Several Types of Intermediate Besov–orlicz Spaces

The Sobolev spaces have played an essential role in the research of partial differential equations ever since the mid thirties. In order to handle boundary value problems, and, in particular, traces, the more general scales of space have been introduced, namely Slobodeckii spaces and Besov spaces. It proved that Besov spaces are a suitable replacement for Sobolev spaces in many situations and that they are remarkably manageable by means of Fourier analysis. The standard general reference concerning these methods is [13]. The aim of this paper is to introduce and investigate several types of Besov spaces related to Orlicz spaces rather than Lp spaces. For a detailed discussion of Orlicz spaces the reader is refered to [4]. In Section 2 we introduce several classes of functions Φ used below as a replacement for tp from the classical Lp case. We also recall some basic useful properties such as characterization by indices, ∆2 conditions etc. In Section 3 we define the first type of Besov–Orlicz spaces, a most natural one, and point out several results analogous to those known from the classical theory. We deal with such properties as description of the spaces by differences or approximation, lifting properties, and real interpolation formulae. Here, and always below, the proofs when completely analogous to the Lp case are considered as superfluous and therefore omitted. We would like to emphasize the fact that no restriction on the growth of Φ such as ∆2 is needed in Section 3. The imbedding problems and the trace problems unlike the above mentioned ones lead to certain peculiar technical difficulties and it turns out that pointwise estimates are useful, which entails us sometimes to work with modulars rather than norms. For this purpose we introduce the so–called Besov–Orlicz classes. The definitions are justified by proving that the new spaces are the trace classes for Sobolev–Orlicz spaces. This all is done in Section 4, mainly in Subsection 4.3. Another interesting fact to observe is that our scale of spaces is considerably richer than as one could expect from the classical theory. We show in Section 5 that even in a remarkably simple case of Φ of power–logarithmic type there is no inclusion between two types of spaces. The products of type 0 · ∞ are considered as zero. The letter C denotes various constants not necessarily the same at each occurrence, but remaining independent of appropriate quantities.