,1)$. A weak factorization theorem for functions belonging to the Hardy space $H^1({ℝ}_{+},d{m}_{\lambda })$ in the sense of Coifman–Rochberg–Weiss in Bessel setting, via $R_{{△}_{\lambda }}$ and its adjoint, is also obtained.

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on $L^p$ space for $1\le p\le \infty$, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied.

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces H_μ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space H_μ(I) and S(I) into itself. An integral representation of p.d.o h_{1, μ, a} is obtained. Using the Hankel convolution it is shown that p.d.o h_{1, μ, a} satisfies a certain -norm inequality. Properties of Sobolev-type space are studied.