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

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

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

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

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.

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.