Almost orthogonal operators on the bitorus II

作者：Prestini Elena^*

来源：Mathematische Zeitschrift, 2012, 271(1-2): 271-291.

DOI：10.1007/s00209-011-0863-1

• Carleson operator
• A.e. convergence of double Fourier series

We prove that the operator Tf(x, y) = integral(pi)(-pi) integral(vertical bar x%26apos;vertical bar%26lt;vertical bar y%26apos;vertical bar) e(iN(x, y)x%26apos;)/x%26apos; e(iN(x, y)y%26apos;)/(y)%26apos; f(x - x%26apos;, y - y%26apos;)dx%26apos;dy%26apos;, with x, y is an element of [0, 2 pi] and where the cut off vertical bar x%26apos;vertical bar %26lt; vertical bar y%26apos;vertical bar is performed in a smooth and dyadic way, is bounded from L-2 to weak-L2-epsilon, any epsilon %26gt; 0, under the basic assumption that the real-valued measurable function N(x, y) is %26quot;mainly%26quot; a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman%26apos;s proof of a.e. convergence of Fourier series of L2 functions.

• 出版日期2012-6