In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable $BM{O}_{\phi }$ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted $L^p$ and Lipschitz integral spaces on ${ℝ}^n$. We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

We deal with the Hardy-Lorentz spaces $H^{p,q}\left(ℝⁿ\right)$ where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then ${(*\left(f\right))}^{\wedge }\left(t\right)=\left(f̂\right)\left(t\right)$ a.e. As...

We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^p\left(ℝ\right)$, 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^p\left(ℝ\right)$, 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^p\left(ℝ\right)$.

In this paper, two important geometric concepts–grapical center and width, are introduced in $p$-adic numbers field. Based on the concept of width, we give the Heisenberg uncertainty relation on harmonic analysis in $p$-adic numbers field, that is the relationship between the width of a complex-valued function and the width of its Fourier transform on $p$-adic numbers field.

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:=d^2/d{x}^2-2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1\left(d\gamma \right)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.