We define and investigate the conjugacy operator for expansions with respect to the orthonormalized Jacobi polynomials. A general transplantation theorem for Jacobi series proved by Muckenhoupt is then applied to obtain weighted norm inequalities for conjugate Jacobi series.
We prove a transplantation theorem for Fourier-Bessel coefficients. Theorems of such type were proved byAskey andWainger  andAskey  for ultraspherical and Jacobi coefficients, respectively. Our theorem can be also seen as a dual result to a transplantation
theorem for Fourier-Bessel series which was proved byGilbert .
We prove weighted Lp-inequalities for the gradient square function associated with the Poisson semigroup in the multi-dimensional Hermite function expansions setting. In the proof a technique of vector valued Calderón-Zygmund operators is used.
We consider higher order Riesz transforms for the multi-dimensional Hermite function expansions. The Riesz transforms occur
to be Caldern--Zygmund operators hence their mapping properties follow by using results from a general theory. Then we investigate
higher order conjugate Poisson integrals showing that at the boundary they approach appropriate Riesz transforms of a given
function. Finally, we consider imaginary powers of the harmonic oscillator by using tools developed for studying Riesz transforms.