CONVOLUTION ESTIMATES AND GENERALIZED DELEEUW THEOREMS FOR MULTIPLIERS OF WEAK TYPE-(1,1)

In the context of a locally compact abelian group, we establish maxima l theorem counterparts for weak type (1, 1) multipliers of the classic al de Leeuw theorems for individual strong multipliers. Special method s are developed to handle the weak type (1, 1) estimates involved sinc e standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal the orem for convolutions with weak type (1, 1) multipliers which opens av enues of approximation. These results complete a recent series of pape rs by the authors which extend the de Leeuw theorems to a full range o f strong type and weak type maximal multiplier estimates in the abstra ct setting.