Commutators and related operators on harmonic Bergman space of \(\mathbb{R}_+^{n+1}\) (Q676202)

Let \(L^p= L^p(\mathbb{R}^{n+1}_+)\) \((1\leq p\leq\infty,\;n>1)\) be the Lebesgue spaces of complex-valued functions on \(\mathbb{R}^{n+1}_+\), and \(H\) be the harmonic Bergman space of all harmonic functions in \(L^2\), and \({\mathcal P}\) be the orthogonal projection from \(L^2\) onto \(H\). Let \(u_0= {\mathcal P}(f)\) for any \(f\in L^2\). Then there exists a unique solution set \(\{u_1,\dots, u_n\}\subset H\) (\(u_j\) will be denoted by \({\mathcal P}_j(f)\)) to the Cauchy-Riemann equations. The author considers the following operators on \(L^2\) together: \({\mathcal C}_b\), \({\mathcal C}_j= [b,{\mathcal P}_j]\), \({\mathcal P}_k{\mathcal C}_j- {\mathcal P}_j{\mathcal C}_k\), \(\sum^n_{j= 1}{\mathcal C}_j{\mathcal P}_j\), \(\sum^n_{j=1}{\mathcal P}_j{\mathcal C}_j\), for a given symbol \(b\), where \({\mathcal C}_b(f)= [b,{\mathcal P}](f)= b{\mathcal P}(f)-{\mathcal P}(bf)\), under the general assumption: ``\(b\in L^1\) is harmonic''. The author proves that the boundedness of each of the above operators in \(L^2\) implies the boundedness of all others and is equivalent to the condition: \(\sup\{| y(\partial b/\partial y)(x, y)|: (x,y)\in \mathbb{R}^{n+1}_+\}< \infty\). Second, he proves that the same statement is valid for the compactness of these operators and it is equivalent to the condition: ``The function \(b\) satisfies \(y(\partial b/\partial y)(x,y)\to 0\) uniformly in \(x\) as \(y\to 0^+\).'' He proves also that the bilinear form \((f,g)\mapsto\langle fg,b\rangle\) is bounded or compact on \(H\times H\) if and only if \(b\in L^\infty\) or \(b\equiv 0\), respectively. The main auxiliary tools of his proofs are certain new properties of \(k\)-powers of Laplacian and results on paracommutators obtained in 1988 by S. Janson, J. Peetre and L. Peng.