A CENTERED NORM INEQUALITY FOR SINGULAR INTEGRAL-OPERATORS

Let K be a standard singular integral kernel on R satisfying the usual Holder continuity condition of order delta, and define w(x) = c(1 + Absolute value of x)-(1+alpha) (where c is chosen so that the integral of w is 1), Tf = K *f, gBAR the mean of g with respect to the measure w(x) dx, and \\.\\p the L(p) norm with respect to w(x) dx. Although the inequality \\Tf\\p less-than-or-equal-to c(p)\\f\\p is not true in general, the centred norm inequality \\Tf - TfBAR\\p less-than-or-equal-to c(p)\\f\\(p) does hold for 1 < p < infinity if alpha < delta.