Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces

Abstract

Let ( X , d , μ ) \left({\mathcal{X}},d,\mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space M p u ( μ ) {M}_{p}^{u}\left(\mu ) , where 1 ≤ p < ∞ 1\le p\lt \infty and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) u\left(x,r):{\mathcal{X}}\times \left(0,\infty )\to \left(0,\infty ) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions u 1 , u 2 {u}_{1},{u}_{2} , and u u belong to W τ {{\mathbb{W}}}_{\tau } with τ ∈ ( 0 , 2 ) \tau \in \left(0,2) , we prove that the bilinear θ \theta -type generalized fractional integral T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , where u 1 u 2 = u {u}_{1}{u}_{2}=u , α ∈ ( 0 , 1 ) \alpha \in \left(0,1) , and 1 q = 1 p 1 + 1 p 2 − 2 α \frac{1}{q}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha with p 1 , p 2 ∈ ( 1 , 1 α ) {p}_{1},{p}_{2}\in \left(1,\frac{1}{\alpha }) , and also show that the T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) , where 1 = 1 p 1 + 1 p 2 − 2 α 1=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha . Meanwhile, we prove that the commutator T ˜ θ , α , b 1 , b 2 {\widetilde{T}}_{\theta ,\alpha ,{b}_{1},{b}_{2}} formed by b 1 , b 2 ∈ RBMO ˜ ( μ ) {b}_{1},{b}_{2}\in \widetilde{{\rm{RBMO}}}\left(\mu ) and T ˜ θ , α {\widetilde{T}}_{\theta ,\alpha } is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M q u ( μ ) {M}_{q}^{u}\left(\mu ) , and it is also bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) {M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M 1 u ( μ ) {M}_{1}^{u}\left(\mu ) .