### A characterization of a two-weight norm inequality for maximal operators

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A new characterization is given for the pairs of weight functions v, w for which the fractional maximal function is a bounded operator from ${L}_{v}^{p}\left(X\right)$ to ${L}_{w}^{q}\left(X\right)$ when 1 < p < q < ∞ and X is a homogeneous space with a group structure. The case when X is n-dimensional Euclidean space is included.

In [P] we characterize the pairs of weights for which the fractional integral operator ${I}_{\gamma}$ of order $\gamma $ from a weighted Lebesgue space into a suitable weighted $BMO$ and Lipschitz integral space is bounded. In this paper we consider other weighted Lipschitz integral spaces that contain those defined in [P], and we obtain results on pairs of weights related to the boundedness of ${I}_{\gamma}$ acting from weighted Lebesgue spaces into these spaces. Also, we study the properties of those classes of weights and compare...

Let ${}_{s}$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_{s}$ differentiates the integral of f if s ∉ S, and $D{\u0305}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in}_{s}}{\left|R\right|}^{-1}{\int}_{R}f=\infty $ almost everywhere if s ∈ S. If the condition $D{\u0305}_{s}f\left(x\right)=\infty $ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a ${G}_{\delta}$ (resp. a ${G}_{\delta \sigma}$).

The object of this note is to generalize some Fourier inequalities.

A real-valued Hardy space $H{\xb9}_{\surd}\left(\right)\subseteq L\xb9\left(\right)$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H{\xb9}_{\surd}\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{\xb9}_{\surd}\left(\right)$, and no Orlicz space...

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\overrightarrow{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the ${L}^{q}(\Omega ,d\mu )$ norm of |∇u| is dominated by the ${L}^{p}(\Omega ,dv)$ norms of $div\overrightarrow{f}$ and $|\overrightarrow{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

We establish the boundedness in ${L}^{q}$ spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

We give characterizations of weighted Besov-Lipschitz and Triebel-Lizorkin spaces with ${A}_{\infty}$ weights via a smooth kernel which satisfies “minimal” moment and Tauberian conditions. The results are stated in terms of the mixed norm of a certain maximal function of a distribution in these weighted spaces.

Let $\mu $ be a nonnegative Radon measure on ${\mathbb{R}}^{d}$ which only satisfies $\mu \left(B(x,r)\right)\le {C}_{0}{r}^{n}$ for all $x\in {\mathbb{R}}^{d}$, $r>0$, with some fixed constants ${C}_{0}>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathrm{RBMO}\left(\mu \right)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.