Fractional integrals and the classical fractional maximal function are smoothing operators, in the sense that they map Lebesgue spaces into first order Sobolev spaces. We show that this phenomenon continues to hold for the fractional spherical maximal function when the dimension of the ambient space is greater than or equal to 5. A key element in the proof is a local smoothing estimate for the wave equation.
This is joint work with Joao P. G. Ramos and Olli Saari.
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