On the dimensional weak-type (1,1) bound for Riesz transforms

Let Rj denote the jth Riesz transform on Double-struck capital Rn. We prove that there exists an absolute constant C > 0 such that |{|R(j)f| > lambda}|<= C (1/lambda parallel to f parallel to(L1(Rn)) +sup(nu)|{|R-j nu| > lambda}| for any lambda > 0 and f is an element of L1(Double-struck capital Rn), where the above supremum is taken over measures of the form nu = Sigma(N)(k=1)a(k)delta(ck) for N is an element of N, c(k) is an element of R-n, and a(k) is an element of R+ with n-ary sumation k=1Na k <= 16 parallel to f parallel to L1(Double-struck capital Rn). This shows that to establish dimensional estimates for the weak-type (1, 1) inequality for the Riesz transforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderon-Zygmund operators.