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In this paper we give a complete description of the power weighted inequalities, of strong, weak and restricted weak type for the pair of Riesz transforms associated with the Laguerre function system $\{\mathcal{L}_k^{\alpha}\}$ , for any given α>-1. We achieve these results by a careful estimate of the kernels: near the diagonal we show that they are local Calderón–Zygmund operators while in the complement they are majorized by Hardy type operators and the maximal heat-diffusion operator. We also show that in all the cases our results are sharp.

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows ($X$,μ,$T$_{$t$}) and$f$∈$L$^{$p$}($X$,μ), there is a set$X$_{$f$}⊂$X$of probability one, so that for all$x$∈$X$_{$f$}, $$\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.

In this paper we study the spectral counting function for the weighted$p$-laplacian in one dimension. First, we prove that all the eigenvalues can be obtained by a minimax characterization and then we show the existence of a Weyl-type leading term. Finally we find estimates for the remainder term.