APDE seminar@BCAM: Power weight inequalities for spherical maximal functions

Abstract

The average of a Schwartz function f over the sphere of radius t and centre x in R^d is 

      A_t f(x) := ∫_{S^{d-1}} f(x + tω) dσ(ω),

where σ is the normalized surface measure on the sphere S^{d-1}. For any set of dilations E contained in (0,∞), the spherical maximal function M_E is defined by taking pointwise the supremum of the averages over t in E.

When E = (0,∞), Stein and Bourgain proved that M_E is bounded on L^p(R^d) if and only if p > d/(d-1) in dimensions d > 2 and d = 2, respectively. In fact, for any E, the lower bound on p depends on the Minkowski dimension of E, as shown by Seeger, Wainger, and Wright. Moreover, when E = (0,∞), Duoandikoetxea and Vega established the weighted L^p estimates for M_E with power weights |x|^α.

Jointly with Roos and Seeger we identified the relevant dimensional parameter for the weighted L^p estimates for any E, its Assouad spectrum, settling an open problem appearing in the work of Duoadikoetxea and Seijo.