Higher order Riesz transforms associated with Bessel operators
The authors study Riesz transforms \(R_\mu^{(k)}\) of order \(k\geq 1\) related to the Bessel operator \[ \Delta_\mu := - \frac{\partial^2}{\partial x^2} -\frac{2\mu+1}{x} \frac{\partial}{\partial x}=D^* D, \quad \mu> -\frac{1}{2}, \] where \(D:=\partial/ \partial x\) and \(D^*= -x^{-2\mu-1}D x^{2\mu+1}\) denotes the adjoint operator of \(D\) in \(L^2(x^{2\mu+1}dx)\). The Riesz transform \[ R_\mu^{(k)} f(x) = \omega_k f(x) + \lim_{\varepsilon\to0}\int_{|x-y|>\varepsilon} f(y) R_\mu^{(k)} (x,y) y^{2\mu+1} \, dy, \quad x\in(0,\infty) \] where \(\omega_k\) are constants, is initially defined on the space of smooth functions having compact supporton \((0,\infty)\). Note that the kernel \(R_\mu^{(k)} (x,y)\) is connected with the Poisson kernel associated with the operator \(\Delta_{\mu}\). One of main results of the paper shows that for every \(p\in(1,\infty)\) the above defined operator \(R_\mu^{(k)}\) can be extended as a bounded operator from \(L^p(x^{2\mu+1}dx)\) into \(L^p(x^{2\mu+1}dx)\), and also as a bounded operator from \(L^{1}(x^{2\mu+1}dx)\) into \(L^{1,\infty}(x^{2\mu+1}dx)\). Moreover, weighted inequalities for \(R_\mu^{(k)}, k\geq 1\) are studied and the boundedness of the operator \(R_\mu^{(k)}\) in appropriate weighted spaces is established.
scientific article; zbMATH DE number 5357903
scientific article