Embeddings and the Growth Envelope of Besov Spaces Involving Only Slowly Varying Smoothness

Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness

Bohum´ırOpic

Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University, Prague

(joint work with A. Caetano and A. Gogatishvili)

Santiago de Compostela, July, 2011

Santiago de Compostela, July, 2011 1 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Basic notation

A - B ... ∃c ∈ (0, +∞) s.t. A ≤ cB A ≈ B ... A - B and B - A Ω ⊂ IRn a domain, f ∈ M(Ω) ∗ f (t) := inf {λ ≥ 0 : |{x ∈ Ω: |f (x)|>λ}|n ≤ t} ∗∗ −1 R t ∗ f (t) := t 0 f (τ) dτ, t > 0 X , Y ... (quasi-) Banach spaces X ,→ Y n n f ∈ Lp(IR ), 1 ≤ p < ∞, h ∈ IR n ∆hf (x) := f (x + h) − f (x), x ∈ IR

ω1(f , t)p := supx∈IRn,|h|≤t k∆hf (x)kp, t ≥ 0

Santiago de Compostela, July, 2011 2 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Slowly varying functions Deﬁnition b ∈ SV = SV (0, 1) ⇐⇒ b ∈ M+(0, 1) , 0 6≡ b 6≡ ∞, and + + ∀ε > 0 ∃ hε ∈ M (0, 1; ↑), ∃ h−ε ∈ M ((0, 1; ↓) such that

ε −ε t b(t) ≈ hε(t) and t b(t) ≈ h−ε(t) ∀ t ∈ (0, 1)

b ∈ SV (0, 1) ⇒ ∃ b˜ ∈ SV (0, 1) ∩ C((0, 1)), b˜ ≈ b on (0, 1)

Examples `(t) := 1 + | log t|, t ∈ (0, 1) `1 := `, ì := `1 ◦ ì−1, i > 1 m α 1 α Q i b(t) = ` (t) := i=1 ì (t), t ∈ (0, 1), m ∈ IN, m α = (α1, . . . , αm) ∈ IR β 2 b(t) = exp |log t| , t ∈ (0, 1), β ∈ (0, 1)

Santiago de Compostela, July, 2011 3 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness 0,b 0,b n The Besov space Bp,r = Bp,r (R )

Deﬁnition Let 1 ≤ p < ∞, 1 ≤ r ≤ ∞ and let b ∈ SV (0, 1) be such that

−1/r kt b(t)kr,(0,1) = ∞. (1)

0,b 0,b n n The Besov space Bp,r = Bp,r (R ) consists of those functions f ∈ Lp(R ) for which the norm

−1/r kf k 0,b := kf kp + kt b(t) ω1(f , t)pkr,(0,1) (2) Bp,r is ﬁnite.

Santiago de Compostela, July, 2011 4 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem 1 Let 1 ≤ p < ∞, 1 ≤ r ≤ ∞, 0 < q ≤ ∞ and let b ∈ SV (0, 1) satisfy (1). Assume that ω is a non-negative measurable function on (0, 1). Then

∗ kω(t)f (t)kq,(0,1) kf k 0,b (3) . Bp,r

0,b n for all f ∈ Bp,r (R ) if and only if

Z t 1/p ∗ −1/r 1/n ∗ p kω(t)f (t)kq,(0,1) . t b(t ) (f (u)) du (4) 0 r,(0,1)

n for all f ∈ M0(R ).

Santiago de Compostela, July, 2011 5 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Main ingredients of the proof

Theorem (Kolyada, (1988)) If 1 ≤ p < ∞, then

Z ∞ Z s 1/p −p/n ∗ ∗ p ds t s (f (u) − f (s)) du . ω1(f , t)p tn 0 s

n for all t > 0 and f ∈ Lp(R ).

Santiago de Compostela, July, 2011 6 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem (Caetano, Gogatishvili, Opic, (2008)) n Let f ∈ Lp(R ) and ( f ∗(V |x|n), x ∈ n, if p = 1, F (x)) := n R ∗∗ n n f (Vn|x| ), x ∈ R , if p ∈ (1, ∞),

where Vn := |Bn(0, 1)|n. Then

Z ∞ Z s 1/p −p/n ∗ ∗ p ds ω1(F , t)p . t s (f (u) − f (s)) du tn 0 s

n for all t > 0 and f ∈ Lp(R ).

Santiago de Compostela, July, 2011 7 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness loc The classical Lorentz space Λq (ω)

Deﬁnition + loc Given q ∈ (0, ∞] and ω ∈ M (0, 1) , the classical Lorentz space Λq (ω) n is deﬁned to be the set of all measurable functions f ∈ R such that

∗ kωf kq;(0,1) < ∞.

In particular, putting ω(t) := t1/p−1/q b(t), t ∈ (0, 1), where loc b ∈ SV (0, 1), we obtain the Lorentz-Karamata spaceL p,q;b.

Note that Lorentz-Karamata spaces involve as particular cases the generalized Lorentz-Zygmund spaces, the Lorentz spaces, the Zygmund classes and Lebesgue spaces.

Santiago de Compostela, July, 2011 8 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Some more notation

1 ≤ p < ∞, 1 ≤ r ≤ ∞, 0 < q ≤ ∞

If b ∈ SV (0, 1), we put b(t) := 1 for t ∈ [1, 2],

−1/r 1/n br (t) := ks b(s )kr,(t,2), t ∈ (0, 1),

Z t ∗∗ −1 b∞(t) := t b∞(τ) dτ, t ∈ (0, 1). 0 If ω ∈ M+(0, 1) , we put

Ωq(t) := kω(s)kq,(0,t), t ∈ (0, 1].

1 1 1 ρ := ∞ if p ≤ q and ρ := q − p if q < p

Santiago de Compostela, July, 2011 9 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem 2 Let 1 ≤ p < ∞, 1 ≤ r ≤ ∞, 0 < q ≤ ∞ and let b ∈ SV (0, 1) satisfy (1). 0,b n (i) Let 1 ≤ r ≤ q ≤ ∞. Then inequality (3) holds for all f ∈ Bp,r (R ) if and only if

− 1 − 1 Ωq(1) + ks p ρ Ωq(s)kρ,(t,1) . br (t) for all t ∈ (0, 1). (5) 0,b n (ii) Let 0 < q < r < ∞. Then inequality (3) holds for all f ∈ Bp,r (R ) if and only if

Z 1 qr 2 − 1 − 1 r−q r 1 r dt Ωq(1) + ks p ρ Ωq(s)kρ,(t,1) br (t) q−r b(t n ) < ∞. (6) 0 t

0,b n (iii) Let 0 < q < r = ∞. Then inequality (3) holds for all f ∈ Bp,r (R ) if and only if

Z 1 1 q − p − ρ ∗∗ −q Ωq(1) + ks Ωq(s)kρ,(t,1) d(b∞(t) ) < ∞. (7) (0,1) Santiago de Compostela, July, 2011 10 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem 3 Let 1 ≤ p < ∞, 1 ≤ r ≤ ∞, 0 < q ≤ ∞. Let b ∈ SV (0, 1) satisfy (1). Deﬁne, for all t ∈ (0, 1),

b (t)1−r/q+r/ max{p,q}b(t1/n)r/q−r/ max{p,q} if r 6= ∞ b˜(t) := r . (8) b∞(t) if r = ∞

Then the inequality

1/p−1/q ˜ ∗ kt b(t)f (t)kq,(0,1) kf k 0,b (9) . Bp,r

0,b n holds for all f ∈ Bp,r (R ) if and only ifq ≥ r.

Santiago de Compostela, July, 2011 11 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem 4 Let 1 ≤ p < ∞, 1 ≤ r ≤ q ≤ ∞ and let b ∈ SV (0, 1) satisfy (1). + (i) Let κ ∈ M0 (0, 1; ↓). Then the inequality 1/p−1/q ˜ ∗ kt b(t)κ(t)f (t)kq,(0,1) kf k 0,b (10) . Bp,r

0,b n holds for all f ∈ Bp,r (R ) if and only if κ is bounded. + (ii) Let κ ∈ M0 (0, 1) and q = ∞. Then inequality (10) holds for all 0,b n f ∈ Bp,r (R ) if and only if kκk∞,(0,1) < ∞.

Santiago de Compostela, July, 2011 12 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Deﬁnition (GE)

n Let (A, k · kA) ⊂ M(R ) be a quasi-normed space s.t. A 6,→ L∞. A functionh ∈ C +((0, ε]; ↓) , ε ∈ (0, 1), is called the (local) growth envelope function of the space A provided that

h(t) ≈ sup f ∗(t) for all t ∈ (0, ε]. (11) kf kA≤1

Given a growth envelope function h of the space A and a number u ∈ (0, ∞], the pair (h, u) is called the (local) growth envelope of the space A when the inequality

Z f ∗(t)q 1/q dµH (t) . kf kA (0,ε) h(t)

(with the usual modification when q = ∞) holds for all f ∈ A if and only if the exponent q satisfies q ≥ u. Here µH is the Borel measure associated with the non-decreasing function H(t) := − ln h(t), t ∈ (0, ε). The componentu in the growth envelope pair is called the fine index. Santiago de Compostela, July, 2011 13 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorem 5 Let 1 ≤ p < ∞, 1 ≤ r ≤ ∞ and let b ∈ SV (0, 1) satisfy (1). Then the 0,b n growth envelope of Bp,r (R ) is the pair

−1/p −1 (t br (t) , max{p, r}).

Note that Z 2 −1/p −1 −1/p−1 −1 t br (t) ≈ s br (s) ds =: h(t) ∀t ∈ (0, 1) t and that the function h is non-increasing and absolutely continuous on the whole interval (0, 1).

Santiago de Compostela, July, 2011 14 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Remark Put H(t) := − ln h(t) for t ∈ (0, ε), where ε ∈ (0, 1) is small enough. 0 1 Since H (t) ≈ t for a.e. t ∈ (0, ε) , the measure µH associated with the dt function H satisﬁes dµH (t) ≈ t . Thus, by Deﬁnition (GE) and Theorem 5,

1/p−1/q ∗ 0,b n kt br (t)f (t)kq,(0,ε) kf k 0,b for all f ∈ B ( ) (12) . Bp,r p,r R if and only if q ≥ max{p, r}. (13) Hence, if (13) holds, then inequality (12) gives the same result as inequality (9) of Theorem 3 (since (13) implies that b˜ = br ). However, if r ≤ q < p, then inequality (12) does not hold, while inequality (9) 0,b n does. This means that the embeddings of Besov spaces Bp,r (R ) given by Theorem 3 cannot be described in terms of growth envelopes when 1 ≤ r ≤ q < p < ∞.

Santiago de Compostela, July, 2011 15 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Theorems 3 and 4 imply that if 1 ≤ p < ∞, 1 ≤ r ≤ q ≤ ∞, then

0,b n Bp,r (R ) ,→ Lp,q;b˜(Ω),

n where Ω is a domain in R of ﬁnite Lebesgue measure, and that this embedding is optimal within the scale of Lorentz-Karamata spaces.

Santiago de Compostela, July, 2011 16 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Compact embeddings

Theorem 6 Let 1 ≤ p < ∞, 1 ≤ r ≤ q ≤ ∞ and let b ∈ SV (0, 1) satisfy (1). Let Ω n be a bounded domain in R and let 0 < P ≤ p. Assume that b¯ ∈ SV (0, 1) and, ifP = p > q, that b¯/b˜ is a non-decreasing function on the interval (0, δ) for some δ ∈ (0, 1). Then

0,b n 1 Bp,r (R ) ,→,→ LP,q;b(Ω) ) if and only if t1/P b¯(t) lim = 0. t→0+ t1/pb˜(t)

0,b n ) This means that the mapping u 7→ u|Ω from Bp,r (R ) into LP,q;b(Ω) is compact.

Santiago de Compostela, July, 2011 17 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness Thank you for your attention

Santiago de Compostela, July, 2011 18 Bohum´ırOpic (Prague, Czech Republic) Embeddings and the growth envelope of Besov spaces involving only slowly varying/ 18smoothness