Abstract
We consider a new Sobolev type function space called the space with multiweighted derivatives $$ W_{p,\bar \alpha }^n $$ , where $$ \bar \alpha $$ = (α 0, α 1,…, α n ), α i ∈ ℝ, i = 0, 1,…, n, and $$ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $$ , $$ D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d} {{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n $$ We establish necessary and sufficient conditions for the boundedness and compactness of the embedding $$ W_{p,\bar \alpha }^n $$ ↪ $$ W_{q,\bar \beta }^m $$ , when 1 ⩽ q < p < ∞, 0 ⩽ m < n.