A counter-example in singular integral theory

作者：Difiore Lawrence B^*; Shapiro Victor L

来源：Studia Mathematica, 2012, 213(2): 157-167.

DOI：10.4064/sm213-2-3

• singular integrals
• spherical harmonic kernels
• singular in the L-p-sense

An improvement of a lemma of Calderon and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(\log r\), let f is an element of C-1(R-N\{0}) and suppose f vanishes outside of a compact subset of R-N, N %26gt;= 2. Also, let k(x) be a Calderon Zygmund kernel of spherical harmonic type. Suppose f (x) = O (\log r\) as r --%26gt; 0 in the L-p-sense. Set %26lt;br%26gt;F(x) = integral(RN) k(x - y)f(y)dy for all x is an element of R-N\{0}. %26lt;br%26gt;Then F(x) = O(log(2) r) as r --%26gt; 0 in the L-p-sense, 1 %26lt; p %26lt; infinity. A counter-example is given in R-2 where the increased singularity O(log(2) r) actually takes place. This is different from. the situation that Calderon and Zygmund faced.

• 出版日期2012