On the limiting case for boundedness of the \(B\)-Riesz potential in \(B\)-Morrey spaces. (English) Zbl 1199.42114
Summary: We consider the generalized shift operator associated with the Laplace-Bessel differential operator
\[
\Delta_B=\sum^n_{j=1}\frac{\partial^2}{\partial x^2_j}+\sum^k_{i=1}\frac{\gamma_i}{x_i}\, \frac{\partial}{\partial x_i},
\]
and study the modified \(B\)-Riesz potential \(\tilde I_{\alpha,\gamma}\) generated by the generalized shift operator acting in the \(B\)-Morrey space in the limiting case.
We prove that the operator \(\tilde I_{\alpha,\gamma}\), \(0 < \alpha < n + | \gamma| \), is bounded from the \(B\)-Morrey space \(L_{(n+| \gamma| -\lambda)/\alpha,\lambda,\gamma} (\mathbb R^n_{k,+} )\) to the \(B\)-BMO space \(\text{BMO}_\gamma (\mathbb R_{k,+}^n )\).
We prove that the operator \(\tilde I_{\alpha,\gamma}\), \(0 < \alpha < n + | \gamma| \), is bounded from the \(B\)-Morrey space \(L_{(n+| \gamma| -\lambda)/\alpha,\lambda,\gamma} (\mathbb R^n_{k,+} )\) to the \(B\)-BMO space \(\text{BMO}_\gamma (\mathbb R_{k,+}^n )\).
MSC:
| 42B30 | \(H^p\)-spaces |
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