October 5, 2025Open Access

Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces

Key Points

The homogeneous fractional integral operator shows boundedness from weighted Lebesgue spaces to Campanato spaces for $n/α<p<∞$.
Under Dini-type smoothness conditions on $Ω$, the operator is bounded from Morrey spaces to weighted Campanato spaces when $p/q<κ<1$.
The study provides key insights into how fractional integral operators behave in various functional spaces, expanding understanding of operator theory.
Implications of these results may enhance functional analysis techniques for dealing with spaces of functions characterized by different smoothness conditions.

Abstract

Let 0<α<n and T⏨, ⏐ be the homogeneous fractional integral operator which is defined by equation* T⏨, ⏐f (x): = ₑ䂞Ω (x-y) |x-y|^{n-α}f (y) \, dy, equation* where Ω is homogeneous of degree zero in Rⁿ for n2, and is integrable on the unit sphere S^n-1. In this paper we study boundedness properties of the homogeneous fractional integral operator T⏨, ⏐ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on Ω, we prove that T⏨, ⏐ is bounded from L^p (ωᵖ) to C^γ, _ω (a class of Campanato spaces) for appropriate indices, when n/α<p<. Moreover, we prove that if Ω satisfies certain Dini-type smoothness condition on S^n-1, then T⏨, ⏐ is bounded from M^p, κ (ωᵖ, ωq) to C^γ, (ωq) (weighted Campanato spaces) for appropriate indices, when p/q<κ<1.

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