Bochner–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.

References

References

1. Fefferman, Charles. (1971). "The multiplier problem for the ball". [[Annals of Mathematics]].
2. (2008). "Topics in Mathematical Analysis". World Scientific.