# Pointwise Estimates for Block-Radial Functions of Sobolev Classes

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## Abstract

The paper gives sharp pointwise estimates for functions belonging to \(\dot{H}^{s,p}(\mathbb {R}^N)\) with radial symmetry in *m* blocks of variables, for \(m<sp<N\). The estimates are formulated in terms of multiradial monomials. The form of the monomials depends on the structure of the group of block-radial symmetries and the distances of the given point to the hyperplanes in \(\mathbb {R}^N\) that contain the singular orbits of the group. For some exceptional set of parameters the logarithmic factor is needed. Weak continuity related to the estimates is also considered.

## Keywords

Multiradial functions Sobolev spaces Strauss inequality Pointwise estimates## Mathematics Subject Classification

Primary 46E35 Secondary 46B50 46N20 42C99## 1 Introduction

*C*such that

A preliminary and more coarse pointwise estimate for multiradial functions in \(\dot{H}^{1,p}(\mathbb {R}^N)\) with the right hand side as in (2.9) below has been previously given by the authors in Corollary 1, [17], for a subspace of \(\dot{H}^{1,p}(\mathbb {R}^N)\). For a large range of parameters this subspace is strictly smaller than \(\dot{H}^{1,p}(\mathbb {R}^N)\), and perhaps more significantly, for a large range of parameters inequality (2.9) is not an optimal estimate. In this paper we give a pointwise estimate for multiradial functions in \(\dot{H}^{1,p}(\mathbb {R}^N)\) by a monomial of block radii \(r_i(x)\), \(i=1,\dots , m\), with exponents that differ in different cones \(\{x\in \mathbb {R}^N: r_{i_1}(x)\ge \dots \ge r_{i_m}(x)\}\). These monomials are dominated by the right hand side of (2.9) (where the exponents in the monomial are the same for the whole space). For exceptional values of parameters, the estimate has to be amended by an additional logarithmic factor. In addition to the pointwise estimate (which we also generalize to the spaces \(\dot{H}^{s,p}\)), and to its optimality, we prove cocompactness of a related (non-compact) embedding relative to the group of dilations at the origin.

*m*-tuple \(\gamma =(\gamma _1,\ldots , \gamma _m)\), \(\gamma _1+\cdots + \gamma _m = |\gamma |= N\). The

*m*-tuple \(\gamma \) describes decomposition of \(\mathbb {R}^{|\gamma |}= \mathbb {R}^{\gamma _{1}}\times \dots \times \mathbb {R}^{\gamma _{m}}\) into

*m*subspaces of dimensions \(\gamma _{1},\dots ,\gamma _{m}\) respectively. Let

We will denote a subspace of any space *X* of functions on \(\mathbb {R}^{|\gamma |}\) consisting of functions invariant with respect to the action the group \(SO(\gamma )\) as \(X_{\gamma }\). If \(SO(\gamma )=SO(N)\), then we will write \(X_\mathrm {rad}\) since in that case the subspace consists of radial functions.

The spaces of our concern here are homogeneous Sobolev spaces of invariant functions \(\dot{H}_\gamma ^{s,p}(\mathbb {R}^{|\gamma |})\), \(s>0\), \(p>1\), defined as the completion of \(C_{0,\gamma }^{\infty }(\mathbb {R}^{|\gamma |})\) in the norm \(\Vert u\Vert _{s,p}=\Vert (-\Delta )^su\Vert _{p} =\Vert \mathcal {F}^{-1}(|\xi |^s \mathcal {F}u)\Vert _{p}\), which generalizes the norm \(\Vert \nabla u\Vert _p\) in the case \(s=1\). The space \(\dot{H}_\gamma ^{s,p}(\mathbb {R}^{|\gamma |})\) can be identified as subspace of homogeneous Sobolev space \(\dot{H}^{s,p}(\mathbb {R}^{|\gamma |})\) defined as the completion of \(C_0^\infty (\mathbb {R}^{|\gamma |})\). The space \(\dot{H}^{s,p}(\mathbb {R}^{|\gamma |})\) is a spaces of functions if \(sp<N\), cf. Theorem 3.11 and Proposition 3.41 in [20].

The main technical tool used in the paper is the method of atomic decompositions. Strauss type inequalities for Sobolev spaces of integer smoothness can be also proved by more elementary methods, and, as mentioned above, such inequalities were obtained by authors in [17] \(s=1\). However, the inequality that could be obtained by a simpler argument, (2.9), is less sharp than (2.8), and is verified for, generally, a more narrow class of functions. The main objective of using the approach of this paper was to refine the estimates of [17] for the classical Sobolev spaces, but as it happened, generalization to Sobolev spaces of the fractional did not complicate the proofs.

Precise pointwise estimates of radial functions belonging to different inhomogeneous spaces with fractional smoothness can be found in [13]. 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].

## 2 Main Results

*x*belonging to principal orbits. (cf. (2.8)), where the following notations are used:

*J*of \(\{1,\dots ,m\}\) we define the associated effective dimension

*x*of order \(s-\frac{N}{p}\).

### Lemma 2.1

### Proof

### Remark 2.2

The notation above brings up the matter of regularity of the functions in \(\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\). Let \(f\in \dot{H}^{s,p}_\gamma (\mathbb {R}^N)\), \(p>1\), \(m<sp<N\). If \(R_m(x)\) is bounded away from zero, then *f* is locally a \(H^{s,p}\)-function of *m* variables \(r_1,\dots ,r_m\) and is therefore continuous in such region since \(m<sp\).

Without loss of generality let us assume that \(r_1(x)\ge \dots \ge r_m(x)\). Whenever \(r_j(x)=\dots =r_k(x)\), also without loss of generality, we assume that \(\gamma _j\le \dots \le \gamma _k\). Consider a region where \(R_n(x)= R_J(x)\), \(J={\{n+1,\ldots , m\}}\), is bounded away from zero. In such region, *f* can be considered locally as a \(\dot{H}^{s,p}\)-function of \(d_n= d_J\) variables \(r_1,\dots ,r_n, x_{1+\sum _{i_=1}^n\gamma _i},\dots , x_N\), and therefore *f* is continuous whenever \(R_n(x)\ne 0\) and \(d_n<sp\). Such region contains all orbits of the form \(\Gamma =\{x:\,r_{n+1}(x)=\dots =r_m(x)=0, r_i(x)=\rho _i>0, i=1,\dots n\}\), and therefore the number \(d_n\) can be called *effective dimension* of such orbits.

Since \(d_n\) is a monotone decreasing function of *n*, \(d_m=m\), and \(d_0=N\), there exists \(n^*\in \{1,\dots ,m\}\), which is the smallest *n* such that \(d_n < sp\). The function *f* is continuous whenever \(d_{n*}< sp\) and \(R_{n^*}(x)\ne 0\), but may be discontinuous at the orbits where \(R_{n^*}\) equals 0. The similar statement holds for any permutation of indices.

*x*can belong to several sets \(\mathcal {C}_J\) if \(r_J(x)\) coincides with several values of \(r_j(x)\), \(j\notin J\). Note however that the expression (2.4) is independent of the ordering by the values of \(r_i(x)\) that coincide.

*J*,

*s*and

*p*we define a subdomain \(\mathcal {C}(J,s,p)\) of \(\mathcal {C}_J\) by

*sp*separates consecutive effective dimensions.

We first give a pointwise estimate for points *x* that belong to principal \(SO(\gamma )\)-orbits, i.e. if \(R_m(x)\not =0\), and when *sp* does not take any of the values \(d_I\), \(I\subsetneqq \{1,\dots ,m\}\), in particular when \(sp\notin \mathbb {N}\).

### Theorem 2.3

Let \(s>0\), \(m\in \mathbb {N}\), \(p>1\), \(m<sp<N\) and assume that \(\gamma _i\ge 2\), \(i=1,\dots ,m\). Assume also that \(sp\ne d_I\) for any \(I\subset \{1,\dots ,m\}\).

(i) If \(R_m(x)\not =0\) then there exists \(J\subsetneqq \{1,\ldots , m\}\) such that \(d_J< sp\) and \(x\in \mathcal {C}(J,s,p)\).

The above theorem is the simplified version of Theorem 4.1 in Sect. 4. We formulate here this version for the convenience of the reader.

We also state a simpler, but more coarse, estimate.

### Theorem 2.4

### Remark 2.5

- (1)
Note that the right-hand side of (2.8) defines on the set \(\cup _{J\subsetneqq \{1,\ldots , m\},d_J<sp}\mathcal C(\gamma ,s,p)=\{x\in \mathbb {R}^N:\,R_m(x)>0\}\) a continuous positive-homogeneous function of degree \(s-\frac{N}{p}\). This is the same homogeneity as \(|x|^{s-N/p}\) that appears in the right hand side of the Strauss inequality, which is in fact the case \(m=1\) of (2.8) and (2.9).

- (2)The above mentioned homogeneity implies, with \(\hat{x}=\frac{x}{\Vert x\Vert }\), thatfor \(x\in \mathcal {C}(J,s,p)\).$$\begin{aligned} |f(x)|\le C R_J(\hat{x})^{-1/p} r_{J}(\hat{x})^{s-d_J/p} \Vert x\Vert ^{s-\frac{N}{p}} \Vert f\Vert _{s,p}. \end{aligned}$$
- (3)
For the case \(s=1\) the estimate (2.9) for a subspace of \(\dot{H}^{1,p}_\gamma (\mathbb {R}^N)\) was given previously as Corollary 1 in [17]. Part (ii) of the corollary extends it to the whole \(\dot{H}^{1,p}_\gamma (\mathbb {R}^N)\) under an additional assumption \(\min _i\gamma _i\ge p\). The authors would also like to bring it to the attention of the reader that Proposition 1 of [17] that alleges that the subspace of the functions considered in [17] cannot be generally enlarged to \(\dot{H}^{1,p}_\gamma (\mathbb {R}^N)\), contains a computational error (a request for publication an erratum has been made) and should be ignored.

For certain ranges of parameters inequality (2.8) takes a simpler form. In particular, if *sp* is sufficiently close to *N* (in particular, if \(m=1\)), the estimate is the same as for radial functions.

### Corollary 2.6

Assume the conditions of Theorem 2.3.

### Proof

*i*such that \(x\in \mathcal {C}(J_i,s,p)\). But in that case \(r_{J_i}(x)=R_{J_i}(x) \sim \Vert x\Vert \). This proofs (i).

Analogously the assumption (ii) implies that \(x\in \mathcal {C}(\emptyset ,s,p)\) if \(R_m(x)\not =0\) since \(d_\emptyset =m<sp< d_{\{i\}}\) for any *i*. But \(R_{\emptyset }(x)=R_{m}(x)\) and \(r_{\emptyset }(x)=r_\mathrm{min}(x)\). This proofs (ii). \(\square \)

The second assertion of Theorem 2.3 is restated in Sect. 4 as Theorem 4.1, and for the exceptional region characterized by \(sp=d_J\) a similar estimate, but with a logarithmic term, is provided by Theorem 4.2. Corollaries 4.3 and 4.4 simplify the respective statements of Theorems 4.1 and 4.2 in the bi-radial case (\(m=2\)).

Further results of the paper are dealing with optimality of the estimate in Theorem 4.1 and with weak continuity properties of the estimate, in Sects. 5 and 6 respectively.

## 3 Preparations: Atomic Decomposition

We assume that \(N\ge 2\), \(\gamma _i\ge 2\) for any \(i=1,\ldots , m\).

To prove our statements we use the approach of atomic decomposition. In the context of Sobolev and Besov spaces the method goes back to the seminal papers by M. Frazier and B. Jawerth, cf. [6, 7]. The atomic decomposition adapted to the radial case was constructed by J. Epperson and M. Frazier in [5]. Here we use slightly different approach described in [16]. It should be pointed out that we use the atomic decomposition of inhomogeneous spaces and afterwards we use the homogeneity argument to transfer the result to homogeneous spaces. This can be done since estimating functions are homogeneous. In contrast to [5] our atoms are supported on balls not on annuli centred at zero. Here we recall the main idea of the method, with the needed modifications, and we refer the reader to [16] for more details.

We start with notions of separation and discretization in \(\mathbb {R}^N\) since they are needed for description of the atomic decomposition. Let *B*(*x*, *r*) denote the ball of radius *r* in \(\mathbb {R}^N\).

### Definition 3.1

Let \(\varepsilon >0\) be a positive number, \(\alpha =1,2,\ldots \) be a positive integer and *X* a nonempty subset of \(\mathbb {R}^N\).

(a) A subset \({\mathcal H}\) of *X* is said to be \(\varepsilon \)–separation of *X*, if the distance between any two distinct points of \(\mathcal H\) is greater than or equal to \(\varepsilon \).

*X*is called an \((\varepsilon ,\alpha )\)–discretization of

*X*if it is an \(\varepsilon \)–separation of

*X*and

### Remark 3.1

1. Both notations are well known and important in geometry eg. cf. [1, Chapter 4]. Please note, that our notion of discretization is a bit different to that one in Chavel’s book.

2. Let *m* be a positive integer. If \(\mathcal H\) is an \((\varepsilon ,\alpha )\)–discretization of \(\mathbb {R}^N\) and \(m\ge \alpha \), then the family \(\{B(x,m\varepsilon )\}_{x\in {\mathcal H}}\) is a uniformly locally finite covering of \(\mathbb {R}^N\) whose multiplicity is bounded from above by a constant depending on *N* and *m*, but independent of \(\varepsilon \).

We describe the needed discretizations related to the group \(SO(\gamma )\). In this case we can proceed in the following way. Let \(\{ x^{(j,i)}_{k,\ell }\}\), \(\ell =1,\ldots , \max \{1,k\}^{\gamma _i-1}\) and \(k\in \mathbb {N}_0\), be a \((2^{-j},\alpha _i)\)-discretization in \(\mathbb {R}^{\gamma _i}\) constructed in [15, Sect. 3.2], with values of \(x^{(j,i)}_{0,0}\) set instead of zero, as in [15], to \(|x^{(j,i)}_{0,0}|=2^{-2j}\).

*p*)-atom centered at the point \(x^{(j)}_{\mathbf {k},\mathbf {l}}\in {\mathcal H}_j\) if:

### Proposition 3.2

Let \(s>0\) and \(1\le p,q\le \infty \). Let \({\mathcal H}_j\) be a sequence of discretizations described above. Then

(ii) Conversely, any distribution represented by (3.4) with (3.5) belongs to \(B^{s,p}_{q,\gamma }(\mathbb {R}^N)\).

For the proof we refer to [16], cf. Theorem 1, Lemma 2, Remark 7 and Step 2 of the proof of Theorem 2 ibidem. A similar proposition holds for the case of non-positive smoothness index (\(s\le 0\)), but in this case additional assumptions concerning the atoms, so call moment conditions, are needed. We refer once more to [16].

## 4 Pointwise Estimate for Multiradial Functions

In this section we will state and prove the main result in full generality, that is, including the exceptional cases when a logarithmic term appears. Let \(s>0\) and \(1<p<\infty \) be such that \(m<sp<N\).

First we consider the case when the value of *sp* is distinct from the effective dimension of any orbit.

### Theorem 4.1

Let \(s>0\), \(m\in \mathbb {N}\), \(p>1\), \(m<sp<N\). Assume that \(\gamma _i\ge 2\), \(i=1,\dots ,m\). Let \(J\subsetneqq \{1,\dots ,m\}\) and \(d_J< sp\). We assume moreover that \(sp\not =d_I\) for any set *I* such that \(J\subset I\subset \{1,\ldots ,m\}\). Then there exists \(C>0\), \(C=C(\gamma ,s,p)\), such that for every \(f\in \dot{H}^{s,p}_\gamma (\mathbb {R}^N)\), and every \(x\in {\mathcal C}(J,s,p)\) the estimates (2.8) hold.

We now formulate the result for the logarithmic pointwise estimate that occurs in the regions whose effective dimension coincides with the value of *sp*.

### Theorem 4.2

Let \(s>0\), \(m\in \mathbb {N}\), \(p>1\), \(m<sp<N\). Assume that \(\gamma _i\ge 2\), \(i=1,\dots , m\). Let \(J\subsetneqq \{1,\dots ,m\}\) and \(d_J< sp\).

As the statements of the theorems above are rather complicated, we would like to give some corollaries with simpler statements. One such corollary is the already stated Theorem 2.4. The following two corollaries are straightforward elaborations, respectively, of Theorem 4.1 and of Theorem 4.2 in the bi-radial case (\(m=2\)).

### Corollary 4.3

Let \(s>0\), \(m=2<sp<N\), and assume that \(\gamma _i\ge 2\), \(i=1,2\).

### Corollary 4.4

Let \(s>0\), \(m=2<sp<N\), and assume that \(\gamma _i\ge 2\), \(i=1,2\).

In order to prove Theorems 4.1 and 4.2 we rewrite their assertions as the theorem below for the case when the values \(r_i(x)\) are ordered in *i*, and prove the latter instead.

### Theorem 4.5

Let \(s>0\), \(p>1\), \(m\in \mathbb {N}\), \(m<sp<N\), and assume that \(\gamma _i\ge 2\), \(i=1,\dots ,m\). Let *n*, \(1\le n\le m\), be the smallest integer such that \(d_n = \sum _{i=n + 1}^{m}\gamma _i+ n < sp\). Let \(x\in \mathbb {R}^N\) with \(r_1(x)\ge r_2(x)\ge \ldots r_m(x)\) and \(R_{n}(x)>0\).

### Proof

We first prove the inequalities (4.8) and (4.9) for \(f \in B^{s,p}_{\infty , \gamma }(\mathbb {R}^N)\) and \(x\not =0\), \(\max \{r_i(x): i=1,\ldots , n\}\le 1\), \(n\le m\).

By the assumption \(x\in {\mathcal C}_J\) where \(J=\{n+1, \ldots ,m\}\). We assume in addition that \(r_i(x)\le 1\), \(i=1,\ldots ,m\). If \(r_i(x)>0\), then one can find \(j_i\in \mathbb {N}\) such that \(2^{-j_i-1}\le r_i(x)\le 2^{-j_i+1}\). The inequality \(r_i(x)>0\) is satisfied for \(i=1,\dots , \nu \), with some \(\nu \), \(n\le \nu \le m\). We assume that \(\nu \) is the largest integer with this property. We may assume that \(j_{1}\le j_2\le \dots \le j_{\nu }\).

*f*with

*x*belongs to \(\text{ supp } a_{j,\mathbf {k},\mathbf {l}}\) if

*j*into \(\nu +1\) intervals \(0=j_0\le j< j_1 \), \(j_{\ell }\le j< j_{\ell +1}\), \(\ell =0,\dots \nu -1\), and \(j_\nu \le j<\infty \). Some of these intervals may be empty, and the corresponding sums are assigned value zero.

*x*, we have

*C*depends only on the atomic decomposition and

*n*, in particular it is independent of \(j,\ell \) and \(j_\ell \). Thus for any \(C_{j,\mathbf {k}}\) that appear in the right hand side of (4.16) we have, given our assumption that \(r_i(x)\le 1\),

*C*is independent of \(j,\ell \) and \(j_\ell \).

If \(0\le \ell < n\) then \(d_n < d_\ell \). Tedious, but elementary computations, which we have confined to Lemma 4.6 below, show that if \(sp\ne d_{n-1}\), then the sums \(\sum _{j=j_{\ell }}^{j_{\ell +1}-1}\) are dominated by the sum \(\sum _{j=j_{n}}^{j_{n+1}-1}\).

Note that the term \(r_1(x)^\frac{p-N}{p}\) is in fact \(r_{\max }(x)^\frac{p-N}{p}\), and since \(\ell ^\infty \)-norm in \(\mathbb {R}^N\) is equivalent to the Euclidean norm, this term is equivalent to \(|x|^\frac{p-N}{p}\).

*Q*, positively homogeneous of degree \(N-sp\) that appears in the right hand side of (4.8) and (4.9), whenever \(|x|\le 1\).

*z*, and taking the limit at \(t\rightarrow \infty \), we have

In order to complete the argument above we have to prove the following elementary technical statement. We recall that any sum over an empty set is understood as zero.

### Lemma 4.6

Let \(1<n\le m\) and \(d_{n} < sp \le d_{n-1}\). Assume that \(j_1\le j_2\le \dots \le j_{n}\), and that for some \(q>0\) we have \(2^{-j_i-q}\le r_i(x)\le 2^{-j_i+q}\), \(i=1,\dots ,n\).

### Proof

The proof is based on the observation that the sums in (4.22) and (4.23) are geometric sums and thus are evaluated by the upper term or by the lower term and by the number of terms if the power is zero. Observe that the mapping \(\ell \mapsto d_\ell \) is monotone decreasing on \(\{0,\dots , n\}\) from *N* to \(d_n\).

### Proof of Theorem 2.4

### Lemma 4.7

### Proof

If in (4.27) all inequalities are strict, then the relations (4.27) are satisfied only by one permutation and in consequence *x* belongs only to one set \(\mathcal {C}(J,s,p)\). However if some equalities occur in (4.27) then another permutation is possible. Let \((i_1,\ldots ,i_m)\) be such permutation. We may assume that the last permutation is an inversion of \((j_1,\ldots ,j_m)\) i.e. \(i_m=j_{m+1}\) and \(i_{m+1}=j_{m}\) for some *m*. If (4.27) holds for \((i_1,\ldots ,i_m)\) then \(r_{i_m}(x)=r_{i_{m+1}}(x)=r_{j_m}(x)=r_{j_{m+1}}(x)\). So for \(m\not = k\) we have \(R_J(x)=R_I(x)\), \(r_J(x)=r_I(x)\) and \(d_{J}=d_{I}\), \(d_{J\cup \{j_x\}}=d_{I\cup \{i_x\}}\). If \(m=k\) then simple calculations prove (4.25) and (4.26). \(\square \)

### Proposition 4.8

Let \(s>0\), \(m\in \mathbb {N}\), \(p>1\), \(m<sp<N\). Assume that \(\gamma _i\ge 2\), \(i=1,\dots , m\). Then for every \(f\in \dot{H}^{s,p}_\gamma (\mathbb {R}^N)\) the function \(W_{sp}f\) is continuous on \(\mathbb {R}^N\).

### Proof

*f*in \(\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\). But then by (4.28) the sequence \(W(x)f_n(x)\) is uniformly convergent to

*W*(

*x*)

*f*(

*x*). \(\square \)

## 5 Optimality of the Estimate

We prove that the estimates (2.8) are optimal.

### Proposition 5.1

### Proof

Without loss of generality, it suffices to prove the optimality of (4.8). This means that we assume that \(r_1(x)\ge \cdots \ge r_m(x)>0\). Let *n* be such that \(d_{n}<sp<d_{n-1}\) \(d_k=\sum _{i=k+1}^{m}\gamma _i+k\). We consider a point \(z\in \mathbb {R}^N\) with \(r_i(z)=2^{-j_i}\) for some \(j_i\in \mathbb {N}\), \(i=1,\dots ,m\) satisfying \(j_{1}\le j_2\le \dots \le j_{n}=j_{n+1}=\dots =j_m\) .

*C*depends on multi-index \(\beta \) but it is independent of \(j_0, \mathbf {k}, \ell \). Let us now choose

*q*such that \(1> \alpha 2^{-q+2}+2^{-q+1}\). If \(x\in B(x_{j_0,\mathbf {k},\ell },\alpha 2^{-j_0})\) and the ball \(B(x_{j_0,\mathbf {k},\ell },\alpha 2^{-j_0})\) has a nonempty intersection with \(\Omega _{j_1,\dots , j_n}\) then \(r_n(x) \ge 2^{-j_0}\).

*z*.

*z*. Now if we put

## 6 Cocompactness and Defect of Weak Convergence

*j*, but converges weakly to zero whenever \(|j|\rightarrow \infty \). On the other hand, we have compactness of the following trace embedding.

### Lemma 6.1

### Proof

Let \(A'_1=\{x\in \mathbb {R}^N:\;1/2<|x|<4\}\) and let \(\chi \in C_0^\infty (A'_1)\) be a function that equals 1 on \(A_1\). Then for every \(s'\in (s/p,s)\) the multiplication operator \((Tu)(x)=\chi (x) u(x)\) is a compact operator \(T:\,\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\rightarrow \dot{H}^{s',p}_\gamma (\mathbb {R}^N)\). Since the embedding \(\dot{H}_\gamma ^{s',p}(\mathbb {R}^N)\hookrightarrow L^\infty (W_{s'p},\mathbb {R}^N)\) is continuous, *T* is a compact operator from \(\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\) to \(L^\infty (W_{s'p},\mathbb {R}^N)\), which implies that the trace embedding \(\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\hookrightarrow L^\infty (W_{s'p},A_1)\) is compact. \(\square \)

The following counterexample shows that the assertion of Lemma 6.1 becomes false if we replace \(s'\) with *s*.

### Proposition 6.2

Let \(m<sp<N\) and \(s\not = d_J\) for any \(J\subset \{1,\ldots , m\}\}\). Let \(A= \{x\in \mathbb {R}^N :\; c_1<|x|<c_2\}\), \(0<c_1< c_2<\infty \). There exists a sequence \((f_k)_k\) of \(SO(\gamma )\)-invariant functions such that:

(a) \((f_k)_k\) is a bounded sequence in \(\dot{H}^{s,p}_\gamma (\mathbb {R}^N)\),

(b) \(\text{ supp } f_k \subset A\) for any *k*,

(c) \((f_k)_k\) does not contain a subsequence convergent in \(L^\infty ( W_{sp},A)\).

### Proof

We recall the notation of cocompact embeddings. Let *G* be a group of bijective linear isometries of a reflexive Banach space *X*. We say that the sequence \((u_n)\) in *X* is *G*-weakly convergent to 0 if \(g_n u_n\rightharpoonup 0\) for any choice of the sequence \((g_n)\subset G\). A continuous embedding of a reflexive Banach space *X* into a normed linear space *Y* is called cocompact relative to the group *G*, if any sequence \((u_n)\) in *X*, that is *G*-weakly convergent to 0, converges to zero in the norm of *Y*. Embedding (6.1) is not cocompact relative to the group of dilations (6.2). Sequence \((f_k)_k\) provided by Proposition 6.2 has the property that for any sequence of dilations \(({g_{j_k}})_k\) of the form (6.2), the sequence \(({g_{j_k}}f_k)_k\) is weakly convergent to zero without vanishing in the \(L^\infty (W_{sp},\mathbb {R}^N)\)-norm. We conjecture that there is a group of bijective linear isometries on \(\dot{H}^{s,p}_\gamma \) (larger than the set of dilations (6.2)) relative to which the embedding (6.1) becomes cocompact.

### Theorem 6.3

### Proof

As a consequence we have the following structural result for bounded multiradial sequences. The theorem is an immediate consequence of profile decomposition in [18](Definition 2.5 and Theorem 2.6), taking into account that the equivalent norm of \(\dot{H}^{s,p}(\mathbb {R}^N)\) defined by means of the Littlewood-Paley decomposition satisfies Opial’s condition.

### Theorem 6.4

*k*.

## Notes

### Acknowledgements

The authors are indebted to the referees for their valuable remarks and corrections that helped to improve the paper.

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