Singular integral operators for rearrangement-invariant Morrey spaces on local fields
Let \(K\) be a local field, \(n\in\mathbb{N}\), \(X\) be a rearrangement-invariant Banach function space on \(K^n\), \(u\) be a positive function on \(\mathbb{Z}\), \(\Omega\in L^{\infty}(K^n)\). By \(M_u^X(K^n)\) denote be the Morray space associated with \(K,n,X\) and \(u\). The singular integral operator associated with \(\Omega\) denote by \(T_{\Omega}\). The main results of the paper are the following theorems. Theorem 1. Let the Boyd's indices of \(X\) satisfy the estimates \(0<\underline{\alpha}_X\leq \overline{\alpha}_X<1\), and let the functions \(u\) and \(\Omega\) satisfy the following conditions: \[ u(k)>C \;\;\; (k\in \mathbb{Z}\setminus\mathbb{N}), \tag{1} \] \[ u(k)> C\|\chi_{B(x,k)}\|_X \;\;\; (x\in K^n, k\in \mathbb{N}), \tag {2} \] where \(C\) is a positive constant; \[ \sum_{j=1}^{\infty}\frac{u(k-j-1)}{\|\chi_{B(x,k-j-1)}\|_X}<\infty\;\;\;(x\in K^n,k\in \mathbb{Z});\tag{3} \] \[ \Omega(\mathfrak{p}^k x)=\Omega(x)\;\;\;(x\in K^n, k\in\mathbb{Z}),\tag{4} \] \[ \sup_{|y|=1} \sum_{j=1}^{\infty}\int_{|x|=1}|\Omega(x+\mathfrak{p}^j y)-\Omega(x)|dx<\infty \tag{5} \] where \(\mathfrak{p}\) is a prime element of \(K\); \[ \int_{|x|=1}|\Omega(x)dx=0. \] Then the singular integral operator \(T_{\Omega}\) is well-defined on \(M_u^X(K^n)\). Theorem 2. Let the Boyd's indices of \(X\) satisfy the estimates \(0<\underline{\alpha}_X\leq \overline{\alpha}_X<1\), the function \(\Omega\) satisfies the conditions (3-5), and the function \(u\) satisfies the conditions (1), (2) and the following ones: \[ u(k-1)\leq C u(k)\;\;\; (k\in \mathbb{Z}), \] \[ \sum_{j=1}^{\infty}\frac{u(k-j-1)\|\chi_{B(x,k)}\|_X}{\|\chi_{B(x,k-j-1)}\|_X}\leq Cu(k)\;\;\;(x\in K^n,k\in \mathbb{Z}), \] where \(C\) is a positive constant. Then the singular integral operator \(T_{\Omega}\) acts boundedly in the space \(M_u^X(K^n)\). An application of Theorem 2 for Morrey spaces built on Lorentz-Karamata spaces is given.
scientific article; zbMATH DE number 7617798
scientific article