Weighted Hardy inequalities of the weak type by Joan Cerd`a and Joaquim Mart´ ın Abstract We consider weak–type Hardy inequalities for weights on the half–line, and we prove that, when restricted to decreasing functions, the weak L p –type of the conjugate Hardy transform does not depend on p and that, under a doubling condition, it is equivalent to the corresponding strong L p –estimates. 1 Introduction All functions are assumed to be Lebesgue–measurable on R + = (0, ∞), ω will be a weight, i.e., a non–negative function, and W its indefinite integral, W (x)=  ∞ 0 ω(t) dt. We write T : X −→ Y to indicate that T is a bounded operator between X and Y , two function spaces on R + . Many weighted inequalities which originate some important classes of weights describe the boundedness of some classical transforms, such as Hardy– Littlewood maximal operator M , Hardy operator, S 1 f (x)= 1 x  x 0 f (t) dt, or its conjugate S 2 f (x)=  ∞ x f (t) dt t . For 1 <p< ∞, it was shown by Muckenhoupt [10] that M : L p (ω) −→ L p (ω) (1 <p< ∞).