Boundedness of Stein’s spherical maximal function in variable Lebesgue spaces and application to the wave equation

Abstract

If \({p(\cdot): \mathbb{R}^{n} {\rightarrow} (0,\infty), n\geq 3}\), is globally log-Holder continuous and its infimum p − and its supremum p + are such that \({\frac{n}{n-1} < p^{-} \leq p^{+} < p^{-} (n-1)}\), then the spherical maximal operator (integral averages taken with respect to the (n − 1)-dimensional surface measure) is bounded. When n = 3, the result is then interpreted as the preservation of the integrability properties of the initial velocity of propagation to the solution of the initial-value problem for the wave equation.