On radial Fourier multipliers and almost everywhere convergence

We study almost everywhere ( a.e.) convergence on Lp and Lorentz spaces Lp,q , for variants of Riesz means at the critical index λ(p)=d(1/2–1/p)–1/2 for p>2d/(d–1) . For the classical Riesz means Stλ(p) , we show a.e. convergence for f∈Lp,1 . We derive more general results for radial and quasi‐radial Fourier multipliers and associated maximal functions, acting on L2 spaces with power weights and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted L2 spaces, and a sharp endpoint bound for Stein's square function associated with the Riesz means.

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