×

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
PDF BibTeX XML Cite