# Weak Type Estimates for Cesaro Sums of Jacobi Polynomial Series

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*Sagun Chanillo; Benjamin Muckenhoupt*

This work completely characterizes the behavior of Cesaro means
of any order of the Jacobi polynomials. In particular,
pointwise estimates are derived for the Cesaro mean kernel. Complete
answers are given for the convergence almost everywhere of partial sums
of Cesaro means of functions belonging to the critical \(L^p\)
spaces. This characterization is deduced from weak type estimates for
the maximal partial sum operator. The methods used are fairly general
and should apply to other series of special functions.

#### Table of Contents

# Table of Contents

## Weak Type Estimates for Cesaro Sums of Jacobi Polynomial Series

- Table of Contents vii8 free
- § 1. Introduction 110 free
- § 2. Facts and definitions 615 free
- § 3. An absolute value estimate for 3(1 y) ≤ 2(1 x) 1221
- § 4. A basic estimate for 3(1 y) ≤ 2(1 x) 1524
- § 5. A kernel estimate for 3(1 y) ≤ 2(1 x) and 1 ≤ θ ≤ 0 2029
- § 6. A reduction lemma 2332
- § 7. A kernel estimate for 3(1 y) ≤ 2(1 x) and 0 ≥ 1 2837
- § 8. A Cesaro kernel estimate for t ≤ s/2 3140
- § 9. A basic estimate for separated arguments 3544
- §10. A reduction lemma for separated arguments 3746
- §11. A kernel estimate for separated arguments 3948
- §12. Cesaro kernel estimate for t ≤ s b 4049
- §13. Cesaro kernel estimate for s near t 4251
- §14. Kernel estimates 4655
- §15. A weak type lemma 5160
- §16. Lemmas for the upper critical value 6271
- §17. Proofs of theorems (1.1) – (1.3) 7281
- §18. Norm estimates for p not between the critical values 7483
- §19. A polynomial norm inequality 7887
- §20. A lower bound for a norm of the kernel 8190
- §21. Some limitations of the basic results 8493
- §22. Growth of Cesaro means 8897
- §23. References 9099