Transference for radial multipliers and dimension free estimates

Abstract

For a large class of radial multipliers on ${L^p}({{\mathbf{R}}^{\mathbf{n}}})$, we obtain bounds that do not depend on the dimension n. These estimates apply to well-known multiplier operators and also give another proof of the boundedness of the Hardy-Littlewood maximal function over Euclidean balls on ${L^p}({{\mathbf{R}}^{\mathbf{n}}})$, $p \geq 2$, with constant independent of the dimension. The proof is based on the corresponding result for the Riesz transforms and the method of rotations