  We prove duality theorems for the {\'e}tale cohomology of logarithmic
Hodge-Witt
  sheaves and split tori on smooth curves over a local field of positive
  characteristic. As an application, we obtain a description of the Brauer
group of the
  function fields of curves over local fields in terms of the characters of the
  idele groups. We also show that the classical Brauer-Manin pairing between
the Brauer
  and Picard groups of smooth projective curves over local fields has analogues
for
  arbitrary smooth curves, smooth projective curves with modulus and singular
projective
  curves over such fields.
