Generalized Stieltjes operators S_beta,mu are bounded on T_p^(alpha)(t^alpha) for 0 < beta - 1/p < mu, commute and factorize with generalized Cesaro operators, have explicitly represented spectra, with analogous results on the real line linking to Hilbert and Fourier transforms.
Duoandikoetxea, The Hilbert transform and Hermite functions: A real variable proof of the L2-isometry , J
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Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces
Generalized Stieltjes operators S_beta,mu are bounded on T_p^(alpha)(t^alpha) for 0 < beta - 1/p < mu, commute and factorize with generalized Cesaro operators, have explicitly represented spectra, with analogous results on the real line linking to Hilbert and Fourier transforms.