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.


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