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


 * $$(\xi)_+ = \begin{cases} \xi, & \mbox{if } \xi > 0  \\ 0,  & \mbox{otherwise}. \end{cases}$$

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


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

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


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

Application to convolution operators
For $$\delta > 0$$ and $$n=1$$, $$S_R^\delta$$ and $$B_R^\delta$$ 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 $$L^p$$ spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to $$\delta = 0$$).

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


 * $$\delta \leq \tfrac{n-1}{2}$$

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

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

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

Define $$\delta (p)$$, the "critical index", as


 * $$\max( n|1/p - 1/2| - 1/2, 0)$$.

Then the Bochner–Riesz conjecture states that


 * $$\delta > \delta (p)$$

is the necessary and sufficient condition for a $$L^p$$ bounded Fourier multiplier operator. It is known that the condition is necessary.