科研之友
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摘要
A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as follows:
Theorem Suppose f is an element of L(Sigma(n-1)), n >= 3. If f satisfies the condition
1/theta(n-1) integral(D(x,theta)) vertical bar f(y) - f(x)vertical bar dy = O(1/vertical bar log theta vertical bar), as theta -> 0+
at every point x in a set E of positive measure in Sigma(n-1), then the Cesaro means of critical order n-2/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.
