Riesz potential on the Heisenberg group and modified Morrey spaces (Q2857161)

In this paper, the authors study the fractional maximal operator \(M_\alpha\), \(0\leq\alpha<Q\), and the Riesz potential operator \(\mathcal I_\alpha\), \(0<\alpha<Q\), in the modified Morrey space \(\tilde L_{p,\lambda}(\mathbb H_n)\) on the Heisenberg group \(\mathbb H_n\), where \(Q=2n+2\) is the homogeneous dimension on \(\mathbb H_n\).NEWLINENEWLINE They give necessary and sufficient conditions for the boundedness of the operators \(M_\alpha\) and \(\mathcal I_\alpha\) in \(\tilde L_{p,\lambda}(\mathbb H_n)\).NEWLINENEWLINE Let \(1\leq p<\infty,\;0\leq\lambda\leq Q\). We denote by \(\tilde L_{p,\lambda}(\mathbb H_n)\) the modified Morrey space, and by \(W\tilde L_{p,\lambda}(\mathbb H_n)\) the modified weak Morrey space, the set of locally integrable functions \(f\) on \(\mathbb H_n\) with NEWLINE\[NEWLINE\| f\|_{\tilde L_{p,\lambda}}=\sup_{u\in\mathbb H_n, t>0}\biggl(\min\{1,t\}^{-\lambda} \int_{B(u,t)}| f(y)|^p dV(y)\biggr)^{1/p}<\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\| f\|_{W\tilde L_{p,\lambda}}=\sup_{r>0}\;r\sup_{u\in\mathbb H_n, t>0}\biggl(\min\{1,t\}^{-\lambda} |\{v\in B(u,t):| f(v)|>r\}|\biggr)^{1/p}<\inftyNEWLINE\]NEWLINE respectively, where \(B(u,t)\) is a ball in \(\mathbb H_n\).NEWLINENEWLINE We define the fractional maximal operator \(M_\alpha\), \(0\leq\alpha<Q\), the fractional integral operator \(I_\alpha\), \(0<\alpha<Q\), the modified fractional integral operator \(\tilde I_\alpha\), \(0<\alpha<Q\), and the Riesz potential operator \(\mathcal I_\alpha\), \(0<\alpha<Q\), by NEWLINE\[NEWLINEM_\alpha f(u)=\sup_{r>0}| B(u,r)|^{-1+\alpha/Q}\int_{B(u,r)}| f(v)| dV(v),NEWLINE\]NEWLINE NEWLINE\[NEWLINEI_\alpha f(u)=\int_{\mathbb H_n}| v^{-1}u|^{\alpha-Q}f(v) dV(v),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\tilde I_\alpha f(u)=\int_{\mathbb H_n}\bigl(| uv^{-1}|^{\alpha-Q}-| v|^{\alpha-Q} \chi_{CB(e,1)}(v)\bigr)f(v) dV(v),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathcal I_\alpha f(z,t)=\mathcal L^{-\alpha/2} f(z,t)NEWLINE\]NEWLINE respectively, where \(CB(e,1)=\mathbb H_n\setminus B(e,1)\) and \(\mathcal L\) is the sub-Laplacian on \(\mathbb H_n\).NEWLINENEWLINE The authors prove that the operators \(M_\alpha\), \(I_\alpha\) and \(\mathcal I_\alpha\) are bounded from \(\tilde L_{p,\lambda}(\mathbb H_n)\) to \(\tilde L_{q,\lambda}(\mathbb H_n)\) if and only if \(\alpha/Q \leq 1/p-1/q \leq \alpha/(Q-\lambda)\) and from \(\tilde{L}_{1,\lambda}(\mathbb H_n)\) to \(W\tilde L_{q,\lambda}(\mathbb H_n)\) if and only if \(\alpha/Q \leq 1-1/q \leq \alpha/(Q-\lambda)\).NEWLINENEWLINE In the limiting case \(\alpha/Q \leq 1/p \leq \alpha/(Q-\lambda)\) they prove that the operator \(M_\alpha\) is bounded from \(\tilde L_{p,\lambda}(\mathbb H_n)\) to \(L_\infty(\mathbb H_n)\) and the operator \(\tilde I_\alpha\) is bounded from \(\tilde L_{p,\lambda}(\mathbb H_n)\) to \(BMO(\mathbb H_n)\).NEWLINENEWLINE As applications of the properties of the fundamental solution of the sub-Laplacian \(\mathcal L\) on \(\mathbb H_n\), they prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces on the Heisenberg group setting. As an another application, they prove the boundedness of \(\mathcal I_\alpha\) from Besov-modified Morrey spaces \(B\tilde L_{p\theta,\lambda}^s(\mathbb H_n)\) to \(B\tilde L_{q\theta,\lambda}^s(\mathbb H_n)\).