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.

Appendices

Appendix A: Homogeneous Besov and Triebel-Lizorkin Spaces

1.1 A.1 Distribution Spaces Modulo Polynomials

General references for homogeneous Besov and Triebel-Lizorkin spaces are [10–12, 19, 30]. For convenience of the reader we recall the definition and a few properties of these spaces.

Let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{d})$ be a radial function such that supp φ⊂{ξ∈ℝ^d:|ξ|≤3/2} and φ(ξ)=1 if |ξ|≤1. Then we define

$$ \varphi_j (\xi):= \varphi(2^{-j}\xi) - \varphi(2^{-j+1}\xi), \xi \in\mathbb{R}^d, \quad j \in\mathbb{Z}.$$

(36)

This leads to a specific homogeneous smooth dyadic decomposition of unity since

$$\sum_{j=-\infty}^\infty\varphi_j (\xi) = 1,\quad\xi\neq0.$$

We shall identify tempered distributions modulo polynomials. In fact, we consider the classes

$$[f]:= \{ f+ p: p\ \mbox{polynomial over}\ \mathbb{R}^d\},\quad f \in S'(\mathbb{R}^d).$$

Definition 2

Let 0<q≤∞ and s∈ℝ.

• (i) Let 0<p≤∞. Then the homogeneous Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ is the collection of all classes [f] such that

  $$\| [f] |\dot{B}^s_{p,q}(\mathbb{R}^d)\| := \Biggl(\sum_{j=-\infty}^\infty2^{jsq}\| \mathcal{F}^{-1}[\varphi_j (\xi)\mathcal{F} f(\xi)]( \cdot ) |L_p(\mathbb{R}^d)\|^q\Biggr)^{1/q}<\infty.$$

• (ii) Let 0<p<∞. Then the homogeneous Triebel-Lizorkin space $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$ is the collection of all classes [f] such that

  $$\| [f] |\dot{F}^s_{p,q}(\mathbb{R}^d)\| := \Biggl\| \Biggl(\sum_{j=-\infty}^\infty2^{jsq}|\mathcal{F}^{-1}[\varphi_j (\xi)\mathcal{F} f(\xi)]( \cdot )|^q \Biggr)^{1/q} \Big|L_p (\mathbb{R}^d)\Biggr\| <\infty.$$

Remark 16

(i) The definition makes sense since

$$\mathcal{F}^{-1}[\varphi_j \mathcal{F}(f+p)] = \mathcal{F}^{-1}[\varphi_j\mathcal{F}f]$$

for all polynomials p, all f∈S′(ℝ^d), and all j∈ℤ. Moreover, the spaces $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ and $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$ are independent of the resolution of unity up to equivalence of quasi-norms. Furthermore, we always have

$$\sum_{j=-\infty}^\infty\mathcal{F}^{-1}(\varphi_j \mathcal{F} g) \in[f],\quad\forall g \in[f].$$

(ii) The spaces $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ and $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$ are quasi-Banach spaces.

(iii) Let 1<p<∞. Define $\dot{H}^{s}_{p}(\mathbb{R}^{d})$ as the collection of all classes [f] such that $\mathcal{F}^{-1}[ |\xi|^{s} \mathcal{F}f(\xi) ](\cdot) \in L_{p} (\mathbb{R}^{d})$ equipped with the induced norm. Usually $\dot{H}^{s}_{p}(\mathbb{R}^{d})$ are called homogeneous potential spaces. Then $\dot{H}^{s}_{p}(\mathbb{R}^{d})$ coincides with $\dot{F}^{s}_{p,2} (\mathbb{R}^{d})$ in the sense of equivalent norms.

The following well-known continuous embeddings are of some use for us.

Lemma 10

Let s,s ₀,s ₁∈ℝ and 0<q,q ₀,q ₁≤∞.

• (i) Let 0<p ₀≤p ₁<∞. We have $\dot{F}^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d}) \hookrightarrow\dot {F}^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d})$ if

  $$ s_0 - \frac{d}{p_0} = s_1 - \frac{d}{p_1}$$

  (37)

  and either p ₀<p ₁ or p ₀=p ₁ and q ₀≤q ₁.

• (ii) Let 0<p ₀,p ₁≤∞. We have $\dot{B}^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d}) \hookrightarrow\dot {B}^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d})$ if (37), p ₀≤p ₁, and q ₀≤q ₁ hold.

• (iii) Let 0<p ₀<p ₁≤∞. We have

  $$\dot{B}^{s_0}_{p_0,q_0} (\mathbb{R}^d)\hookrightarrow\dot{F}^{s}_{p,q} (\mathbb{R}^d)\hookrightarrow \dot{B}^{s_1}_{p_1,q_1}(\mathbb{R}^d)$$

  if q ₀≤p≤q ₁.

• (iv) Let 0<p<∞. We have

  $$\dot{B}^{s}_{p,q_0} (\mathbb{R}^d)\hookrightarrow \dot{F}^{s}_{p,q} (\mathbb{R}^d)\hookrightarrow \dot{B}^{s}_{p,q_1}(\mathbb{R}^d)$$

  if q ₀≤min(p,q) and max(p,q)≤q ₁.

Remark 17

(i) Observe, for fixed s and p the Besov space $\dot{B}^{s}_{p,\infty}(\mathbb{R}^{d})$ is the largest in the both scales $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ and $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$.

(ii) For proofs we refer, e.g., to [13]. This reference does not cover the second embedding in part (iii). For this part we refer to Franke [9], but see also [32].

1.2 A.2 Function Spaces Modulo Polynomials of a Certain Degree

First of all we wish to mention that the mapping

$$[f] \to\sum_{j=-\infty}^\infty\, \mathcal{F}^{-1}[\varphi_j \mathcal{F} f]$$

is an isomorphism which maps $\dot{F}^{0}_{p,2} (\mathbb{R}^{d})$ onto L _p (ℝ^d) if 1<p<∞, see [18] or [29, Theorem 3.2.1]. Next we turn to homogeneous Sobolev spaces $\dot{W}^{m}_{p} (\mathbb{R}^{d})$. This time we consider classes of functions modulo polynomials of degree m−1, i.e. we put

$$ [f]_m:=\biggl\{g \in\mathcal{S}' (\mathbb{R}^d): g = f+ \sum_{|\alpha|<m} a_\alpha x^\alpha \biggr\}.$$

(38)

Then $\dot{W}^{m}_{p} (\mathbb{R}^{d})$ is the collection of all classes [f] _m such that D ^α f∈L _p (ℝ^d), |α|=m. For m∈ℕ and 1<p<∞ there exists an isomorphism of $\dot{F}^{m}_{p,2} (\mathbb{R}^{d})$ onto $\dot{W}^{m}_{p} (\mathbb{R}^{d})$. Proofs can be found in [18] or in [29, Theorem 3.2.1]. A fractional order version is given by the following definition.

Definition 3

Let 0<q≤∞.

• (i) Let 0<p≤∞. Assume σ _p <s<m, for some natural number m. Then the class [f] _m of regular distributions belongs to $\dot{B}^{s,m}_{p,q}(\mathbb{R}^{d})$ if

  $$N^s_{p,q}(f):= \biggl( \int_0^\infty t^{-sq} \sup_{|h|<t} \|\Delta_h^m f(x)|L_p (\mathbb{R}^d)\|^{q}\frac{dt}{t}\biggr)^{1/q} < \infty.$$

• (ii) Let 0<p<∞. Assume σ _p,q (d)<s<m, for some natural number m. Then the class [f] _m of regular distributions belongs to $\dot {F}^{s,m}_{p,q}(\mathbb{R}^{d})$ if

  $$ M^s_{p,q}(f):= \biggl(\int_{{\mathbb{R}}^d} \biggl(\int_0^\infty t^{-sq}\biggl( t^{-d}\int_{|h|\leq t} |\Delta_h^mf(x)|\, dh\biggr)^{q}\frac{dt}{t}\biggr)^{p/q} dx \biggr)^{1/p}< \infty.$$

  (39)

Proposition 3

Under the restrictions from Definition 3 we have the following: there exists an isomorphism of $\dot{A}^{s,m}_{p,q} (\mathbb{R}^{d})$ onto $\dot{A}^{s}_{p,q} (\mathbb{R}^{d})$.

Remark 18

With A=B Proposition 3 can be found in [30, Theorem 5.2.3/2]. With A=F it is proved in [5]. Both references cover the case of Banach spaces only. However, the methods from [5] extend to the quasi-Banach space case.

The Regularity of Some Test Functions

Now we turn to the regularity of g _α,δ , see (10), with respect to the fractional order spaces. There are several possibilities to attack this problem. We decided for using differences, see Proposition 3. The following two lemmas will cover Proposition 1.

Lemma 11

Let 0<p<∞, 0<q≤∞ and σ _p,q <s<d/p.

• (i) The function g _α,0 belongs to $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$ if, and only if $\alpha> \frac{d}{p} - s$.

• (ii) Let δ>0. Then g _α,δ belongs to $\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$ if either $\alpha> \frac{d}{p} - s$ (δ≥0 arbitrary) or $\alpha= \frac{d}{p} - s$ and δ>1/p.

Proof

Step 1. Proof of (i) in case q=∞. For given s>0 there exists an integer M∈ℕ₀ and a real positive number τ∈(0,1] such that s=M+τ. The function g _α,0 is smooth. Hence we may apply the Mean Value Theorem. This implies

$$\Delta_h^{M} g_{\alpha,0} (x) = |h|^{M}D_e^{M} (g_{\alpha,0})(\xi),$$

where e is the direction from x to x+Mh and ξ=x+te for some t∈(0,M|h|), $D_{e}^{M}$ denotes the M-th order derivative of g _α,0 (restricted to this line) in direction e. We obtain

(40)

Mathematical induction yields

$$D^{\beta} g_{\alpha,0}(\xi) =(1+|\xi|^2)^{-\frac{\alpha}{2} - |\beta|} p_\beta(\xi) ,\quad\xi\in\mathbb{R}^d,$$

where p _β is a polynomial of degree at most |β|. Let |x|=r>2max((M+1)|h|,1) and |β|=M. Then

(41)

as long as |ξ−x|,|η−x|≤(M+1)|h|. Hence, with (40) and (41) we get

(42)

This can be complemented by the obvious inequality

(43)

since g _α,0 is a bounded C ^∞ function. According to (39) it remains to check

(44)

Combining (42)–(44) we have proved sufficiency in (i) with q=∞. Now, let q<∞. We choose 0<p ₀<p and define

$$s_0:= s - \frac{d}{p} + \frac{d}{p_0}.$$

From the arguments used above we conclude that $g_{\alpha,0} \in\dot{F}^{s_{0}}_{p_{0},\infty}(\mathbb{R}^{d})$, but $\dot{F}^{s_{0}}_{p_{0},\infty}(\mathbb{R}^{d}) \hookrightarrow\dot{F}^{s}_{p,q}(\mathbb{R}^{d})$, see Lemma 10(i). Observe, that also in case s=M+1 the above estimate is sufficient to guarantee $g_{\alpha,0} \in\dot{F}^{s}_{p,\infty}(\mathbb{R}^{d})$.

Step 2. Sketch of the proof of (ii). We only describe the needed modifications for the limiting situation s+α=d/p.

Substep 2.1. Let q=∞. First observe, that Leibniz formula for derivatives of products yields

where p _ϱ are polynomials of degree at most |ϱ|. The chain rule yields

for appropriate constants $c_{\ell, \gamma^{1}, \ldots , \gamma^{\ell}}$. Obviously

$$|D^{\gamma} (\log(e+|\xi|^2)) | \lesssim|\xi|^{-|\gamma|},\quad|\xi |\ge1,\ |\gamma|>0,$$

and consequently

$$ |D^\vartheta(\log(e+|\xi|^2))^{-\delta}| \lesssim|\xi|^{-|\vartheta|}(\log(e+|\xi|^2))^{-\delta-1},\quad|\xi|\ge1,\ |\vartheta|>0.$$

(45)

As in (41) we derive

(46)

where |ξ−x|, |η−x|≤(M+1)|h|, |x|=r>2max((M+1)|h|,1) and |β|=M. Now, let $P_{\alpha,\varrho} (\xi):= (1+|\xi|^{2})^{-\frac{\alpha}{2} - |\varrho|}p_{\varrho}(\xi)$. Similarly as in (46), see also (41), and using (45) we find

under the same restrictions as in (46). This leads to the following modification of (42)

The term

$$\int_{|x|<2}\biggl(\sup_{t>0} t^{-s}t^{-d}\int_{|h|<t} |\Delta_h^{M+1} g_{\alpha,\delta}(x)|\, dh \biggr)^p \, dx$$

can be estimated as in (43). For the modification of (44) observe

$$\int_{0}^{t} r^{d-1}(1+ r^2)^{- \alpha/2}(\log(e + r^2))^{-\delta} \, dr \lesssim t^{d-\alpha} (\log(e+t))^{-\delta}$$

uniformly in t>1/M, since α<d. This proves $g_{\alpha,\delta} \in\dot{F}^{\frac{d}{p} - \alpha}_{p,\infty}(\mathbb{R}^{d})$ if δ>1/p.

Substep 2.2. Let 0<q≤∞, 0<p<∞ and $0 < s := \frac{d}{p}- \alpha< \frac{d}{p}$. We choose 0<p ₀<p and define

$$s_0:= s - \frac{d}{p} + \frac{d}{p_0}= \frac{d}{p_0} - \alpha.$$

From Substep 2.1 and Lemma 10(i) we conclude that

$$g_{\alpha,\delta} \in\dot{F}^{s_0}_{p_0,\infty}(\mathbb{R}^d)\hookrightarrow\dot{F}^{s}_{p,q}(\mathbb{R}^d)$$

if δ>1/p ₀. For p ₀↑p the claim follows.

Step 3. Necessity in (i). Let us assume $g_{\alpha,0} \in\dot{F}^{s}_{p,\infty} (\mathbb{R}^{d})$ with $\alpha=\frac{d}{p} -s$. Then, by Lemma 10(i), it follows that the class [g _α,0] contains at least one element which belongs to L _t (ℝ^d), $t= \frac{d}{\frac{d}{p} - s}$. Since there is at most one element in such a class which decays near infinity we get g _α,0∈L _t (ℝ^d). By Lemma 4(i) we conclude δ>1/t>0. This is a contradiction. □

Lemma 12

Let 0<p<∞, 0<q≤∞ and σ _p <s<d/p.

• (i) The function g _α,0 belongs to $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ if $\alpha> \frac{d}{p} - s$.

• (ii) Let δ>0. Then g _α,δ belongs to $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ if either $\alpha> \frac{d}{p} - s$ or $\alpha= \frac{d}{p} - s$ and δ>1/q.

Proof

Both assertions follow from Lemma 11 by using Lemma 10(iii). □

Appendix B: Radial Distributions and Atomic Decompositions

We recall a construction of Epperson and Frazier [8]. We will do that with certain detail because we are going to use it with a different normalization.

Let J _ν denote the Bessel function of order ν, $\nu\ge-\frac{1}{2}$, defined by

$$J_\nu(t) := \left\{\begin{array}{@{}l@{\quad}l}\frac{(t/2)^\nu}{\sqrt{\pi} \Gamma(\nu+ \frac{1}{2})}\int_{-1}^1 (1-y^2)^{\nu-1/2} e^{i ty}\, dy& \mbox{if}\ \nu> -\frac{1}{2}, \\[2mm](\frac{2}{\pi t})^{1/2}\cos t & \mbox{if}\ \nu= - \frac{1}{2},\end{array}\right.\quad t \in\mathbb{R}.$$

Let μ _ν,0<μ _ν,1<⋯ be the positive zeros of J _ν . We put μ _ν,−1:=0. Then

$$\mu_{\nu,k}= \pi\biggl(k+\frac{\nu}{2}\biggr) + O\biggl(\frac{1}{k+1}\biggr)$$

and

$$\mu_{\nu,k} - \mu_{\nu,k-1}= \pi + O\biggl(\frac{1}{k+1}\biggr) .$$

For k=0,1,2,… we introduce associated annuli (balls, if k=0)

From now on we fix $\nu= \frac{d-2}{2}$ and drop it in notation.

Next we recall the definition of smooth radial atoms from [8].

Definition 4

Let s∈ℝ and 0<p<∞. A radial function a is called a smooth radial atom associated to ${\mathbb{A}}_{j,k}$ if it satisfies the following conditions:

(47)

(48)

Here c _γ :=1 if |γ|≤s+1 and c _γ must be independent of j and k if |γ|>s+1.

As usual one has to introduce associated sequence spaces as well. Let $\chi_{{\mathbb{A}}_{j,k} }$ denote the characteristic function of the set ${\mathbb{A}}_{j,k}$. Then we define $\widetilde{\chi}^{(p)}_{j,k} :=2^{\frac{jd}{p}}\chi_{\mathbb{A}_{j,k}}$. The announced sequence spaces are then given by

and

Again we will use these notation with $\dot{a}_{p,q}$ in place of $\dot {b}_{p,q}$ or $\dot{f}_{p,q}$ if there is no need to distinguish these cases. Now we are in position to formulate the result of Epperson and Frazier [8], see Theorem 4.1 and the comments in Sect. 5.

Theorem 6

Suppose 0<p<∞, 0<q≤∞ and either s>σ _p,q −1 if A=F or s>σ _p −1 if A=B. For $[f] \in R\dot{A}^{s}_{p,q}(\mathbb{R}^{d})$ there exist smooth radial atoms a _j,k associated to ${\mathbb{A}}_{j,k}$, j∈ℤ, k∈ℕ₀, and a sequence $(s_{j,k})_{j,k} \in\dot{a}_{p,q}$, such that

$$ \sum_{j \in\mathbb{Z}} \sum_{k=0}^\infty s_{j,k}\,a_{j,k}\in[f]$$

(49)

and

$$\|[f] |R\dot{A}^s_{p,q}(\mathbb{R}^d)\|\asymp \| (s_{j,k})_{j,k}|\dot{a}_{p,q}\|.$$

Remark 19

The identity (49) should be interpreted in the following way. The sequence (f _n ) _n , where

$$f_n =\sum_{j=-n}^{n} \sum_{k=0}^\infty s_{j,k}a_{j,k},$$

converges to some g∈[f] with respect to the quasi-norm in $R\dot {A}^{s}_{p,q}(\mathbb{R}^{d})$ as n tends to infinity, if q<∞, and in $S'(\mathbb{R}^{d})/{\mathcal{P}}$ if q=∞.

We need another result of Epperson and Frazier, see Theorem 3.1 and the comments in Sect. 5 in [8].

Lemma 13

Suppose 0<p<∞, 0<q≤∞ and either s>σ _p,q if A=F or s>σ _p if A=B. There exists a positive constant c such that for any sequence $(a_{j,k})_{j\in\mathbb{Z}, k\in\mathbb{N}_{0}}$ of radial functions satisfying the conditions (47), (48) (restricted to values of γ such that |γ|≤s+1) and any sequence $(s_{j,k})_{j,k} \in\dot{a}_{p,q}$ the inequality

$$\Biggl\|\sum_{j=-\infty}^\infty\sum_{k=0}^\infty s_{j,k}a_{j,k} \Big|R\dot{A}^s_{p,q}(\mathbb{R}^d)\Biggr\|\le c \|(s_{j,k})_{j,k} |\dot{a}_{p,q}\|$$

holds.

Remark 20

Radial subspaces of homogeneous Besov spaces have been characterized in a wavelet-style by Rauhut [20] and Rauhut and Rösler [21]. These methods could be used here as well.