Jacobi polynomial estimates and Fourier-Laplace convergence (Q1384998)

The second named author [J. Beijing Norm. Univ., Nat. Sci. 29, No. 2, 143-154 (1993; Zbl 0799.43002)] reduced the convergence problems for Fourier-Laplace series on the sphere to the consideration of a certain equiconvergence operator defined by convolution with certain Jacobi polynomials. In the present paper, the authors derive new estimates for the Jacobi polynomials \(P^{(\alpha,\beta)}_k(x)\) with complex indices \(\alpha\), \(\beta\), where \(\text{Re }\alpha> -1\), \(\text{Re }\beta> -1\). By making use of Stein's interpolation theorem, they obtain \(L^p\) estimates for the corresponding maximal equiconvergence operator. As a consequence, they deduce the following remarkable Theorem 2: If \(n\geq 3\) and \[ \int_{\Omega_n}| f(x)|(1+ \log^2_+| f(x)|)dx< \infty, \] where \[ \Omega_n:= \{(x_1,\dots, x_n)\in \mathbb{R}^n: x^2_1+\cdots+ x^2_n= 1\}, \] then the Cesàro means \(S_N(f)\) at critical index of the Fourier-Laplace series of \(f\) converge almost everywhere.