Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces

Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces J. S. Neves CMUC/University of Coimbra Coimbra, 15th December 2010 (Joint work with A. Gogatishvili and B. Opic, Mathematical Institute, Academy of Sciences of Czech Republic)

Outline of the talk 1 Introduction Classical Results Better target spaces 2 r.i. Banach function spaces (r.i. BFS) Banach function norm ρ Banach function space X(IR n ) Non-increasing rearrangement f Representation space Lorentz-Karamata spaces 3 Bessel-potential-type spaces and optimal embeddings Bessel-potential-type spaces Generalized Hölder spaces Optimal Embeddings: super-limiting case Optimal Embeddings: limiting case Bessel-potential spaces H σ X(IR n ) and optimal embeddings Optimal Embeddings: limiting case-again! Optimal Embeddings: super-limiting case-again! 4 References

Classical results Sobolev s classical embedding theorem Ω IR n is a domain with a sufficient smooth boundary Limiting case, i.e., p = n k 1 (q = +, if p = 1 so that k = n). Wp k (Ω) Lq(Ω), for all q [p,+ ), Super-limiting case, i.e., p > n/k, W k p (Ω) C B(Ω). When k = 1 + n/p IN, W k p (Ω) C 0,α (Ω), for all α (0, 1). If p = 1, we can have α = 1 - Lipschitz space.

Better target spaces Super-Limiting case: almost Lipschitz functions Brézis-Wainger [1980], H 1+n/p p (IR n ) C 0,λ( ) (IR n ) with λ(t) := t log t 1 p, t (0, 1 ), which implies, for some positive constant c, 2 f(x) f(y) c f H 1+n/p p x y log x y 1 p for all f H 1+n/p p (IR n ) and x, y IR n such that 0 < x y < 1 2.

Banach function norm ρ ρ : M + (IR n ) [0, ] ρ(f) = 0 f = 0 a.e. in IR n ρ(αf) = αρ(f) if α 0 ρ(f + g) ρ(f) + ρ(g) 0 g f a.e. = ρ(g) ρ(f) (lattice property) 0 f n ր f a.e. = ρ(f n) ր ρ(f) (Fatou property) E n < = ρ(χ E ) < E n < = R f(x) dx C E Eρ(f) f M + (IR n ), 0 < C E <

Banach function space X(IR n ) ρ Banach function norm on M(IR n ) BFS X(IR n ) = X ρ(ir n ) := {f M(IR n ) : ρ( f ) < } f X := ρ( f ) associate BFS X (IR n ) := X ρ (IR n ) associate norm ρ ρ (g) := sup{ R f(x)g(x) dx: ρ(f) 1} IR n Example 8 < Z 1/p f p (x) dx if 1 p < L p (IR n ), 1 p, ρ p(f) := IR : n ess sup IR nf(x) if p = (L p (Ω)) = L p (IR n ), 1/p + 1/p = 1

Non-increasing rearrangement f Definition (the distribution function) µ f (λ) := {x IR n : f(x) > λ} n, λ 0 Theorem Z f(x) q dx = q Z Ω 0 λ q 1 µ f (λ) dλ q (0, ). g equimeasurable with f if µ g(λ) = µ f (λ), λ 0 Definition (the non-increasing rearrangement) f (t) := inf{λ 0 : µ f (λ) t} = {λ 0; µ f (λ) > t} 1, t 0 f and f are equimeasurable: µ f (λ) = µ f (λ), λ 0.

Non-increasing rearrangement f : Example a 1 µ(b 3 ) µ f (s) a 2 a 3 f(x) µ(b 2 ) µ(b 1 ) A 2 A 3 A 1 x a 3 a 2 a 1 s 3X f(x) = a j χ Aj (x) j=1 µ f (s) = 3X kx µ(a j ) χ [ak+1,a k )(s) (a 4 := 0) k=1 j=1 {z } =:µ(b k ) a 1 f (t) a 2 3X f (t) = a j χ [µ(bj 1 ),µ(b j ))(t) (µ(b 0 ) := 0) a 3 j=1 µ(b 1 ) µ(b 2 ) µ(b 3 ) t

Non-increasing rearrangement f : Example t I 1 (x) = x 1, x IR 2 \{0} (I 1 ) (t) = ( t π ) 1/2, t > 0. I σ(x) = x σ n, x IR n \{0}, 0 < σ < n (I σ) (t) = ( t ω n ) σ/n 1, t > 0.

r.i. Banach function space BFS X is rearrangement invariant (r.i.) f X, g equimeasurable with f, then g X and g X = f X Luxemburg representation theorem = If X(IR n ) is a r.i. BFS, then a r.i. BFS X over ((0, ), µ 1 ) s.t. f X = f X f X(IR n ). X representation space of X(IR n ) Example Z f(x) p dx = IR n Z 0 f (t) p dt, 0 < p <, f M(IR n ) ess sup IR n f = f (0), f M(IR n ) L p (IR n ), 1 p are r.i. Banach function spaces

Lorentz-Karamata spaces Definition b slowly varying function on (0,+ ) (b SV(0,+ )): Lebesgue measurable function b : (0, + ) (0, + ); for each ǫ > 0, t ǫ b(t) f ǫ(t) ր and t ǫ b(t) f ǫ(t) ց.

Lorentz-Karamata spaces Definition b slowly varying function on (0,+ ) (b SV(0,+ )): Lebesgue measurable function b : (0, + ) (0, + ); for each ǫ > 0, t ǫ b(t) f ǫ(t) ր and t ǫ b(t) f ǫ(t) ց. Examples Let α, β R: b(t) = (1 + log t ) α (1 + log(1 + log t )) β ; b(t) = exp( log t α ), 0 < α < 1.

Lorentz-Karamata spaces Definition b slowly varying function on (0,+ ) (b SV(0,+ )): Lebesgue measurable function b : (0, + ) (0, + ); for each ǫ > 0, t ǫ b(t) f ǫ(t) ր and t ǫ b(t) f ǫ(t) ց. Lorentz-Karamata space L p,q;b (R n ) f p,q;b := t 1 p 1 q b(t) f (t) q;(0,+ ) Example b = l α - product of powers of iterated logs Logarithmic Bessel-potential-type spaces (Edmunds, Gurka and Opic, 1997).

Bessel-potential-type spaces Let σ > 0, p (1,+ ), q [1,+ ] and b SV(0,+ ). Let g σ be Bessel kernel. H σ L p,q;b (IR n ) = u : u = g σ f, f L p,q;b (IR n ) endowed with the (quasi-)norm u σ;p,q;b := f p,q;b. Notation: H 0 L p,q;b (IR n ) = L p,q;b (IR n ). Bessel Kernel g σ, com σ > 0: cg σ(ξ) = (2π) n/2 (1 + ξ 2 ) σ/2, ξ IR n. where the Fourier transform ˆf of a function f is given by ˆf(ξ) R := (2π) n/2 e iξ x f(x) dx IR n

Bessel-potential-type spaces Let σ > 0, p (1,+ ), q [1,+ ] and b SV(0,+ ). Let g σ be Bessel kernel. H σ L p,q;b (IR n ) = u : u = g σ f, f L p,q;b (IR n ) endowed with the (quasi-)norm u σ;p,q;b := f p,q;b. Notation: H 0 L p,q;b (IR n ) = L p,q;b (IR n ). Example b = l α - product of powers of iterated logs Logarithmic Bessel-potential-type spaces (Edmunds, Gurka and Opic, 1997).

Bessel-potential-type spaces Let σ > 0, p (1,+ ), q [1,+ ] and b SV(0,+ ). Let g σ be Bessel kernel. H σ L p,q;b (IR n ) = u : u = g σ f, f L p,q;b (IR n ) endowed with the (quasi-)norm u σ;p,q;b := f p,q;b. Notation: H 0 L p,q;b (IR n ) = L p,q;b (IR n ). Theorem [Neves 2004 (Edmunds, Gurka and Opic, 1997 - Logarithmic Bessel-potential-type spaces)] If k N, p (1, + ) and q (1, + ) H k L p,q;b (R n ) = W k L p,q;b (R n ) and the corresponding (quasi-)norms are equivalent, where W k L p,q;b (R n ) := {u : D α u L p,q;b (R n ), α k} and u W k L p,q;b (R n ) = X α k D α u p,q;b.

Notation h IR n, f C B (IR n ) h f(x) := f(x + h) f(x), x IR n k+1 h f(x) := h ( k hf)(x), x IR n, k IN ω k (f, t) := sup h t k hf, t 0. ω(f, t) := ω 1 (f, t). Definition (the class L k r ) the class L k r, k IN, r (0,+ ], consists of all continuous functions λ : (0, 1) (0,+ ) such that t 1/r 1 λ(t) = + (1) r;(0,1) t 1/r t k λ(t) < + (2) r;(0,1) k, r and λ are connected by (1) and (2)

the generalized Hölder space Definition k IN, r (0,+ ], λ L k r (IR n ) := {f C B (IR n ); f Λ k,λ( ),r (IR n ) < + } f Λ k,λ( ),r (IR n ) := f + t 1/r ω k (f, t) λ(t) Λ k;λ( ),r r;(0,1) If k = 1, we sometimes use Λ λ( ),r(ir n ) := Λ 1;λ( ),r (IR n ) and C 0,λ( ) (IR n ) := Λ λ( ), (IR n ). conditions (1) and (2) are natural the scale Λ k,λ( ),r contains both Hölder and Zygmund spaces λ(t) t α, α (0, 1] Λ 1,λ( ), (IR n ) C 0,α (IR n ) λ(t) t α, α > 0, k > α Λ k,λ( ), (IR n ) Z α (IR n )

Optimal Embeddings: super-limiting case Theorem [Gogatishvili, N. & Opic (2005)] Let σ [1, n + 1), max{1, n/σ} < p < n/(σ 1), q (1, + ), r [q,+ ] and let b SV(0, + ). Ω IR n a nonempty domain. λ(t) = t σ n/p [b(t n )] 1, t (0, 1]. (i) Then H σ L p,q;b (R n ) Λ λ( ),r(r n ). (ii) If lim t 0 + t λ(t) τ 1/r τ µ(τ) r;(0,t) = 0, H σ L p,q;b (R n ) / Λ µ(.),r(ω) (iii) Let q (0, q). H σ L p,q;b (R n ) / Λ λ( ),q (Ω). (i) - N. (2004); (i),(ii)- Logarithmic Bessel potential space-edmunds, Gurka and Opic (1997,2000) - r = + ;

Optimal Embeddings: super-limiting case Theorem [Gogatishvili, N. & Opic (2005)] Let σ (1, n + 1), p = n/(σ 1), q (1,+ ), r [q,+ ] and b SV(0, + ) such that t 1/q [b(t)] 1 q ;(0,1) = +. Z 2 «1/q +1/r λ r(t) = t [b(t n )] q /r τ 1 [b(τ)] q dτ, t (0, 1]. t n (i) Then H σ L n/(σ 1),q;b (R n ) Λ λr(.),r (R n ). τ 1/r τ λ r(τ) r;(0,t) t 0 + τ 1/r τ µ(τ) r;(0,t) (ii) If lim = 0, H σ L n/(σ 1),q;b (R n ) / Λ µ(.),r(ω) (iii) Let q (0, q). H σ L n/(σ 1),q;b (R n ) / Λ λ q (.),q (Ω). (i) - N. (2004); (i),(ii) - Logarithmic Bessel potential space-edmunds, Gurka and Opic (1997,2000) - r = + ;

Optimal Embeddings: super-limiting case [(i)] general ideas S(IR n ) is dense in H σ L p,q;b (IR n ). Let u S(R n ) H σ L p,q;b (IR n ). (Lifting argument) u x i H σ 1 L p,q;b (R n ), for i = 1,..., n. Use embedding results for limiting case and inequality of DeVore and Sharpley inequality Z t ω(u, t) u (σ n ) dσ, t > 0. 0 Use of appropriate Hardy-type inequalities. Hence u Λ λ( ),r u σ;p,q;b for all u S(R n ). Proofs: [(ii), (iii)] general ideas Use of appropriate extremal functions!

Optimal Embeddings: super-limiting case-examples Example Let n IN such that n 2 and let k IN such that 2 k n. Let p = n k 1, q (1,+ ) and r [q,+ ]. Let α IR and let b SV(0, + ) be defined by b(t) = (1 + log t ) α, t > 0. If α < 1 q, then with W k L p,q (log L) α (IR n ) Λ λq( ),q (IR n ) Λ λr( ),r (IR n ) C 0,λ ( ) (IR n ), λ r(t) = t(1 + log t ) 1 r + 1 q α and λ (t) = t(1 + log t ) 1 q α, t (0, 1]. If α = 1 q, then with W k L p,q (log L) α (IR n ) Λ λq( ),q (IR n ) Λ λr( ),r (IR n ) C 0,λ ( ) (IR n ), λ r(t) = t(1 + log t ) 1 r (1 + log(1 + log t )) 1 r + 1 q β, t (0, 1]; λ (t) = t(1 + log(1 + log t )) 1 q β, t (0, 1]. If α > 1 q then W k L p,q (log L) α (IR n ) Lip(IR n ).

Optimal Embeddings: limiting case Theorem (Gogatishvili, N. and Opic (2007)) Let 0 < σ < n, p = n/σ, q (1, + ) and b SV(0, + ) such that t 1/q [b(t)] 1 q ;(0,1) < +. Then where λ(t) := H σ L p,q;b (IR n ) C 0,λ( ) (IR n ). Z t n 0 [b(τ)] q! dt 1/q, t > 0. t Logarithmic Bessel potential space - Edmunds, Gurka and Opic (2005); 1 with αm > 1 q q and λ(t) = l αm m (t). H σ L p,q; 1 q,, 1 q,αm (IRn ) C 0,λ( ) (IR n ),

Bessel-potential spaces H σ X(IR n ) Let σ > 0 X = X(IR n ) = X(IR n, µ n)... r. i. BFS over (IR n, µ n) = X L 1 (IR n ) + L (IR n ) = u = f g σ is well defined for all f X H σ X(IR n ) := {u : u = f g σ, f X(IR n )} u H σ X := f X Notation: H 0 X(IR n ) = X(IR n ).

Optimal Embeddings: smoothness σ (0, n) Theorem [Gogatishvili, N. & Opic (2009) σ < 1, (2010) σ (0, n)] Let σ (0, n) and let X = X(IR n ) be a r. i. BFS. Assume that Ω is a domain in IR n. Then H σ X(IR n ) C(Ω) if and only if g σ X < +. Lemma [Gogatishvili, N. & Opic (2009) σ < 1, (2010) σ (0, n) ] Let σ (0, n), p (1, + ), q [1,+ ] and b SV(0, + ). If X = L p,q;b (IR n ), then if and only if either or p = n σ g σ X p > n σ and t 1 q (b(t)) 1 q ;(0,1) < +. (4) (3)

Optimal Embeddings: smoothness σ (0, n) Theorem [Gogatishvili, N. & Opic (2010)] Let σ (0, n) and let X = X(IR n ) = X(IR n, µ n) be a r. i. BFS such that g σ X < +. Put k := [σ] + 1, assume that r (0,+ ] and µ L k r. Then H σ X(IR n ) Λ k,µ( ),r (IR n ) if and only if Z t n t 1 r (µ(t)) 1 τ σ n 1 f (τ) dτ f X for all f X. r;(0,1) 0

Optimal Embeddings: smoothness σ (0, n) Proof: uses key estimates Let σ (0, n) and let X = X(IR n ) be a r. i. BFS such that g σ X < +. Then f g σ C(IR n ) for all f X and ω k (f g σ, t) Z t n 0 s σ n 1 f (s) ds t (0, 1), f X, where k [σ] + 1. Moreover, this estimate is sharp: given k IN, there are δ (0, 1) and α > 0 such that ω k (f g σ, t) Z t n 0 s σ n 1 f (s) ds t (0, 1), f X, where f(x) := f (β n x n )χ Cα(0,δ)(x), x = (x 1,, x n) IR n, C α(0, δ) := C α B(0, δ) with C α := {y IR n : y 1 > 0, y 2 1 > α P n i=2 y 2 i }.

Optimal Embeddings: limiting case-example Theorem [Gogatishvili, N. & Opic (2009) σ < 1, [Gogatishvili, N. & Opic (2010) σ (0, n) ] Let σ (0, n), p = n σ, q (1,+ ], r (0, + ], k = [σ] + 1, µ Lk r and let b SV(0, + ): t 1 q (b(t)) 1 q ;(0,1) < +. Let λ qr(x) := b q /r (x n ) If 1 < q r +, then Z x n 0! 1 b q (t) dt q + 1 r, x (0, 1]. t H σ L p,q;b (IR n ) Λ k,µ( ),r (IR n t 1 r (µ(t)) 1 r;(x,1) ) lim < +. x 0 + t 1 r (λ qr(t)) 1 r;(x,1) Remark When r [1,+ ], then Λ k,λqr( ),r (IR n ) = Λ 1,λqr( ),r (IR n ).

Embeddings: Examples in the limiting case Example Let σ (0, n), p = n, q (1,+ ] and r [q,+ ]. σ If α > 1 q, β IR, then H σ L p,q (log L) α (log log L) β (IR n ) Λ 1,λqr( ),r (IR n ) C 0,λq ( ) (IR n ), λ qr(t) = (1 + log t ) 1 r + 1 q α (1 + log(1 + log t )) β, t (0, 1]; λ q (t) = (1 + log t ) 1 q α (1 + log(1 + log t )) β, t (0, 1]. If α = 1 q, β > 1 q, then H σ L p,q (log L) α (log log L) β (IR n ) Λ 1,λqr( ),r (IR n ) C 0,λq ( ) (IR n ), λ qr(t) = (1 + log t ) 1 r (1 + log(1 + log t )) 1 r + 1 q β, t (0, 1]; λ q (t) = (1 + log(1 + log t )) 1 q β, t (0, 1].

Optimal Embeddings: super-limiting case-again! Theorem [Gogatishvili, N. & Opic (2009) σ < 1, [Gogatishvili, N. & Opic (2010) σ (0, n) ] Let σ (0, n), p ( n,+ ), q [1,+ ], b SV(0, + ), r (0, + ], k = [σ] + 1, σ and µ L k r. Let λ(x) := x σ p n (b(x n )) 1, x (0, 1]. If 1 q r +, then H σ L p,q;b (IR n ) Λ k,µ( ),r (IR n t ) lim 1 r (µ(t)) 1 r;(x,1) λ(x) < +. x 0 + Remark When r [1,+ ], then If σ = n + 1, then p Λ k,λ( ),r (IR n ) = Λ [σ p n ]+1,λ( ),r (IR n ). Λ k,λ( ),r (IR n ) = Λ 2,λ( ),q (IR n ).

Optimal Embeddings: super-limiting case-again! Corollary [Gogatishvili, N. & Opic (2010)] If σ (1, n), p = n, q (1, + ], r [q, + ] and b SV(0, + ) be such that σ 1 t 1/q [b(t)] 1 q ;(0,1) = +. Let λ(t) := t (b(t n )) 1, t (0, 1]. and let Then λ qr(t) := t [b(t n )] q /r Z 2 t n «1/q +1/r τ 1 [b(τ)] q dτ, t (0, 1]. H σ L p,q;b (IR n ) Λ 2,λ( ),q (IR n ) Λ 1,λqr( ),r (IR n ). Remark Previous Corollary improves Theorem 3.2 of (GNO, 2005), provided σ = 1 + n p < n and shows that the Brézis-Wainger embedding of the Sobolev space H 1+ n p p (IR n ), σ = 1 + n < n, into the space of almost Lipschitz functions is a p consequence of a better embedding whose target is the Zygmund space.

Example if n > 1, p = q, p > with n n 1, b(t) = lα 1 (t), t (0, + ), α < 1 p, H 1+ n p L p (log L) α (IR n ) Λ 2,λ( ),p (IR n ) Λ 1,λr( ),r (IR n ) Λ 1,λ ( ), (IR n ), and λ(x) := x l α 1 (x) for all x (0, 1], λ pr(x) = x (l 1 (x)) 1/p +1/r α, x (0, 1], r [p,+ ]. If α = 0, this example shows that the Brézis-Wainger embedding of the Sobolev space H 1+ n p p (IR n ), σ = 1 + n < n, into the space of almost Lipschitz functions is a p consequence of a better embedding whose target is the Zygmund space Λ 2,Id( ),n/k (IRn ) (Id( ) stands for the identity map).

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