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.

Define

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

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

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

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

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

For ${\displaystyle \delta >0}$ and ${\displaystyle n=1}$, ${\displaystyle S_{R}^{\delta }}$ and ${\displaystyle B_{R}^{\delta }}$ 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 ${\displaystyle L^{p}}$ spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to ${\displaystyle \delta =0}$).

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

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

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

Another question is that of for which ${\displaystyle \delta }$ and which ${\displaystyle p}$ the Bochner–Riesz means of an ${\displaystyle L^{p}}$ function converge in norm. This issue is of fundamental importance for ${\displaystyle n\geq 2}$, since regular spherical norm convergence (again corresponding to ${\displaystyle \delta =0}$) fails in ${\displaystyle L^{p}}$ when ${\displaystyle p\neq 2}$. This was shown in a paper of 1971 by Charles Fefferman.^[1]

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

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

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

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

Then the Bochner–Riesz conjecture states that

${\displaystyle \delta >\delta (p)}$

is the necessary and sufficient condition for a ${\displaystyle L^{p}}$ bounded Fourier multiplier operator. It is known that the condition is necessary.^[2]

• 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.