Spherical means, wave equations, and Hermite-Laguerre expansions
Let \(\mu_r\) denote the normalized surface measure of the sphere of radius \(r\) in \(\mathbb C^n\), and let \(W(\mu_r)\) denote the Weyl transform of this measure, which is thus an operator on functions on \(\mathbb R^n\). The authors show that the maximal operator \(\sup_{r > 0} | W(\mu_r) f|\) is bounded in \(L^p\) for all \(p > 2n/(2n-1)\), in analogy with the spherical maximal function theorems of Stein and Bourgain. The operator \(W(\mu_r)\) can be expanded in terms of the Hermite projections \(P_k\), and the coefficients, as a function of \(r\), are Laguerre functions of type \((n-1)\). The authors generalize their previous result by also considering operators whose Hermite projection coefficients are Laguerre functions of order \(\alpha\) for arbitrary \(\alpha\). As a consequence the authors prove the following pointwise convergence result. If \(u(x,t)\) solves the one-dimensional Cauchy problem \[ u_{tt} - u_{xx} + x^2 u + 1/2 u(x,t) = 0 \] with initial data \(u=0\), \(u_t = f\), then \(u(x,t)/t\) converges to \(f(x)\) as \(t \to 0\) for a.e. \(x\) provided that \(f \in L^1 \cap L^p\) for some \(p > 2\). Some other variants of the above results are also discussed.
scientific article; zbMATH DE number 1216761
scientific article