## Limiting imbeddings. The case of missing derivatives.(English)Zbl 0930.46029

The main result (Theorem 3.2) is given in the form of an inequality $\|f|L_{\overline q}\|\leq c \prod^m_{j= 1} q^{1-1/p_j}\|f\mid S^{\overline r}_{\overline p,\overline 2} F\|,$ where $$\|f|L_{\overline q}\|$$ is the norm of the function $$f\in L_{\overline q}(\mathbb{R}^{n_1},\dots, \mathbb{R}^{n_m})$$ in case of mixed powers $$\overline q= (q_1,\dots, q_m)$$ and $$\|f|S^{\overline r}_{\overline p,\overline 2} F\|$$ is the norm of the respective tempered distribution $$f\in{\mathcal S}'$$ in the sense of the generalized Sobolev space $$S^{\overline r}_{\overline p,\overline 2} F(\mathbb{R}^{n_1},\dots, \mathbb{R}^{n_m})$$ defined by means of the inverse and direct Fourier transform with application of a standard resolution of unity associated to dyadic balls in $$\mathbb{R}^{n_j}$$, $$1\leq j\leq m$$. It is supposed that $$\overline p= (p_1,\dots, p_m)$$, $$1\leq p_j\leq q_j<\infty$$ and $$p_j= n_j$$, $$j= 1,2,\dots, m$$. This inequality produces the respective imbedding. Inequalities of the above form may be applied to the case of Orlicz spaces generated by a function $$\Phi$$ of exponential order.
There are also obtained interpolation results, both complex and real. There are deduced imbedding theorems for Sobolev spaces with selected derivatives.

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B70 Interpolation between normed linear spaces