PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 144, Number 4, April 2016, Pages 1543–1552 http://dx.doi.org/10.1090/proc/12767 Article electronically published on December 21, 2015 ESTIMATES FOR RADIAL SOLUTIONS TO THE WAVE EQUATION JAVIER DUOANDIKOETXEA, ADELA MOYUA, AND OSANE ORUETXEBARRIA (Communicated by Joachim Krieger) Abstract. We consider the Cauchy problem for the wave equation with null initial position and radial initial velocity ψ. With the solution u of this prob- lem we define the operator S(ψ) = sup t>0 t −1 |u(x, t)|. Using simple one- dimensional operators to bound pointwise the operator S we obtain weighted L p estimates with power weights. Even for the unweighted estimates the result for dimension higher than 3 differs from the one with general functions. 1. Introduction Let us consider the Cauchy problem for the wave equation in R d for d ≥ 2: (1.1)  u tt − Δu =0, u(x, 0) = φ(x),u t (x, 0) = ψ(x). When d is odd a formula for the solution u(x, t) in terms of φ and ψ can be written using means on spheres centered at x with radius t and their derivatives. For even d the corresponding formula can be obtained from Duhamel’s method of descent. See two different ways of obtaining the solution in [5] and [11]. When the initial data φ and ψ are radial, the solution u is also radial in the spatial variable and the formula for u can be given in terms of one-dimensional integrals involving hypergeometric functions (see [2]). In this paper we limit ourselves to the case φ(x) ≡ 0 and ψ(x)= g(|x|). Following [2] the solution of the equation can be written in the form (1.2) u(r, t)=  ∞ 0 K(r, t; s)g(s)ds, where the kernel K is given in terms of Legendre functions. (With abuse of notation here and throughout we write u(r, t) instead of u(x, t) for |x| = r.) The odd- dimensional case is particularly simple, (1.3) K(r, t; s)= 1 2 r −(d−1)/2 s (d−1)/2 P d−3 2  s 2 + r 2 − t 2 2rs  χ (|r−t|,r+t) (s), where P k is the Legendre polynomial of degree k. Received by the editors October 16, 2014 and, in revised form, February 25, 2015. 2010 Mathematics Subject Classification. Primary 35L05; Secondary 35B05. Key words and phrases. Wave equation, radial solutions, weighted inequalities. This work was supported by grant MTM2011-24054 of the Ministerio de Econom´ıa y Compet- itividad (Spain) and grant IT-641-13 of the Basque Gouvernment. Adela Moyua (1956-2013). The author passed away during the production of this work. c 2015 American Mathematical Society 1543