On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

Abstract

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J. Fourier Anal Appl. 1:311-353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.

Pointwise Estimates for Block-Radial Functions of Sobolev Classes

“…The similar estimates in the case of homogeneous spaces can be found in [2] and [14]. The applications of block-radial functions to semi-linear elliptic equations can be found for example in [4,8,12] and [9].…”

“…We refer also to [14] and [13] for comparison of the behaviour at infinity of the radial function belonging to homogeneous and inhomogeneous Sobolev-Besov type spaces respectively.…”

The radial lemma of Strauss in the context of Morrey spaces

“…In many papers, we refer, for example, to the survey [30] or the articles [33], [38], [6], [36] and [34], it has been shown that one gets a better insight into the behavior of radial functions, if one replaces the Sobolev space W 1 p (R n ) by spaces of fractional order of smoothness, for instance, Bessel potential or Besov spaces. In such a framework, H 1 (R n ) can be replaced either by H s (R n ) with s > 1/2 or by B 1/2 2,1 (R n ) to guarantee the same conclusions as in the Strauss radial lemma above.…”

Weighted embedding theorems for radial Besov and Triebel–Lizorkin spaces

“…In the case of unweighted radial subspaces of Besov and Triebel-Lizorkin spaces, Sickel and Skrzypczak [26,27] and Sickel, Skrzypczak and Vybiral [28] obtained compactness of the related embeddings and an extension of Strauss' radial lemma.…”