123 research outputs found

### New fractional-type weights and the boundedness of some operators

The purpose of this study is twofold. On the one hand, we establish a new
class of fractional-type multiple weights ${A_{\vec p( \cdot ),q( \cdot )}}$,
which can be regarded as a variable exponents generalization of ${A_{\vec
p,q}}$, a multilinear generalization of ${A_{p( \cdot ),q( \cdot )}}$, and a
fractional-type generalization of ${A_{\vec p( \cdot )}}$. Moreover, it can be
characterized by multilinear fractional-type operators ${\mathscr T}_\alpha$.
This generalizes the work by Moen (2009), Cruz-Uribe and Wang (2017), and
Cruz-Uribe and Guzm\'an (2020), which is also called multilinear
Hardy-Littlewood-Sobolev theorem on weighted variable exponents Lebesgue
spaces. On the other hand, we also set up a new class of fractional-type
variable matrix weights ${{\mathbb{A}}_{p( \cdot ),q( \cdot )}}$ that are
characterized by fractional-type averaging operators ${\mathscr A}_{\alpha,Q}$
and fractional Christ-Goldberg maximal operator ${{\mathcal M}_{\alpha ,W}}$.Comment: 41 pages; 1 figure

### Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper, the authors establish the two-weight boundedness of the local
fractional maximal operators and local fractional integrals on Gaussian measure
spaces associated with the local weights. More precisely, the authors first
obtain the two-weight weak-type estimate for the local-$a$ fractional maximal
operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with 1\leq
p\leq qa}
A_{p,q,\alpha}^{b'}, and then obtain the two-weight weak-type estimate for the
local fractional integrals. In addition, the authors obtain the two-weight
strong-type boundedness of the local fractional maximal operators under a
condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the
two-weight strong-type boundedness of the local fractional integrals. These
estimates are established by the radialization method and dyadic approach.Comment: 25 pages, 2 figure

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