We investigate the weighted boundedness for a class of local lacunary maximal operators on arbitrary homogeneous Lie groups. On \(\mathbb {R}^n,\) we analyse the \(L^p\) - \(L^q\) -boundedness for a class of local lacunary maximal functions on weighted spaces. Our analysis covers weighted dyadic maximal functions as the following ones: \( M_{D,\alpha }^{d\sigma }f(x)=\sup _{j\in \mathbb {Z}:j<0}|\frac{1}{2^{j\alpha }}\smallint \limits _{\mathbb {S}^{n-1}}f(x-2^{j}y)|x-2^{j}y|^{\alpha }d\sigma (y)|,\,\alpha <0, \) and also dyadic maximal operators of the form \( M_{D,\delta ,\rho }^{d\sigma }f(x)=\sup _{j\in \mathbb {Z}:j<0}|\smallint \limits _{\mathbb {S}^{n-1}}f(2^{j(\delta -\rho )}x-2^{j\delta }y)d\sigma (y)|,\,\,\delta <\rho , \) where \(d\sigma \) denotes the normalised surface measure on the sphere \(\mathbb {S}^{n-1}\subset \mathbb {R}^n\) . We also consider the weighted boundedness for a class of local lacunary maximal functions on homogeneous Lie groups, and we present applications of our results on the Heisenberg group for weighted local lacunary maximal functions defined by spherical averages associated with the Korányi sphere.
