The paper by Elias M. Stein aims to generalize a well-known convexity theorem of M. Riesz, originally deduced using "real-variable" techniques. The author builds on the work of Thorin, Tamarkin, Zygmund, and Calderón and Zygmund, focusing on the approach of Calderón and Zygmund. The main result is an interpolation theorem for linear operators, which allows for variations in both the Lebesgue spaces and the operators themselves. This theorem is then applied to two main areas: 1. **Mean Convergence Below the Critical Index**: The paper proves that Bochner-Riesz means of multiple Fourier series and integrals converge in the $L_p$ norm for $1 < p < \infty$ below the critical index. This is achieved by showing that the operators satisfy the conditions of Theorem 1, which guarantees the desired convergence. 2. **Pitt's Theorem for Orthonormal Systems**: The paper generalizes Pitt's theorem for Fourier series to uniformly bounded orthonormal systems. This generalization is significant because it holds under more general conditions and includes well-known inequalities such as those of F. Riesz and R. E. A. C. Paley, as well as a recent result by I. I. Hirschman. The proof of these results relies on the interpolation theorem and various inequalities, including those involving Bessel functions and Paley's inequalities. The paper concludes with detailed proofs and discussions of the implications of these results.The paper by Elias M. Stein aims to generalize a well-known convexity theorem of M. Riesz, originally deduced using "real-variable" techniques. The author builds on the work of Thorin, Tamarkin, Zygmund, and Calderón and Zygmund, focusing on the approach of Calderón and Zygmund. The main result is an interpolation theorem for linear operators, which allows for variations in both the Lebesgue spaces and the operators themselves. This theorem is then applied to two main areas: 1. **Mean Convergence Below the Critical Index**: The paper proves that Bochner-Riesz means of multiple Fourier series and integrals converge in the $L_p$ norm for $1 < p < \infty$ below the critical index. This is achieved by showing that the operators satisfy the conditions of Theorem 1, which guarantees the desired convergence. 2. **Pitt's Theorem for Orthonormal Systems**: The paper generalizes Pitt's theorem for Fourier series to uniformly bounded orthonormal systems. This generalization is significant because it holds under more general conditions and includes well-known inequalities such as those of F. Riesz and R. E. A. C. Paley, as well as a recent result by I. I. Hirschman. The proof of these results relies on the interpolation theorem and various inequalities, including those involving Bessel functions and Paley's inequalities. The paper concludes with detailed proofs and discussions of the implications of these results.