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We prove duality theorems for the {\'e}tale cohomology of logarithmicHodge-Witt  sheaves and split tori on smooth curves over a local field of positive  characteristic. As an application, we obtain a description of the Brauergroup 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 betweenthe Brauer  and Picard groups of smooth projective curves over local fields has analoguesfor  arbitrary smooth curves, smooth projective curves with modulus and singularprojective  curves over such fields.

Duality for cohomology of split tori on curves over local fields