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Bilinear \(\theta\)-type generalized fractional integral and its commutator on nonhomogeneous metric measure spaces. (English) Zbl 07731149

Recently there have so many papers focusing on the various integral operators and function spaces on \((X,d,\mu)\). In the present paper, the authors establish the boundedness of the bilinear \(\theta\)-type generalized fractional integral and its commutator on nonhomogeneous metric measure space. Under the assumption that the functions \(\Phi\) and \(\lambda\) satisfy certain conditions, the authors prove that the bilinear \(\theta\)-type generalized fractional integral \(B\tilde{T}_{\theta,\alpha}\) is bounded from the product of generalized Morrey spaces \(L^{p_1,\Phi,k}(\mu)\times L^{p_2,\Phi,k}(\mu)\) into spaces \(L^{q,\Phi q/p,k}(\mu)\) and it is also bounded from the product of spaces \(L^{p_1,\Phi,k}(\mu)\times L^{p_2,\Phi,k}(\mu)\) into generalized weak Morrey spaces \(WL^{1,\Phi 1/p,k}(\mu)\). Furthermore, the boundedness of the commutator of \(B\tilde{T}_{\theta,\alpha}\) is also obtained.
Generally speaking, this paper is interesting and well-written. Particularly, the outline of this paper and the proofs are clear.
Reviewer: Feng Liu (Qingdao)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
30L99 Analysis on metric spaces
26A33 Fractional derivatives and integrals
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References:

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