Real interpolation with variable exponent

We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable Besov and Lorentz spaces as introduced recently in Almeida and H"ast"o (J. Funct. Anal. 258 (5) 1628–2655, 2010) and L. Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407–420).