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Tomas
Tomas
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Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example of a function in $L^1[-\pi, \pi]$ with an a.e. divergent Fourier series. The problem is famously solved by Carleson in 1966, and it has at least twosome different proofs (Feffermann, Lacey-Thiele).

My question. What about higher dimension? It seems that this problem is currently out of reach. Can anyone explain why it's so difficult intuitively? And is there any related (positive or negative) result or notes on the higher dimensional Lusin problem?

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example of a function in $L^1[-\pi, \pi]$ with an a.e. divergent Fourier series. The problem is famously solved by Carleson in 1966, and it has at least two different proofs (Feffermann, Lacey-Thiele).

My question. What about higher dimension? It seems that this problem is currently out of reach. Can anyone explain why it's so difficult intuitively? And is there any related (positive or negative) result or notes on the higher dimensional Lusin problem?

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example of a function in $L^1[-\pi, \pi]$ with an a.e. divergent Fourier series. The problem is famously solved by Carleson in 1966, and it has at some different proofs (Feffermann, Lacey-Thiele).

My question. What about higher dimension? It seems that this problem is currently out of reach. Can anyone explain why it's so difficult intuitively? And is there any related (positive or negative) result or notes on the higher dimensional Lusin problem?

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Tomas
Tomas
  • 879
  • 1
  • 7
  • 12

High dimensional Lusin conjecture

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example of a function in $L^1[-\pi, \pi]$ with an a.e. divergent Fourier series. The problem is famously solved by Carleson in 1966, and it has at least two different proofs (Feffermann, Lacey-Thiele).

My question. What about higher dimension? It seems that this problem is currently out of reach. Can anyone explain why it's so difficult intuitively? And is there any related (positive or negative) result or notes on the higher dimensional Lusin problem?