Inequalities for two type potential operators on differential forms (Q483024)

The authors consider potential operators of convolution type acting on differential forms. For \(\Phi \in L^1_{\text{loc}}({\mathbb R}^n)\) nonnegative, the potential operator \(T_\Phi\) in question acts on differential forms as follows: \[ (T_\Phi u)(x) = \sum_I \left( \int_{{\mathbb R}^n} \Phi(x-y)u_I(y) \, dy \right) \, dx_I, \qquad u = \sum_I u_I \, dx_I, \] where the sum is taken over all ordered \(k\)-tuples \(I\). The function \(\Phi\) is assumed to satisfy a weak growth condition (condition D): there exist constants \(\delta>0\), \(C<\infty\) and \(0\leq\varepsilon<1\) so that \[ \sup_{\{x\,:\,2^k < |x| < 2^{k+1}\}} \Phi(x) \leq \frac{C}{2^{kn}} \int_{y\,:\,\{\delta(1-\varepsilon)2^k < |y| < 2 \delta (1+\varepsilon) 2^k\}} \Phi(y) \, dy \] for all \(k \in {\mathbb Z}\). Condition (D) was previously considered by \textit{C. Pérez} [Indiana Univ. Math. J. 43, No. 2, 663--683 (1994; Zbl 0809.42007)] in the context of two-weight norm inequalities for the action of \(T_\Phi\) on functions. The authors prove a two-weight norm inequality for the action of operators \(T_\Phi\) on differential forms, where \(\Phi\) satisfies the growth condition (D). Examples of these operators include Riesz and Bessel potentials. The authors also prove Cacciopoli-type inequalities for the action of certain integral type potential operators on \({\mathcal A}\)-harmonic tensors.