The Discretization Of Weak Type Inequalities Of Maximal Convolution Operators

Many of the operators in Fourier are operators of convolution type. In order to study the properties of almost everywhere convergence of operator series,we consider weak type inequalities of the corresponding maximal operators. For the general maximal convolution operator K*,this paper has discussed the relations between weak type of K* and weak type of K* over finite sums of Dirac deltas,that is to say,the discretization of weak type inequalities of K*.When p=1,they are equivalent in the case of an ordinary sequence of convolution operators. However, p>1,the weak type (p,p) of K* over finite sums of Dirac deltas implies the weak type (p,p) of K*.The previous results have been obtained by Guzman.By implying Fejer kernel which has the properties of approximate identities,we give a new proof of a necessary condition of Guzman theorem.