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Continuity properties of k-plane integrals and Besicovitch sets

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Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π denote its orthogonal complement. If xRn we shall write x = x0 + x with x0 ∈ Πand x ∈ Π . If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x )is defined as the integral of f over the affine subspace Π + x with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x ) regarded as a function of x for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x) with respect to x for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.

References
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(1)Besicovitch A. S.On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312320.
(2)Besicovitch A. S.On fundamental geometric properties of line sets. J. London Math. Soc. 39 (1964), 441448.
(3)Fisher B.On a problem of Besicovitch. Amer. Math. Monthly 80 (1975), 785787.
(4)Smith K. T., Solmon D. C. and Wagner S. L.Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Amer. Math. Soc. 83 (1977), 12271270.
(5)Solmon D. C.The X-ray transform. J. Math. Anal. Appl. 56 (1976), 6183.
(6)Solmon D. C.A note on k-plane integral transforms. J. Math. Anal. Appl. 71 (1979), 351358.
(7)Stein E. M. and Weiss G.Introduction to Fourier analysis on Euclidean space (Princeton University Press, 1971).
(8)Titchmarsh E. C.Introduction to the theory of Fourier integrals (Oxford, Clarendon Press, 1948).
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Mathematical Proceedings of the Cambridge Philosophical Society
• ISSN: 0305-0041
• EISSN: 1469-8064
• URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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