BILINEAR θ-TYPE GENERALIZED FRACTIONAL INTEGRAL AND ITS COMMUTATOR ON NONHOMOGENEOUS METRIC MEASURE SPACES

Abstract

We establish the boundedness of the bilinear $𝜃$-type generalized fractional integral and its commutator on nonhomogeneous metric measure space. Under the assumption that the functions $\mathrm{\Phi }$ and $\lambda$ satisfy certain conditions, we prove that the bilinear $𝜃$-type generalized fractional integral $B{\tilde{T}}_{𝜃,\alpha }$ is bounded from the product of generalized Morrey spaces ${\mathcal{ℒ}}^{p_1,\mathrm{\Phi },\kappa }(\mu )\times {\mathcal{ℒ}}^{p_2,\mathrm{\Phi },\kappa }(\mu )$ into spaces ${\mathcal{ℒ}}^{q,{\mathrm{\Phi }}^{q∕p},\kappa }(\mu )$, and it is also bounded from the product of spaces ${\mathcal{ℒ}}^{p_1,\mathrm{\Phi },\kappa }(\mu )\times {\mathcal{ℒ}}^{p_2,\mathrm{\Phi },\kappa }(\mu )$ into generalized weak Morrey spaces $W{\mathcal{ℒ}}^{1,{\mathrm{\Phi }}^{1∕p},\kappa }(\mu )$. Furthermore, the boundedness of the commutator $[b_1,b_2,B{\tilde{T}}_{𝜃,\alpha }]$ formed by $b_1,b_2\in \widetilde{\text{RBMO\hspace{0.17em}}}(\mu )$ and $B{\tilde{T}}_{𝜃,\alpha }$ on spaces ${\mathcal{ℒ}}^{q,{\mathrm{\Phi }}^{q∕p},\kappa }(\mu )$ and on spaces $W{\mathcal{ℒ}}^{1,{\mathrm{\Phi }}^{1∕p},\kappa }(\mu )$ is also obtained.