Bochner–Riesz operator

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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

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

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

B R δ f ( θ ) = ∑ k ∈ Z n | k | ≤ R ( 1 − | k | 2 R 2 ) + δ f ^ ( k ) e 2 π i k ⋅ θ . {\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 }.} {\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 f {\displaystyle f} {\displaystyle f} on R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} with Fourier transform f ^ ( ξ ) {\displaystyle {\hat {f}}(\xi )} {\displaystyle {\hat {f}}(\xi )}, the Bochner–Riesz means of complex order δ {\displaystyle \delta } {\displaystyle \delta }, S R δ f {\displaystyle S_{R}^{\delta }f} {\displaystyle S_{R}^{\delta }f} (where R > 0 {\displaystyle R>0} {\displaystyle R>0} and Re ( δ ) > 0 {\displaystyle {\mbox{Re}}(\delta )>0} {\displaystyle {\mbox{Re}}(\delta )>0}) are defined as

S R δ f ( x ) = ∫ | ξ | ≤ R ( 1 − | ξ | 2 R 2 ) + δ f ^ ( ξ ) e 2 π i x ⋅ ξ d ξ . {\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 .} {\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 .}

Application to convolution operators

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

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

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

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

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

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

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

Then the Bochner–Riesz conjecture states that

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

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

References

  1. Fefferman, Charles (1971). "The multiplier problem for the ball". Annals of Mathematics. 94 (2): 330–336. doi:10.2307/1970864. JSTOR 1970864.
  2. Sogge, C. D. (2008). "Lectures on eigenfunctions of the Laplacian". Topics in Mathematical Analysis. By Ciatti, Paolo; Gonzalez, Eduardo; Lanza de Cristoforis, Massimo; Leonardi, Gian Paolo. World Scientific. p. 347. ISBN 9789812811066.

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.