Chin. Ann. Math. 30B(3), 2009, 311–320 DOI: 10.1007/s11401-007-0471-x

Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and

Springer-Verlag Berlin Heidelberg 2009

Some Aspects of Lrw1 (G) ∩ L(p, q, w2 dµ)(G) ˙ Ilker ERYILMAZ∗

Cenap DUYAR∗

Abstract Let G be a locally compact Abelian group with Haar measure. The authors discuss some basic properties of Lrw1 (G) ∩ L(p, q, w2 dµ)(G) spaces. Then the necessary conditions for compact embeddings of the spaces Lrw1 (Rd ) ∩ L(p, q, w2 dµ)(Rd ) are showed. Keywords Weighted Lebesgue spaces, Weighted Lorentz spaces, Compact embeddings 2000 MR Subject Classification 46E30, 46E40, 54C25

1 Introduction and Preliminaries Throughtout this work, G will denote a non-compact and non-discrete locally compact Abelian group with Haar measure µ. Also, we will use Beurling’s weight function, i.e., a measurable, locally bounded function on G satisfying w(x) ≥ 1 and w(x + y) ≤ w(x)w(y) for all x, y ∈ G. For two weight functions w1 and w2 , we write w1 ≺ w2 if there exists C > 0, such that w1 (x) ≤ Cw2 (x) for all x ∈ G. We write w1 ≈ w2 if and only if w1 ≺ w2 and w2 ≺ w1 . Certain well-known terms such as Banach module, Banach ideal, translation and character invariance, compact embedding will be used frequently in the sequel; their definitions may be found, e.g., in [3, 4, 10, 12]. For 1 ≤ r < ∞, we set weighted Lebesgue spaces as Lrw (G) = {f | f w ∈ Lr (G)}, which are Banach spaces under the naturel norm nZ o r1 kf kr,w = |f (x)|r wr (x)dµ(x) . G

Recall that one has Lrw1 (G) ⊂ Lrw2 (G) if and only if w2 ≺ w1 (see [6, 8]). The Lorentz spaces over weighted measure spaces L(p, q, wdµ) are defined and discussed in [5, 13]. Instead of Haar measure µ, let us take the measure as wdµ. Then the distribution function of f which is considered complex-valued measurable and defined on the measure space (G, wdµ) is Z λf,w (y) = w{x ∈ G : |f (x)| > y} = w(x)dµ(x), y ≥ 0. {x∈G:|f (x)|>y}

The nonnegative rearrangement of f is given by fw∗ (t) = inf{y > 0 : λf,w (y) ≤ t} = sup{y > 0 : λf,w (y) > t},

t ≥ 0,

Manuscript received November 15, 2007. Published online April 16, 2009. of Mathematics, Sciences and Arts Faculty, Ondokuz Mayıs University, Samsun 55139, Kurupelit, Turkey. E-mail: [email protected] [email protected]

∗ Department

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where we assume that inf φ = ∞ and sup φ = 0. Also the average function of f on (0, ∞) is given by Z 1 t ∗ fw∗∗ (t) = f (s)ds. t 0 w Note that λf,w ( · ), fw∗ ( · ) and fw∗∗ ( · ) are nonincreasing and right continuous functions. The weighted Lorentz space L(p, q, wdµ) is the collection of all the functions f such that kf k∗p,q,w < ∞, where kf k∗p,q,w =

q Z p

∞ 0

q

t p −1 [fw∗ (t)]q dt

1

kf k∗p,∞,w = sup t p fw∗ (t),

 1q

,

0 < p, q < ∞,

0 < p < q = ∞.

t>0

In general, however, k · k∗p,q,w is not a norm since the Minkowski inequality may fail. But by replacing fw∗ with fw∗∗ in the above definition, we get that L(p, q, wdµ) is a Banach space, with the norm k · kp,q,w defined by q Z ∞ q  q1 kf kp,q,w = t p −1 [fw∗∗ (t)]q dt , 0 < p, q < ∞, p 0 1

kf kp,∞,w = sup t p fw∗∗ (t),

0 < p < q = ∞.

t>0

If 1 < p ≤ ∞ and 1 ≤ q ≤ ∞, then kf k∗p,q,w ≤ kf kp,q,w ≤

p kf k∗p,q,w , p−1

where the first inequality is an immediate consequence of the fact that fw∗ ≤ fw∗∗ . The second follows from the Hardy inequality. The plan of the paper is as follows. In Section 2, we will define the intersection of weighted Lebesgue and weighted Lorentz spaces and give some unmentioned properties of these spaces. Then the compact embedding of this intersected spaces will be discussed in Section 3.

2 Some Results in Lrw1 (G) ∩ L(p, q, w2 dµ)(G) For 1 ≤ r ≤ ∞ and 0 < p, q ≤ ∞, we will write the intersection of weighted Lebesgue and w1 ,w2 (G). If we equip this space with weighted Lorentz spaces Lrw1 (G) ∩ L(p, q, w2 dµ)(G) as Br,p,q the sum norm 1 ,w2 k · kw (2.1) r,p,q = k · kr,w1 + k · kp,q,w2 , w1 ,w2 1 ,w2 then it is easy to see that (Br,p,q (G), k · kw r,p,q ) is a normed space. Now we will give some properties of these spaces without their complete proofs. w1 ,w2 1 ,w2 Theorem 2.1 (Br,p,q (G), k · kw r,p,q ) is a Banach space for 1 ≤ r ≤ ∞ and p = q = 1, p = q = ∞ or 1 < p < ∞, 1 ≤ q ≤ ∞. w1 ,w2 1 ,w2 Proof Let (fn ) be a (Br,p,q (G), k · kw r,p,q ) Cauchy sequence. Clearly (fn ) is a Cauchy r sequence in both Lw1 (G) and L(p, q, w2 dµ)(G). Therefore, (fn ) converges to some f ∈ Lrw1 (G) and g ∈ L(p, q, w2 dµ)(G). To prove the theorem, we need to show that f = g(µ − a.e.). Since the convergence in both Lrw1 (G) and L(p, q, w2 dµ)(G) implies the convergence in measure, we get f = g(µ − a.e.) (see [10, 16]).

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w1 ,w2 1 ,w2 Theorem 2.2 The space (Br,p,q (G), k · kw r,p,q ) is translation invariant and the function w1 ,w2 w1 ,w2 f → Lx f is continuous from Br,p,q (G) to Br,p,q (G) for all x ∈ G, where Lx f ( · ) = f ( · − x). w1 ,w2 Proof Let us take any x ∈ G and f ∈ Br,p,q (G). Since λLx f,w2 (y) ≤ w2 (x)λf,w2 (y) for 1

all y ≥ 0, we have kLx f kp,q,w2 ≤ (w2 (x)) p kf kp,q,w2 (see [5]). With the inequality kLx f kr,w1 ≤ w1 (x)kf kr,w1 , we get 1 ,w2 kLxf kw r,p,q = kLx f kr,w1 + kLx f kp,q,w2 1

≤ w1 (x)kf kr,w1 + (w2 (x)) p kf kp,q,w2 1

1 ,w2 ≤ max{w1 (x), (w2 (x)) p }kf kw r,p,q .

w1 ,w2 w1 ,w2 Also, the continuity from Br,p,q (G) to Br,p,q (G) follows from the linearity of Lx for all x ∈ G. w1 ,w2 1 ,w2 Theorem 2.3 The space (Br,p,q (G), k · kw r,p,q ) is strongly character invariant and the w1 ,w2 w1 ,w2 b function f → Mt f is continuous from Br,p,q (G) to Br,p,q (G) for all t ∈ G.

b and f ∈ B w1 ,w2 (G). Since Proof Let us take any t ∈ G r,p,q

λMt f,w2 (y) = w2 {x ∈ G : |Mt f (x)| > y} = w2 {x ∈ G : |hx, tif (x)| > y} = w2 {x ∈ G : |f (x)| > y} = λf,w2 (y),

∗∗ we get (Mt f )∗w2 = fw∗ 2 and (Mt f )∗∗ w2 = fw2 , so kMt f kp,q,w2 = kf kp,q,w2 . By the equality w1 ,w2 1 ,w2 kMt f kr,w1 = kf kr,w1 , we have kMt f kw r,p,q = kf kr,p,q . w1 ,w2 w1 ,w2 Theorem 2.4 For every f ∈ Br,p,q (G), the function x → Lx f , G → Br,p,q (G) is continuous where 1 < p < ∞, 1 ≤ q < ∞ and 1 ≤ r < ∞.

Proof We know that, for 1 ≤ r < ∞, x → Lx f is continuous in Lrw1 (G) (see [8]). Also the continuity of x → Lx f in L(p, q, w2 dµ)(G) was shown for 1 < p < ∞, 1 ≤ q < ∞ in [5]. Thus the proof is easily completed by combining the two results mentioned above. w1 ,w2 1 ,w2 Theorem 2.5 If the weight functions are constant, then (Br,p,q (G), k · kw r,p,q ) is a homogeneous Banach space.

Proof Let the weight functions w1 and w2 be constant. Therefore, the spaces Lrw1 (G) and L(p, q, w2 dµ)(G) become strongly translation invariant spaces. Then by Theorem 2.4, we get the result. w1 ,w2 1 1 ,w2 Theorem 2.6 If w0 ≻ w1 and w0 ≻ w2 , then (Br,p,q (G), k · kw r,p,q ) is a Banach Lw0 (G)module.

Proof Let w0 ≻ w1 and w0 ≻ w2 . Then we know that L1w0 (G) ⊂ L1w1 (G) and L1w0 (G) ⊂ Therefore for any f ∈ L1w0 (G), there exist c1 , c2 > 0 such that kf k1,w1 ≤ c1 kf k1,w0 and kf k1,w2 ≤ c2 kf k1,w0 . Since L(p, q, w2 dµ)(G) is a Banach L1w2 (G)-module for 1 < p < ∞, q 6= ∞ (see [5]) and Lrw1 (G) is a Banach L1w1 (G)-module for 1 ≤ r < ∞, we have L1w2 (G).

1 ,w2 kf ∗ gkw r,p,q = kf ∗ gkr,w1 + kf ∗ gkp,q,w2

≤ kf kr,w1 kgk1,w1 + kf kp,q,w2 kgk1,w2 ≤ kf kr,w1 c1 kgk1,w0 + kf kp,q,w2 c2 kgk1,w0 1 ,w2 ≤ kf kw r,p,q max{c1 , c2 }kgk1,w0

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w1 ,w2 for any f ∈ Br,p,q (G) and g ∈ L1w0 (G). If we define a new norm |k · k| on L1w0 (G) such that |k · k| = max{c1 , c2 }k · k1,w0 , then this norm is equivalent to the norm k · k1,w0 on L1w0 (G). So w1 ,w2 1 1 ,w2 (Br,p,q (G), k · kw r,p,q ) is a Banach (Lw0 (G), |k · k|)-module. w1 ,w2 Theorem 2.7 If 1 < p < ∞, q 6= ∞ and w2 ≻ w1 or w1 ≻ w2 , then the space B1,p,q (G) is a Banach algebra.

Proof Assume that w1 ≻ w2 . Then we know that L1w1 (G) ⊂ L1w2 (G) and for any f ∈ 1 Lw1 (G), there exists c > 0 such that kf k1,w2 ≤ ckf k1,w1 . If we define a new function on w1 ,w2 w1 ,w2 1 ,w2 B1,p,q (G) such that |k·k|w 1,p,q = max{1, c}k·k1,p,q , then it is easy to see that it is a norm. Also, w1 ,w2 w1 ,w2 1 ,w2 these two norms on B1,p,q (G) are equivalent and the identity map i : (B1,p,q (G), k · kw 1,p,q ) → w1 ,w2 w1 ,w2 w1 ,w2 1 ,w2 (B1,p,q (G), |k · k|w 1,p,q ) is a homeomorphism on B1,p,q (G). Now take any f, g ∈ B1,p,q (G). Since L1w1 (G) is a Beurling algebra and the space L(p, q, w2 dµ)(G) is a Banach L1w2 (G)-module for 1 < p < ∞, q 6= ∞, we get w1 ,w2 1 ,w2 |kf ∗ gk|w r,p,q = max{1, c}kf ∗ gkr,p,q

= max{1, c}(kf ∗ gk1,w1 + kf ∗ gkp,q,w2 ) ≤ max{1, c}(kf k1,w1 kgk1,w1 + kf k1,w2 kgkp,q,w2 ) ≤ max{1, c}(max{1, c}kf k1,w1 {kgk1,w1 + kgkp,q,w2 }) w1 ,w2 w1 ,w2 1 ,w2 ≤ max{1, c}kf k1,w1 max{1, c}kgkw r,p,q ≤ |kf k|r,p,q |kgk|r,p,q .

Similarly, if w2 ≻ w1 , then the same way may be followed. w1 ,w2 Theorem 2.8 The space Br,p,q (G) has a bounded approximate identity with compact support for 1 < p < ∞, 1 ≤ q < ∞.

Proof Let K be a compact neighbourhood of the identity of G. Then wi (y) ≤ A for all y ∈ K and i = 1, 2. Let F be the family of all neighbourhoods of the identity contained in K. For U, V ∈ F , define V ≺ U if U ⊂ V . Then, clearly (F, ≺) is a directed set. For every U ∈ F , R there exists a positive continuous function hU on G such that G hU (x)dλ(x) = 1 and the support of hU is contained in U . If we consider the net {hU }U∈F , then we have khU k1,wi ≤ A for each U ∈ F. Therefore we find a bounded approximate identity for L1wi (G) for i = 1, 2. It is shown in [14] that this bounded approximate identity is also a bounded approximate identity for Lrw1 (G). In [5], it was showed that this bounded approximate identity is also a bounded w1 ,w2 approximate identity for L(p, q, w2 dµ)(G) for 1 < p < ∞, 1 ≤ q < ∞. So Br,p,q (G) possesses a bounded approximate identity. The next theorem follows from Theorems 2.6 and 2.8. w1 ,w2 Theorem 2.9 The space Br,p,q (G) is an essential Banach L1w0 (G)-module, if w0 ≻ w1 , w0 ≻ w2 and 1 < p < ∞, 1 ≤ q < ∞.

Theorem 2.10 Let w1 , w2 , w3 and w4 be weight functions on G and 1 ≤ r ≤ ∞, 0 < p, q < ∞. Then w1 ,w2 w3 ,w4 3 ,w4 (1) Br,p,q (G) ⊂ Br,p,q (G) if and only if there exists a constant c > 0 such that kf kw r,p,q ≤ w1 ,w2 w1 ,w2 ckf kr,p,q for all f ∈ Br,p,q (G), w1 ,w2 w3 ,w2 (2) Br,p,q (G) ⊂ Br,p,q (G) if w1 ≻ w3 and 0 < q1 ≤ q2 ≤ ∞. 1 2 w1 ,w2 w3 ,w4 w1 ,w2 Proof (1) Let Br,p,q (G) ⊂ Br,p,q (G). We define a norm k · k on Br,p,q (G) such that

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w1 ,w2 w1 ,w2 3 ,w4 kf k = kf kr,p,q + kf kw r,p,q . It is easy to show that (Br,p,q (G), k · k) is a Banach space. Since w1 ,w2 w1 ,w2 1 ,w2 the unit function i from (Br,p,q (G), k · kw r,p,q ) onto (Br,p,q (G), k · k) is continuous, i is a 1 ,w2 homeomorphism by the closed graph theorem. This shows that the norms k · k and k · kw r,p,q are 1 ,w2 equivalent. Then there is a constant c > 0 such that kf k ≤ ckf kw r,p,q . This gives the inequality w1 ,w2 3 ,w4 kf kw r,p,q ≤ kf k ≤ ckf kr,p,q . w3 ,w4 w1 ,w2 w1 ,w2 w1 ,w2 Conversely, if kf kr,p,q ≤ ckf kr,p,q for all f ∈ Br,p,q (G), then the inclusion Br,p,q (G) ⊂ w3 ,w4 Br,p,q (G) is easy to see. (2) It is known that L(p, q1 , w2 dµ)(G) ⊂ L(p, q2 , w2 dµ)(G), where 0 < q1 ≤ q2 ≤ ∞ (see [11]). Also, since Lrw1 (G) ⊂ Lrw3 (G) if w1 ≻ w3 , the proof is completed. w1 ,w2 Theorem 2.11 Let p = 1, 0 < q ≤ 1 and 1 ≤ r < ∞. Then for any f ∈ Br,1,q (G), the w1 ,w2 ′ function x → kLx f kr,1,q is equivalent to the weight function w = w1 + w2 , i.e., there exist c1 (f ), c2 (f ) > 0 such that ′ 1 ,w2 c1 (f )w′ (x) ≤ kLx f kw r,1,q ≤ c2 (f )w (x).

(2.2)

w1 ,w2 Proof Let f ∈ Br,1,q (G). Then it is known that the function x → kLx f kr,w is equivalent to the weight function w, i.e., there exist k1 (f ), k2 (f ) > 0 such that

k1 (f )w(x) ≤ kLx f kr,w ≤ k2 (f )w(x)

(2.3)

for all x ∈ G (see [8]). Also by Theorem 2.2, we write kLx f k1,q,w2 ≤ w2 (x)kf k1,q,w2

(2.4)

kLxf k1,q,w2 ≥ kLx f k∗1,q,w2 ≥ kLx f k∗1,1,w2 = kLx f k1,w2 ,

(2.5)

and where q ≤ 1 (see [11]). By using (2.3)–(2.5), there are s1 (f ), s2 (f ) > 0 such that s1 (f )w2 (x) ≤ kLx f k1,w2 ≤ kLx f k1,q,w2 ≤ w2 (x)kf k1,q,w2 ,

(2.6)

where s2 (f ) = kf k1,q,w2 . If we combine (2.3) with (2.6), then we have 1 ,w2 s1 (f )w2 (x) + k1 (f )w1 (x) ≤ kLx f kw r,1,q ≤ k2 (f )w1 (x) + s2 (f )w2 (x).

Therefore ′ 1 ,w2 c1 (f )w′ (x) ≤ kLxf kw r,1,q ≤ c2 (f )w (x)

for all x ∈ G. Theorem 2.12 Let w1 , w2 , w3 and w4 be weight functions on G and w′ = w1 + w2 , w1 ,w2 w3 ,w4 w = w3 + w4 . If Br,1,q (G) ⊂ Br,1,q (G), then w′ ≻ w′′ for p = 1, 0 < q ≤ 1. Conversely, w1 ,w2 w3 ,w4 Br,1,q (G) ⊂ Br,1,q (G), if w1 ≻ w3 and w2 ≻ w4 . ′′

w1 ,w2 w3 ,w4 Proof Assume that Br,1,q (G) ⊂ Br,1,q (G). By Theorem 2.11, there are k, l > 0 such that ′ 1 ,w2 k −1 w′ (x) ≤ kLx f kw (2.7) r,1,q ≤ kw (x)

and ′′ 3 ,w4 l−1 w′′ (x) ≤ kLxf kw r,1,q ≤ lw (x),

(2.8)

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w1 ,w2 w3 ,w4 where w′ = w1 + w2 and w′′ = w3 + w4 . Since Br,1,q (G) ⊂ Br,1,q (G) implies that there is a w3 ,w4 w1 ,w2 constant C > 0 such that kLx f kr,1,q ≤ CkLx f kr,1,q , we have w1 ,w2 ′ 3 ,w4 l−1 w′′ (x) ≤ kLxf kw r,1,q ≤ CkLx f kr,1,q ≤ Ckw (x)

(2.9)

and w′′ (x) ≤ Cklw′ (x). This shows that w′ ≻ w′′ . The second part is seen from [5, Proposition 2.7]. Theorem 2.13 Let w1 , w2 , w3p and w4 be weight functions on G and 1 ≤ r ≤ ∞, 1p ≤ w3 ,w4 w3 ,w4 w1 ,w2p ≤ (G), if and only if there exists c > 0 such that kf kr,p,q (G) ⊂ Br,p,q p, q < ∞.p Then Br,p,q w1 ,w2p w1 ,w2 ckf kr,p,q for all f ∈ Br,p,q (G). Proof The proof is similar to that of Theorem 2.10. w ,w p

1 2 (G), the function x → Theorem 2.14 Let 1 ≤ q ≤ p < ∞. Then for any f ∈ Br,p,q w1 ,w2p kLx f kr,p,q is equivalent to the weight function w′ = w1 + w2 , i.e., there exist c1 (f ), c2 (f ) > 0 such that

w ,w p

1 2 ≤ c2 (f )w′ (x). c1 (f )w′ (x) ≤ kLx f kr,p,q

(2.10)

w ,w p

1 2 Proof Let f ∈ Br,p,q (G). Since the function x → kLx f kr,w is equivalent to the weight function w, there exist k1 (f ), k2 (f ) > 0 such that

k1 (f )w(x) ≤ kLx f kr,w ≤ k2 (f )w(x)

(2.11)

for all x ∈ G (see [8]). Also by Theorem 2.2 , we write kLx f kp,q,w2p ≤ w2 (x)kf kp,q,w2p

(2.12)

kLx f kp,q,w2p ≥ kLx f k∗p,q,wp ≥ kLxf k∗p,p,wp = kLxf kp,w2 ,

(2.13)

and 2

2

where q ≤ p (see [11]). By (2.11)–(2.13), there are s1 (f ), s2 (f ) > 0 such that s1 (f )w2 (x) ≤ kLx f kp,w2 ≤ kLxf kp,q,w2p ≤ w2 (x)kf kp,q,w2p ,

(2.14)

where s2 (f ) = kf kp,q,w2p . If we combine (2.11) with (2.14), then we have w ,w p

1 2 ≤ k2 (f )w1 (x) + s2 (f )w2 (x). s1 (f )w2 (x) + k1 (f )w1 (x) ≤ kLx f kr,p,q

Therefore w ,w p

1 2 c1 (f )w′ (x) ≤ kLxf kr,p,q ≤ c2 (f )w′ (x)

(2.15)

for all x ∈ G. Theorem 2.15 Let w w4 be weight functions on G and w′ = w1 + w2 , 1 , w2 , w3 and p p w ,w w ,w 1 3 ′ ′′ 2 4 w′′ = pw3 + w4 . If B r,p,q (G) ⊂ Br,p,q (G), then w ≻ w for 1 ≤ q ≤ p < ∞. Conversely, w1 ,w2 w3 ,w4p Br,p,q (G) ⊂ Br,p,q (G), if w1 ≻ w3 and w2 ≻ w4 . Proof The proof is similar to that of Theorem 2.12.

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w1 ,w2 3 Compact Embeddings of the Spaces Br,p,q (Rd)

In this section, we will work on Rd with Lebesgue measure dx. We denote by Cc (Rd ) the space of complex-valued, continuous functions with compact support. w1 ,w2 Theorem 3.1 Let (fn )n∈N be a sequence in Br,p,q (Rd ). If (fn )n∈N converges to zero in then (fn )n∈N also converges to zero in the vague topology (see [3]), i.e., for n → ∞, Z fn (x)k(x)dx → 0

w1 ,w2 Br,p,q (Rd ),

Rd

for all k ∈ Cc (Rd ). Proof Let k ∈ Cc (Rd ). We write Z 1 ,w2 fn (x)k(x)dx ≤ kkkr′ kfn kr ≤ kkkr′ kfn kw r,p,q

(3.1)

Rd

by H¨older’s inequality where vague topology by (3.1).

1 r

+

1 r′

= 1. Hence the sequence (fn )n∈N converges to zero in the

Theorem 3.2 Let w1 , w2 and υ be Beurling weight functions on Rd . If w1 ≻ w2 , w1 ≻ υ w1 ,w2 and wυ(x) does not tend to zero in Rd for x → ∞, then the embedding of the space Br,p,q (Rd ) 1 (x) r d into Lυ (R ) is never compact. Proof First of all, since w1 ≻ υ, there is a constant C > 0 such that υ(x) ≤ Cw1 (x). w1 ,w2 This implies that Br,p,q (Rd ) ⊂ Lrυ (Rd ). Let (tn )n∈N be a sequence in Rd such that tn → ∞ as n → ∞. Since wυ(x) does not tend to zero in Rd as x → ∞, there exists δ > 0 such that 1 (x) υ(x) w1 (x)

w1 ,w2 ≥ δ > 0 for x → ∞. To proof that the embedding of the space Br,p,q (Rd ) into Lrυ (Rd ) w1 ,w2 is never compact, let us take any fixed f ∈ Br,p,q (Rd ) and define a sequence (fn )n∈N , where −1 fn = w1 (tn )Ltn (f ). Since w1 ≻ w2 , there exists C ′ > 0 such that w2 (x) ≤ C ′ w1 (x) and so w1 ,w2 the sequence (fn )n∈N is bounded in Br,p,q (Rd ). Indeed, we have −1 −1 w1 ,w2 w1 ,w2 1 ,w2 kfn kw r,p,q = kw1 (tn )Ltn (f )kr,p,q = w1 (tn )kLtn (f )kr,p,q 1 ,w2 ≤ w1−1 (tn ) max{w1 (tn ), w2 (tn )}kf kw r,p,q 1 ,w2 ≤ w1−1 (tn ) max{w1 (tn ), C ′ w1 (tn )}kf kw r,p,q 1 ,w2 = max{1, C ′ }kf kw r,p,q .

Now, we will prove that there would not exist a subsequence of (fn )n∈N which is convergent in Lrυ (Rd ). The sequence in the above certainly converges in the vague topology. Indeed, for all k ∈ Cc (Rd ), we get Z Z fn (x)k(x)dx = w1−1 (tn )Ltn f (x)k(x)dx Rd Rd Z 1 = Ltn f (x)k(x)dx w1 (tn ) Rd 1 1 1 ,w2 ≤ kkkr′ kfn kr ≤ kkkr′ kfn kw (3.2) r,p,q . w1 (tn ) w1 (tn ) Since the right-hand side of (3.2) tends to zero as n → ∞, we have Z fn (x)k(x)dx → 0. Rd

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Finally by Theorem 3.1, the only possible limit of (fn ) in Lrυ (Rd ) is zero. It is known that the function x → kLx f kr,υ is equivalent to the weight function υ, i.e., there exist c1 (f ), c2 (f ) > 0 depending on f such that c1 υ(x) ≤ kLx f kr,υ ≤ c2 υ(x) for all x ∈ G. Therefore kfn kr,υ = w1−1 (tn )kLtn (f )kr,υ ≥ c1 w1−1 (tn )υ(x). Since

υ(tn ) w1 (tn )

(3.3)

≥ δ > 0 for all tn , by using (3.3) we write kfn kr,υ ≥ c1 w1−1 (tn )υ(x) ≥ c1 δ.

This means that it is not possible to find a norm convergent subsequence of (fn )n∈N in Lrυ (Rd ). Theorem 3.3 For 1 ≤ q ≤ p < ∞, let w1 , w2 and w3 be Beurling weight functions on Rd . If w3 (x) d w1 (x) does not tend to zero in R as x → ∞, or w3 (x) (2) w1 ≺ w2 , w3 ≺ w2 and w does not tend to zero in Rd as x → ∞, 2 (x) w1 ,w2p then the embedding of the space Br,p,q (Rd ) into L(p, q, w3p dµ)(Rd ) is never compact.

(1) w1 ≻ w2 ≻ w3 and

Proof (1) First of all, since w1 ≻ w2 ≻ w3 , there are constants C1 , C2 > 0 such that w1 ,w2p d w3 (x) ≤ C2 w2 (x) and w2 (x) ≤ C1 w1 (x). This implies that Br,p,q (R ) ⊂ L(p, q, w3p dµ)(Rd ). 3 (x) Let (tn )n∈N be a sequence such that tn → ∞ as n → ∞. Since w w1 (x) does not tend to zero w3 (x) w1 (x) ≥ δ > 0 for x → ∞. To proof the embedding of the space Br,p,q (Rd ) into L(p, q, w3p dµ)(Rd ) is never compact, let us w1 ,w2p any fixed f ∈ Br,p,q (Rd ) and define a sequence (fn )n∈N , where fn = w1−1 (tn )Ltn (f ). w1 ,w2p sequence is bounded in Br,p,q (Rd ). Indeed, we write

in Rd as x → ∞, there exists δ > 0 such that w1 ,w2p

w ,w p

w ,w p

that take This

w ,w p

1 1 1 2 2 2 kfn kr,p,q = kw1−1 (tn )Ltn (f )kr,p,q = w1−1 (tn )kLtn (f )kr,p,q

w ,w p

1 2 ≤ w1−1 (tn ) max{w1 (tn ), w2 (tn )}kf kr,p,q

w ,w p

1 2 ≤ w1−1 (tn ) max{w1 (tn ), C1 w1 (tn )}kf kr,p,q

w ,w p

1 2 = max{1, C1 }kf kr,p,q .

Now, we will prove that there would not exist a subsequence of (fn )n∈N which is convergent in L(p, q, w3p dµ)(Rd ). The sequence in the above certainly converges in the vague topology. Indeed, for all k ∈ Cc (Rd ), we get Z Z fn (x)k(x)dx = w1−1 (tn )Ltn f (x)k(x)dx Rd Rd Z 1 = Ltn f (x)k(x)dx w1 (tn ) Rd 1 1 w1 ,w2p ≤ kkkr′ kfn kr ≤ kkkr′ kfn kr,p,q . (3.4) w1 (tn ) w1 (tn ) Since the right-hand side of (3.4) tends to zero as n → ∞, we have Z fn (x)k(x)dx → 0. Rd

Some Aspects of Lrw1 (G) ∩ L(p, q, w2 d µ)(G)

319

Finally by Theorem 3.1, the only possible limit of (fn ) in L(p, q, w3p dµ)(Rd ) is zero. By Theorem 2.14, it is known that the function x → kLx f kp,q,w3p is equivalent to the weight function w3 , i.e., there exist c1 (f ), c2 (f ) > 0 such that c1 w3 (x) ≤ kLxf kp,q,w3p ≤ c2 w3 (x) for all x ∈ G. Therefore kfn kp,q,w3p = w1−1 (tn )kLtn (f )kp,q,w3p ≥ c1 w1−1 (tn )w3 (x). Since

w3 (x) w1 (x)

(3.5)

≥ δ > 0 for all tn , by using (3.5) we write kfn kp,q,w3p ≥ c1 w1−1 (tn )w3 (x) ≥ c1 δ.

This means that it is not possible to find a norm convergent subsequence of (fn )n∈N in Lrυ (Rd ). (2) This part is similar to part (1). Briefly, the sequence will be formed as fn = w2−1 (tn ) ·Ltn (f ) and the rest. Now, we will introduce a proposition whose proof is easy. Theorem 3.4 Let w1 ≈ w2 , wp3 and w4 be Beurling weight functions on Rd and 1 ≤ q ≤ w1 ,w2 w3 ,w4p d d p < ∞. Then the embedding Br,p,q (R ) into Br,p,q (R ) is continuous if and only if w3 ≺ w1 , w4 ≺ w2 . Theorem 3.5 Let w1 , w2 , w3 and w4 be Beurling weight functions on Rd and 1 ≤ q ≤ p < ∞. If 3 (x) d (1) w4 ≺ w2 ≺ w1 , w3 ≺ w1 and w w1 (x) does not tend to zero in R as x → ∞, or w3 (x) d w2 (x) does not tend to zero in R as x → ∞, w4 (x) (3) w4 ≺ w2 , w3 ≺ w1 ≺ w2 and w does not tend to zero in Rd as x → ∞, 2 (x) w4 (x) (4) w4 ≺ w2 ≺ w1 , w3 ≺ w1 and w does not tend to zero in Rd as x → ∞, 1 (x) p w1 ,w2 w3 ,w4p then the embedding of the space Br,p,q (Rd ) into Br,p,q (Rd ) is never compact.

(2) w3 ≺ w1 ≺ w2 , w4 ≺ w2 and

or or

Proof (1) Let us assume that w4 ≺ w2 ≺ w1 , w3 ≺ w1 . Then there are constants C1 , C2 > 0 psuch that w4 (x) ≤ C1 w2 (x) and w3 (x) ≤ C2 w1 (x). By Theorem 2.15, thisp implies w1 ,w2 w3 ,w4p w1 ,w2p w3 ,w4 that Br,p,q (Rd ) ⊂ Br,p,q (Rd ) and the unit function i from Br,p,q (Rd ) into Br,p,q (Rd ) is w3 (x) continuous. Now assume that w1 (x) does not tend to zero in Rd as x → ∞ and (fn )n∈N is a w ,w p

w ,w p

1 3 2 4 bounded sequence in Br,p,q (Rd ). If any subsequence of (fn )n∈N is convergent in Br,p,q (Rd ), then this subsequence is also convergent in Lrw3 (Rd ). However, this is not possible by Theorem

w ,w p

1 2 3.2, since the embedding of the space Br,p,q (Rd ) into Lrw3 (Rd ) is never compact. (2) This part is similar to part (1). (3) Let us assume that w4 ≺ w2 , w3 ≺ w1 ≺ w2 . Then there are constants C1 , C2 > 0 such w1 ,w2p that wp4 (x) ≤ C1 w2 (x) and w3 (x) ≤ C2 w1 (x). By Theorem 2.15, thispimplies that Br,p,q (Rd ) ⊂ w3 ,w4 w1 ,w2p w3 ,w4 d d Br,p,q (R ) and the unit function i from Br,p,q (R ) into Br,p,q (Rd ) is continuous. Now w4 (x) d assume that w2 (x) does not tend to zero in R as x → ∞ and (fn )n∈N is a bounded sequence in

w ,w p

w ,w p

1 3 2 4 Br,p,q (Rd ). If any subsequence of (fn )n∈N is convergent in Br,p,q (Rd ), then this subsequence p d is also convergent in L(p, q, w4 dµ)(R ). However this is not possible by Theorem 3.3, since the w1 ,w2p embedding of the space Br,p,q (Rd ) into L(p, q, w4p dµ)(Rd ) is never compact. (4) This part is similar to part (3).

˙ Eryilmaz and C. Duyar I.

320

Theorem 3.6 For 0 < q ≤ 1, let w1 , w2 and w3 be Beurling weight functions on Rd . If 3 (x) d w1 ≻ w2 ≻ w3 and w w1 (x) does not tend to zero in R as x → ∞, then the embedding of the w1 ,w2 d d space Br,1,q (R ) into L(1, q, w3 dµ)(R ) is never compact. Theorem 3.7 Let w1 , w2 , w3 and w4 be Beurling weight functions on Rd and 0 < q ≤ 1. If w3 (x) d w1 (x) does not tend to zero in R as x → ∞, w3 (x) (2) w3 ≺ w1 ≺ w2 , w4 ≺ w2 and w does not tend to zero in Rd as x → ∞, 2 (x) w4 (x) (3) w4 ≺ w2 , w3 ≺ w1 ≺ w2 and w does not tend to zero in Rd as x → ∞, 2 (x) w4 (x) (4) w4 ≺ w2 ≺ w1 , w3 ≺ w1 and w does not tend to zero in Rd as x → ∞, 1 (x) w1 ,w2 w3 ,w4 then the embedding of the space Br,1,q (Rd ) into Br,1,q (Rd ) is never compact.

(1) w4 ≺ w2 ≺ w1 , w3 ≺ w1 and

or or or

The proofs of Theorem 3.6 and 3.7 can be derived from Theorems 3.3–3.5.

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