Hausdorff type operators and wavelet transform associated with the modified Whittaker transform

In the present paper, we introduce the Hausdorff type operators associated with the Whittaker operator $ L_{\alpha }=- \frac {1}{4}\left [x<^>2\frac {{\rm d}<^>2}{{\rm d}x<^>2} +\frac {[x<^>2m(x)]'}{m(x)}\frac {{\rm d}}{{\rm d}x}\right ] $ L alpha=-14[x2d2dx2+[x2m(x)]' m(x)ddx], where $ m(x)=x<^>{1-4\alpha }e<^>{-\frac {1}{2x<^>2}} $ m(x)=x1-4 alpha e-12x2; and we prove the boundedness of these operators in space $ L<^>{2}(\mathbb {R}_{+}, m(x)\,{\rm d}x) $ L2(R+,m(x)dx). We investigate continuous Whittaker wavelet transform, and obtain some useful results. The relation between Whittaker wavelet transform and the Hausdorff type operators here is also established. The properties of the adjoint of the Hausdorff type operators are discussed.