On directional Hilbert operators for regular quaternionic functions on \({\mathbb R}^3\) (Q605297)

In this paper, directional quaternionic Hilbert operators on the three-dimensional space \(\mathbb{H}_0 = \left< i,j,k \right> \cong \mathbb{R}^3\) are defined. Let \(\Omega\) be a smooth domain in \(\mathbb{C}^2\). The space \(\mathbb{C}^2\) is identified with the set \(\mathbb{H}\) of real quaternions by means of the mapping that associates the pair \((z_1,z_2) = (x_0 + i x_1, x_2+ix_3)\) with the quaternion \(q=z_1+z_2 j\). Moreover, \(\mathcal{R}(\Omega)\) denotes the class of (left-)regular (also called hyperholomorphic) functions \(f=f_1 + f_2 j : \Omega \rightarrow \mathbb{H}\) in the kernel of the Cauchy-Riemann operator \[ \mathcal{D} = 2 \left( \frac{\partial}{\partial \overline{z}_1} + j \frac{\partial}{\partial \overline{z}_2} \right) = \frac{\partial}{\partial x_0} + i \frac{\partial}{\partial x_1} + j \frac{\partial}{\partial x_2} -k \frac{\partial}{\partial x_3}, \] a variant of the Cauchy-Fueter operator. The space \(\mathcal{R}(\Omega)\) contains the identity mapping, and every holomorphic mapping \((f_1,f_2)\) on \(\Omega\) defines a regular function \(f=f_1+f_2j\). The space \(\mathcal{R}(\Omega)\) exhibits other interesting links with the theory of two complex variables. In particular, it contains the spaces of holomorphic maps with respect to any constant complex structure on \(\mathbb{H}\), not only the standard one. Let \(J_1\), \(J_2\) be the complex structures on the tangent bundle \(T\mathbb{H} \cong \mathbb{H}\) defined by left multiplication by \(i\) and \(j\). Let \(J_1^*\), \(J_2^*\) be the dual structure on the cotangent bundle \(T^*\mathbb{H} \cong \mathbb{H}\), and set \(J_3^* = J_1^* J_2^*\), which is equivalent to \(J_3 = - J_1 J_2\). Let \(J_p = p_1 J_1 + p_2 J_2 +p_3 J_3\) be the orthogonal complex structure on \(\mathbb{H}\) defined by a unit imaginary quaternion \(p = p_1 i +p_2 j +p_3 k\) in the sphere \(\mathbb{S}^2 = \{ p \in \mathbb{H} \;| \;p^2=-1 \}\). In particular, \(J_1\) is the standard complex stucture. Let \(L_p\) be the complex structure defined by left multiplication by \(p\), let \(d\) be the exterior derivative, and let \[ \overline{\partial}_p = \frac{1}{2} \Big( d + p J_p^* \circ d\Big) \] be the Cauchy-Riemann operator w.r.t. the structures \(J_p\) and \(L_p\). Moreover, let \(\text{Hol}_p(\Omega,\mathbb{H}) = \text{Ker\,}\overline{\partial}_p\) be the space of holomorphic maps from \((\Omega, J_p)\) to \((\mathbb{H},L_p)\). Then every element of the space \(\text{Hol}_p(\Omega,\mathbb{H}) \) is a regular function. The quaternionic Cayley transformation \(\psi(q)= (q+1)(1-q)^{-1}\) maps the unit ball \(B\) diffeomorphically to the right half-space \(\mathbb{H}^+ = \{ q \in \mathbb{H} \;| \;\text{Re\,}(q) >0 \}\), where the real part \(\text{Re\,}(q)\) of a quaternion \(q = x_0 + i x_1 + j x_2 + k x_3\) is \(x_0\). This Cayley transformation is combined with the directional, \(p\)-dependent, Hilbert operators \(H_p\) on the unit sphere \(S = \partial B\) of \(\mathbb{H}\) in order to define the three-dimensional directional Hilbert operator \(H_p^3\). A Sobolev-type space \(W \frac{1}{\partial_p} (\mathbb{H}_0,\mathbb{H})\) of \(\mathbb{H}\)-valued functions \(f\) of class \(L^2(\mathbb{H}_0)\) is defined in terms of the Cayley transformation. It is proved that for every \(p \in \mathbb{S}^2\), there exists a \(\mathbb{H}\)-linear bounded Hilbert operator \(H_p^3\) on the space \(W \frac{1}{\partial_p} (\mathbb{H}_0,\mathbb{H})\). For every \(f \in W \frac{1}{\partial_p} (\mathbb{H}_0,\mathbb{H})\), the function \(R_p^3(f):= f+ H_p^3(f)\) is the trace of a regular function on \(\mathbb{H}^+\). The functions \(f\) in the kernel of \(H_p^3\) are in one-to-one correspondence with the \(CR_p\)-functions on \(S\), the boundary values of holomorphic functions on \(B\) with respect to the operator \(\overline{\partial_p}\).