Improved endpoint bounds for the lacunary spherical maximal operator

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->L{(\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->L)}^{1+𝜖}$ for any $𝜖>0$.