Bilinear Calderón-Zygmund operator and its commutator on some variable exponent spaces of homogeneous type

Let (X, d, mu) be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that a bilinear Calder & oacute;n-Zygmund operator is bounded from the product of variable exponent Lebesgue spaces L p 1 ( <middle dot> ) (X) x L p 2 ( <middle dot> ) (X) into spaces L p ( <middle dot> ) (X) , and it is bounded from the product of variable exponent generalized Morrey spaces L p 1 ( <middle dot> ) ,phi 1 (X) x L p 2 ( <middle dot> ) , phi 2 (X) into spaces L p ( <middle dot> ),phi (X) , where the Lebesgue measure functions So( <middle dot> , <middle dot> ), So 1 ( <middle dot> , <middle dot> ) and co 2 ( <middle dot> , <middle dot> ) satisfy co 1 x c0 2 = cO , and 1 p( <middle dot> ) = 1 p 1 ( <middle dot> ) + 1 p 2 ( <middle dot> ) . Furthermore, by establishing sharp maximal estimate for the commutator [b 1 , b 2 , BT] generated by b 1 , b 2 E BMO(X) and BT , the author shows that the [b 1 , b 2 , BT] is bounded from the product of spaces L p 1 ( <middle dot> ) (X) x L p 2 ( <middle dot> ) (X) into spaces L p ( <middle dot> ) (X ) , and it is also bounded from product of spaces L p 1 ( <middle dot> ) ,phi 1 (X) x L p 2 ( <middle dot> ) ,phi 2 (X) into spaces L p ( <middle dot> ) , phi (X) .