Bochner–Riesz mean

The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modification of the Riesz mean.

Definition

Define

[math]\displaystyle{ (\xi)_+ = \begin{cases} \xi, & \mbox{if } \xi \gt 0 \\ 0, & \mbox{otherwise}. \end{cases} }[/math]

Let [math]\displaystyle{ f }[/math] be a periodic function, thought of as being on the n-torus, [math]\displaystyle{ \mathbb{T}^n }[/math], and having Fourier coefficients [math]\displaystyle{ \hat{f}(k) }[/math] for [math]\displaystyle{ k \in \mathbb{Z}^n }[/math]. Then the Bochner–Riesz means of complex order [math]\displaystyle{ \delta }[/math], [math]\displaystyle{ B_R^\delta f }[/math] of (where [math]\displaystyle{ R \gt 0 }[/math] and [math]\displaystyle{ \mbox{Re}(\delta) \gt 0 }[/math]) are defined as

[math]\displaystyle{ B_R^\delta f(\theta) = \underset{|k| \leq R}{\sum_{k \in \mathbb{Z}^n}} \left( 1- \frac{|k|^2}{R^2} \right)_+^\delta \hat{f}(k) e^{2 \pi i k \cdot \theta}. }[/math]

Analogously, for a function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ \mathbb{R}^n }[/math] with Fourier transform [math]\displaystyle{ \hat{f}(\xi) }[/math], the Bochner–Riesz means of complex order [math]\displaystyle{ \delta }[/math], [math]\displaystyle{ S_R^\delta f }[/math] (where [math]\displaystyle{ R \gt 0 }[/math] and [math]\displaystyle{ \mbox{Re}(\delta) \gt 0 }[/math]) are defined as

[math]\displaystyle{ S_R^\delta f(x) = \int_{|\xi| \leq R} \left(1 - \frac{|\xi|^2}{R^2} \right)_+^\delta \hat{f}(\xi) e^{2 \pi i x \cdot \xi}\,d\xi. }[/math]

Application to convolution operators

For [math]\displaystyle{ \delta \gt 0 }[/math] and [math]\displaystyle{ n=1 }[/math], [math]\displaystyle{ S_R^\delta }[/math] and [math]\displaystyle{ B_R^\delta }[/math] may be written as convolution operators, where the convolution kernel is an approximate identity. As such, in these cases, considering the almost everywhere convergence of Bochner–Riesz means for functions in [math]\displaystyle{ L^p }[/math] spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to [math]\displaystyle{ \delta = 0 }[/math]).

In higher dimensions, the convolution kernels become "worse behaved": specifically, for

[math]\displaystyle{ \delta \leq \tfrac{n-1}{2} }[/math]

the kernel is no longer integrable. Here, establishing almost everywhere convergence becomes correspondingly more difficult.

Bochner–Riesz conjecture

Another question is that of for which [math]\displaystyle{ \delta }[/math] and which [math]\displaystyle{ p }[/math] the Bochner–Riesz means of an [math]\displaystyle{ L^p }[/math] function converge in norm. This issue is of fundamental importance for [math]\displaystyle{ n \geq 2 }[/math], since regular spherical norm convergence (again corresponding to [math]\displaystyle{ \delta = 0 }[/math]) fails in [math]\displaystyle{ L^p }[/math] when [math]\displaystyle{ p \neq 2 }[/math]. This was shown in a paper of 1971 by Charles Fefferman.

By a transference result, the [math]\displaystyle{ \mathbb{R}^n }[/math] and [math]\displaystyle{ \mathbb{T}^n }[/math] problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular [math]\displaystyle{ p \in (1, \infty) }[/math], [math]\displaystyle{ L^p }[/math] norm convergence follows in both cases for exactly those [math]\displaystyle{ \delta }[/math] where [math]\displaystyle{ (1-|\xi|^2)^{\delta}_+ }[/math] is the symbol of an [math]\displaystyle{ L^p }[/math] bounded Fourier multiplier operator.

For [math]\displaystyle{ n=2 }[/math], that question has been completely resolved, but for [math]\displaystyle{ n \geq 3 }[/math], it has only been partially answered. The case of [math]\displaystyle{ n=1 }[/math] is not interesting here as convergence follows for [math]\displaystyle{ p \in (1, \infty) }[/math] in the most difficult [math]\displaystyle{ \delta = 0 }[/math] case as a consequence of the [math]\displaystyle{ L^p }[/math] boundedness of the Hilbert transform and an argument of Marcel Riesz.

Define [math]\displaystyle{ \delta (p) }[/math], the "critical index", as

[math]\displaystyle{ \max( n|1/p - 1/2| - 1/2, 0) }[/math].

Then the Bochner–Riesz conjecture states that

[math]\displaystyle{ \delta \gt \delta (p) }[/math]

is the necessary and sufficient condition for a [math]\displaystyle{ L^p }[/math] bounded Fourier multiplier operator. It is known that the condition is necessary.

Further reading

• Lu, Shanzhen (2013). Bochner-Riesz Means on Euclidean Spaces (First ed.). World Scientific. ISBN 978-981-4458-76-4. 
• Grafakos, Loukas (2008). Classical Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09431-1. 
• Grafakos, Loukas (2009). Modern Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09433-5. 
• Stein, Elias M.; Murphy, Timothy S. (1993). Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press. ISBN 0-691-03216-5.