We work with Besov spaces with Lorentz smoothness \(B^s _{q} L_{p,r} (\mathbb {R}^n )\) . Here \(-\infty<s< \infty \) and \(0<p,q,r<\infty \) . We determine the dual of \(B^s _{q} L_{p,r} (\mathbb {R}^n )\) with the help of its characterization in terms of wavelets. In particular, when \(p=1\) and \(1<r<\infty \) , the dual spaces are new Besov spaces defined by using the limiting Lorentz sequence spaces \(\ell _{\infty ,r}\) . We apply the results to determine the dual of certain Triebel-Lizorkin-Lorentz spaces \(F^s _{q} L_{p,r} (\mathbb {R}^n )\) .
