Analysis and Geometry in Metric Spaces (Nov 2023)
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 Mpu(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 u1,u2{u}_{1},{u}_{2}, and uu belong to Wτ{{\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 Mp1u1(μ)×Mp2u2(μ){M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces Mqu(μ){M}_{q}^{u}\left(\mu ), where u1u2=u{u}_{1}{u}_{2}=u, α∈(0,1)\alpha \in \left(0,1), and 1q=1p1+1p2−2α\frac{1}{q}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha with p1,p2∈(1,1α){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 Mp1u1(μ)×Mp2u2(μ){M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M1u(μ){M}_{1}^{u}\left(\mu ), where 1=1p1+1p2−2α1=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}-2\alpha . Meanwhile, we prove that the commutator T˜θ,α,b1,b2{\widetilde{T}}_{\theta ,\alpha ,{b}_{1},{b}_{2}} formed by b1,b2∈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 Mp1u1(μ)×Mp2u2(μ){M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces Mqu(μ){M}_{q}^{u}\left(\mu ), and it is also bounded from the product of spaces Mp1u1(μ)×Mp2u2(μ){M}_{{p}_{1}}^{{u}_{1}}\left(\mu )\times {M}_{{p}_{2}}^{{u}_{2}}\left(\mu ) into spaces M1u(μ){M}_{1}^{u}\left(\mu ).
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