Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces
Jingliang Du, Hua Wang
TL;DR
This work analyzes the boundedness of the homogeneous fractional integral operator $T_{Ω,α}$ with kernels $Ω(x-y)/|x-y|^{n-α}$ on weighted function spaces. Under an $L^{s;β}$-Dini smoothness condition on $Ω$, the authors establish sharp mappings from weighted Lebesgue spaces $L^{p}(ω^p)$ to Campanato-type spaces $\mathcal{C}^{γ,ℓ}_{ω}$ for $n/α<p<∞$ with $γ=α-n/p$, and from weighted Morrey spaces $\mathcal{M}^{p,κ}(ω^p,ω^q)$ to weighted Campanato spaces $\mathcal{C}^{γ_*,ℓ}(ω^q)$ for $p/q<κ<1$ with $γ_*=α-(1-κ)n/p$, including endpoint cases that land in $BMO_{ω}$ or $BMO(\mathbb{R}^n)$. The results rely on precise kernel estimates under the Dini condition, a careful decomposition of functions, and leveraging $A_p$ and $A(p,q)$ weight properties. This provides a unified weighted framework for fractional integrals with homogeneous kernels acting on Lebesgue and Morrey-type spaces and yielding Campanato endpoint spaces. The methods have potential implications for related PDE and harmonic analysis problems in weighted settings.
Abstract
Let $0<α<n$ and $T_{Ω,α}$ be the homogeneous fractional integral operator which is defined by \begin{equation*} T_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} where $Ω$ is homogeneous of degree zero in $\mathbb R^n$ for $n\geq2$, and is integrable on the unit sphere $\mathbb{S}^{n-1}$. In this paper we study boundedness properties of the homogeneous fractional integral operator $T_{Ω,α}$ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on $Ω$, we prove that $T_{Ω,α}$ is bounded from $L^{p}(ω^p)$ to $\mathcal{C}^{γ,\ell}_ω$(a class of Campanato spaces) for appropriate indices, when $n/α<p<\infty$. Moreover, we prove that if $Ω$ satisfies certain Dini-type smoothness condition on $\mathbb{S}^{n-1}$, then $T_{Ω,α}$ is bounded from $\mathcal{M}^{p,κ}(ω^p,ω^q)$ to $\mathcal{C}^{γ,\ell}(ω^q)$(weighted Campanato spaces) for appropriate indices, when $p/q<κ<1$.