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For any Calderón–Zygmund operator   T  , any weight   w  , and    α  >  1   , the operator   T   is bounded as a map fromL  1    (   M   L    log    log    L    (  log    log    log    L  )   αw  )into weak‐    L  1    (  w  )    . The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman–Fefferman, Pérez, and Hytönen–Pérez, on the    L    (  log    L  )   ϵscale. Also, for square functions    S  f   , and weights    w  ∈   A  p    , the norm of   S   fromL  p    (  w  )to weak‐    L  p    (  w  )    ,    2  ⩽  p  <  ∞   , is bounded by[  w  ]A  p1  /  2(  1  +  log    [  w  ]    A  ∞    )    1  /  2, which is a sharp estimate.

Borderline weak‐type estimates for singular integrals and square functions