Symmetric Bessel multipliers (Q2884108)

The authors prove analogues of two classical results, namely, the Hörmander multiplier theorem and the Coifman-Meyer bilinear multiplier theorem, in the context of the symmetric Bessel transform.NEWLINENEWLINEConsider the matrix space \(M_{p, q}(\mathbb F)\), where \(\mathbb F\) is one of the fields \(\mathbb R, \mathbb C, \mathbb H.\) The action of \(U_p \times U_q\) (maximal compact for \(U(p, q)\) ) on \(M_{p, q}\) (identified as the tangent space at the identity coset for \(U(p ,q) / U_p \times U_q \)) leads to an orbit space which can be realized as NEWLINE\[NEWLINE \Xi_q = \{ \xi = (\xi_1, \dotsc, \xi_q) \in \mathbb R^q: \xi_1 \geq \dotsb \geq \xi_q \geq 0 \},NEWLINE\]NEWLINE which is a Weyl chamber of type \(B_q.\) Fundamental results of Rösler imply that \(\Xi_q\) (for certain values of \(q\)) has a convolution structure so that \(\Xi_q\) is a hypergroup. The characters of this hypergroup are identified with Bessel functions of Dunkl type which are associated with a root system of type \(B_q\), NEWLINE\[NEWLINE \xi \rightarrow J_k^{B_{q}}(\xi, i\eta),\quad \eta \in \Xi_q.NEWLINE\]NEWLINE A symmetric Bessel transform can be defined using these characters which lead to a Fourier theory on \(\Xi_q.\) The authors prove a Hörmander-type multiplier theorem and the boundedness of the bilinear multipliers associated to this transform, following the classical methods.