Besov and Triebel-Lizorkin spaces of variable smoothness ...msekce.karlin.mff.cuni.cz/~vybiral/Vort/Conf/Tabarz.pdf · Lebesgue spaces (Lp) Continuous and continuously diﬀerentiable

Outline History and motivation Recent & new function spaces Properties Outlook

Besov and Triebel-Lizorkin spacesof variable smoothness and integrability

Jan Vybıral

Austrian Academy of Sciences

RICAM, Linz, Austria

September 2011FSDONA-2011, Germany

joint work with Henning Kempka (University of Jena, Germany)


Outline

◮ History and motivation◮ Besov and Triebel-Lizorkin spaces◮ Isotropic vs. anisotropic spaces

◮ Recent and new function spaces◮ Function spaces of variable integrability◮ Function spaces of variable smoothness◮ Function spaces of variable integrability and smoothness

◮ Properties of Fs(·)p(·),q(·)(R

n) and Bs(·)p(·),q(·)(R

n)

◮ Traces◮ Embeddings◮ Local means◮ Differences◮ Decompositions

◮ Outlook


History and motivation

Classical spaces of functions:

Lebesgue spaces (Lp)Continuous and continuously differentiable functions (C ,C k)Hardy spaces (Hp)

Classical spaces of distributions:

Sobolev spaces

W kp (R

n) = {f ∈ S ′(Rn) : ‖Dαf ‖p <∞, |α| ≤ k}

1950’s and 1960’s: Slobodecki spaces, Besov spaces defined bydifferences, Bessel potential spaces


Typical properties:

Sobolev embedding theorem:

W k0p0(Rn) → W k1

p1(Rn)

if 0 ≤ k1 ≤ k0 are natural numbers, 1 ≤ p0 ≤ p1 <∞ and

k0 −n

p0= k1 −

n

p1

Trace embedding theorem: (tr f )(x ′) = f (x ′, 0)

tr : W 1p (R

n) → W1−1/pp (Rn−1), 1 < p <∞

...and many others...


Fourier-analytic function spaces

Besov and Triebel-Lizorkin spaces:smooth dyadic resolution of unity

ϕ ∈ S(Rn) : ϕ(x) = 1 if |x | ≤ 1 and ϕ(x) = 0 if |x | ≥ 32

ϕ0 := ϕ, ϕj(·) := ϕ(2−j ·)− ϕ(2−j+1·)

∞∑

j=0

ϕj = 1

f =

∞∑

j=0

(ϕj f )∨, convergence in S ′(Rn)


Definition:

(i) s ∈ R, 0 < p, q ≤ ∞

‖f |B spq(R

n)‖ =

( ∞∑

j=0

2jsq‖(ϕj f )∨|Lp(R

n)‖q)1/q

(ii) s ∈ R, 0 < p <∞, 0 < q ≤ ∞

‖f |F spq(R

n)‖ =

∥∥∥∥( ∞∑

j=0

2jsq |(ϕj f )∨(·)|q

)1/q

|Lp(Rn)

∥∥∥∥


Advantages:

◮ Two (closely related) scales including many special cases(W k

p , 1 < p <∞, Zygmund spaces, Slobodecki spaces)

◮ Many tools and results available (like embeddings, traces,Fourier multipliers, pointwise multipliers, . . . )

◮ Many equivalent characterisations (differences, local means,...)

◮ Good decomposition properties (atoms, molecules, wavelets)

◮ Applications to PDE’s, stochastic, numerics, . . .

◮ Many generalisations (anisotropic spaces, spaces withdominating mixed smoothness, modulation spaces, . . . )

Disadvantages:

◮ Rather complicated definition(three parameters, decomposition of unity, Fourier transform)

◮ Some important spaces are not included (L1, L∞,W11 ,BV )

◮ Spaces on domains?


Isotropy vs. anisotropy

B and F spaces are isotropic. . . i.e. invariant under shifts and rotations

sometimes inconvenient → anisotropic spaces, weighted spaces,spaces of dominating mixed smoothness, . . .

h − p Finite Elements Method (Babuska,. . . )piecewise analytic functions


Recent function spaces

Function spaces with variable integrability

p : Rn → (0,∞] - measurable functionLp(·)(R

n) : all f : Rn → [−∞,∞], such that there is a λ > 0

p(·)(f /λ) =

∫

Rn

ϕp(x)

(|f (x)|

λ

)dx <∞

is finite, where

ϕp(t) =

tp if p ∈ (0,∞),

0 if p = ∞ and t ≤ 1,

∞ if p = ∞ and t > 1

. . . the Minkowski functional of {f :∫Rn |f (x)|

p(x)dx ≤ 1}. . .


Rn∞ := {x ∈ R

n : p(x) = ∞} and Rn0 := R

n \Rn∞

‖f |Lp(·)(Rn)‖ = inf{λ > 0 : p(·)(f /λ) ≤ 1}

= inf

{λ > 0 :

∫

Rn0

(f (x)

λ

)p(x)

dx < 1 and |f (x)| < λ for a.e. x ∈ Rn∞

}

Norm if p(·) ≥ 1; a quasi-norm if p− := infz∈Rn p(z) > 0

W 1p(·)(R

n) = {f ∈ Lp(·)(Rn) : ∇f ∈ Lp(·)(R

n)}

Orlicz (1931), Kovacik & Rakosnık (1991)Diening & Ruzicka (≈ 2000)


Maximal operator in Lp(·)(Rn) & regularity conditions

Definition:(Regularity assumptions): g ∈ C (Rd )

(i) g is locally log-Holder continuous (g ∈ Clogloc (R

n))

|g(x) − g(y)| ≤c

log(e + 1/|x − y |), x , y ∈ R

n

(ii) g is globally log-Holder continuous (g ∈ C log (Rn)) if it islocally log-Holder continuous and

∃c > 0 and g∞ ∈ R : |g(x)− g∞| ≤c

log(e + |x |), x ∈ R

n

If 1/p(·) ∈ C log (Rn) and p− > 1, then M is bounded on Lp(·)(Rn)

. . . recent book Lebesgue and Sobolev spaces with variable

exponents by L. Diening, P. Hasto, P. Harjulehto, and M. Ruzicka.


Function spaces with generalised smoothness

Many different approaches, starting already in 1960’s

Spaces with generalised smoothness:

Gol’dman, Kalyabin, Leopold, Farkas, Moura, . . .Replace 2js by σj

Spaces of variable smoothness:

Unterberger, Leopold, Besov, Almeida, Samko, . . .Replace s by s(x)

2-microlocal spaces:

Peetre, Bony, Jaffard, Kempka, . . .Replace 2js by wj(x)


New function spaces:

Variable smoothness AND integrability

0 < p− := infz∈Rn

p(z) ≤ p(x) ≤ supz∈Rn

p(z) =: p+ <∞, x ∈ Rn

Definition of Fs(·)p(·),q(·)(R

n) and Bs(·)p(·),q(R

n)

(L. Diening, P. Hasto, and S. Roudenko, 2009)s : Rn → R, p, q : Rn → (0,∞] - measurable functions

‖f |Fs(·)p(·),q(·)(R

n)‖ =

∥∥∥∥( ∞∑

j=0

2js(·)q(·)|(ϕj f )∨(·)|q(·)

)1/q(·)

|Lp(·)(Rn)

∥∥∥∥

‖f |Bs(·)p(·),q(R

n)‖ =

( ∞∑

j=0

‖2js(·)(ϕj f )∨(·)|Lp(·)(R

n)‖q)1/q


Definition of Bs(·)p(·),q(·)(R

n) (A. Almeida and P. Hasto, 2010)

‖f |Bs(·)p(·),q(·)(R

n)‖ = ‖2js(·)(ϕj f )∨(·)|ℓq(·)(Lp(·))‖

(fν)ν∈N0 . . . sequence of Lp(·)(Rn) functions

ℓq(·)(Lp(·))(fν) =∞∑

ν=0

inf

{λν > 0 : p(·)

(fν

λ1/q(·)ν

)≤ 1

}

If q+ <∞:

ℓq(·)(Lp(·))(fν) =∑

ν

‖|fν |q(·)|L p(·)

q(·)

‖

The (quasi-)norm in the ℓq(·)(Lp(·)) spaces is defined as usual by

‖fν |ℓq(·)(Lp(·))‖ = inf{µ > 0 : ℓq(·)(Lp(·))(fν/µ) ≤ 1}


Properties of As(·)p(·),q(·)(R

n)

Hardy-Littlewood maximal operator M is not bounded on

Lp(·)(ℓq(·)) and ℓq(·)(Lp(·)) for q(·) variable!

Instead - convolutions with radial decaying kernels!

ην,m(x) := 2nν(1 + 2ν |x |)−m


Theorem:(i) (DHR09) For m > n and 1/p, 1/q ∈ C log(Rn) with1 < p− ≤ p+ <∞ and 1 < q− ≤ q+ <∞:

‖(ην,m ∗ fν)ν∈N0 |Lp(·)(ℓq(·))‖ . ‖(fν)ν∈N0 |Lp(·)(ℓq(·))‖

(ii) (AH10 & KV11) For m > n + clog(1/q) and1/p, 1/q ∈ C log(Rn) with p(·) ≥ 1:

‖(ην,m ∗ fν)ν∈N0 |ℓq(·)(Lp(·))‖ . ‖(fν)ν∈N0 |ℓq(·)(Lp(·))‖

r -trick:Theorem: Let r > 0, ν ≥ 0 and m > n. Then

|g(x)| ≤ c(r ,m, n)(ην,m ∗ |g |r )1/r (x)

for all g ∈ S ′(Rd ) with supp g ⊂ {ξ : |ξ| ≤ 2ν+1}.


Properties of the new function spaces

◮ Independence on the decomposition of unity

◮ Boundedness of the ϕ-transform (. . . Frazier & Jawerth . . . )

◮ Atomic & molecular decomposition

◮ Wavelet decomposition

◮ Traces

◮ Sobolev embeddings

◮ Local means

◮ Differences

◮ Fourier multipliers


Traces

Traces on hyperplanes

Theorem(DHR): Under regularity conditions on p, q and s and if

s(·)−1

p(·)− (n − 1)

(1

min(p(·), 1)− 1

)> 0

then

tr Fs(·)p(·),q(·)(R

n) = Fs(·)− 1

p(·)

p(·),p(·) (Rn−1)


Sobolev Embeddings

(V09, AH10)

s1(x) ≤ s0(x), s0(x)−n

p0(x)= s1(x)−

n

p1(x), x ∈ R

n

Fs0(·)p0(·),q(·)

(Rn) → Fs1(·)p1(·),q(·)

(Rn)

Bs0(·)p0(·),q(·)

(Rn) → Bs1(·)p1(·),q(·)

(Rn)

infx∈Rn

(s0(x)− s1(x)) > 0 =⇒ Fs0(·)p0(·),q0(·)

(Rn) → Fs1(·)p1(·),q1(·)

(Rn)


Local means

Definition of B- and F-spaces works with ϕ∨j ∗ f

‖f |B sp,q(R

n)‖ = ‖2js(ϕ∨j ∗ f )|ℓq(Lp)‖

Under several conditions (smoothness, vanishing moments,Tauberian conditions) this may be replaced by ψj ∗ f

Especially, ψj may have compact support → local means

Connected with the boundedness of the Peetre maximal operator

Both these techniques extended to Bs(·)p(·),q(·)(R

n) and Fs(·)p(·),q(·)(R

n)

in (KV11)


Characterisation by differencesFirst order differences:

∆1hf (x) = f (x + h)− f (x), x ∈ R

n

Higher order differences:

∆Mh f (x) = ∆1

h(∆M−1h f )(x), M = 2, 3, . . .

Ball means of differences

dMt f (x) = t−n

∫

|h|≤t

|∆Mh f (x)|dh =

∫

B

|∆Mth f (x)|dh,

B = {y ∈ Rn : |y | < 1}, t > 0 and M ∈ N

σp := n

(1

min(p, 1)− 1

)and σp,q := n

(1

min(p, q, 1)− 1

)


Theorem: (cf. Triebel, Theory of function spaces; many forerun-ners: Nikol’skij, Lizorkin, Stein, Strichartz, Kalyabin, Besov, ...)(i) Let 0 < p <∞, 0 < q ≤ ∞ and σpq < s < M. Then

‖f |F sp,q(R

n)‖∗ := ‖f ‖p +

∥∥∥∥∥

(∫ ∞

0t−sq

(dMt f (x)

)q dt

t

)1/q ∣∣∣∣Lp(Rn)

∥∥∥∥∥

is an equivalent quasinorm on F sp,q(R

n).(ii) Let 0 < p, q ≤ ∞ and σp < s < M. Then

‖f |B sp,q(R

n)‖∗∗ := ‖f |Lp(Rn)‖+

∥∥∥∥(2ksdM

2−k f (x))∞k=−∞

|ℓq(Lp)

∥∥∥∥

is an equivalent quasinorm on B sp,q(R

n).


Triebel-Lizorkin spaces

‖f |Fs(·)p(·),q(·)(R

n)‖∗ := ‖f ‖p(·)

+

∥∥∥∥∥

(∫ ∞

0t−s(x)q(x)

(dMt f (x)

)q(x) dtt

)1/q(x) ∣∣∣∣Lp(·)(Rn)

∥∥∥∥∥

Discretized counterpart

‖f |Fs(·)p(·),q(·)

(Rn)‖∗∗ := ‖f ‖p(·) +

∥∥∥∥(2ks(x)dM

2−k f (x))∞k=−∞

∣∣∣∣Lp(·)(ℓq(·))∥∥∥∥

Besov spaces

‖f |Bs(·)p(·),q(·)(R

n)‖∗∗ := ‖f ‖p(·) +

∥∥∥∥(2ks(x)dM

2−k f (x))∞k=−∞

|ℓq(·)(Lp(·))

∥∥∥∥


Theorem (KV11): Let 1/p, 1/q ∈ C log(Rn) with p+, q+ <∞,

s ∈ Clog

loc (Rn), M ∈ N with M > s+ and

s− > σp−,q− ·

[1 +

clog(s)

n·min(p−, q−)

]

Then

Fs(·)p(·),q(·)(R

n) = {f ∈ Lp(·)(Rn) ∩ S ′(Rn) : ‖f |F

s(·)p(·),q(·)(R

n)‖∗ <∞}

and ‖ · |Fs(·)p(·),q(·)(R

n)‖ and ‖ · |Fs(·)p(·),q(·)(R

n)‖∗ are equivalent on

Fs(·)p(·),q(·)(R

n). The same holds true for ‖f |Fs(·)p(·),q(·)(R

n)‖∗∗.


Theorem (KV11): Let 1/p, 1/q ∈ C log(Rn), s ∈ Clogloc (R

n), M ∈ N

with M > s+ and

s− > σp− ·

[1 +

clog(1/q)

n+

clog(s)

n· p−

]

Then

Bs(·)p(·),q(·)(R

n) = {f ∈ Lp(·)(Rn)∩S ′(Rn) : ‖f |B

s(·)p(·),q(·)(R

n)‖∗∗ <∞}

and ‖ · |Bs(·)p(·),q(·)(R

n)‖ and ‖ · |Bs(·)p(·),q(·)(R

n)‖∗∗ are equivalent on

Bs(·)p(·),q(·)(R

n).


Decomposition techniquesSequence spaces

m ∈ Zn, j ∈ N0: Qj ,m cube in R

n with sides parallel to thecoordinate axes, centred at 2−jm and with side length 2−j

χj ,m := χQj,mits characteristic function

λ = {λj ,m : j ∈ N0,m ∈ Zn}

‖λ|fs(·)p(·),q(·)‖ :=

∥∥∥∥( ∞∑

j=0

∑

m∈Zn

|2js(·)λj ,mχj ,m(·)|q(·)

)1/q(·)

|Lp(·)(Rn)

∥∥∥∥

=

∥∥∥∥∑

m∈Zn

2js(·)|λj ,m|χj ,m(·)|Lp(·)(ℓq(·))

∥∥∥∥

‖λ|bs(·)p(·),q(·)

‖ :=

∥∥∥∥∑

m∈Zn

2js(·)|λj ,m|χj ,m(·)|ℓq(·)(Lp(·))

∥∥∥∥


ϕ-transform: Sϕf =(〈f , ϕj ,m〉

)j ,m

ϕj ,m. . . shifts and dilations of basic function(s) ϕ

Theorem: (DHR09)

Sϕ : Fs(·)p(·),q(·)(R

n) → fs(·)p(·),q(·)

Theorem: (K10): Atomic, molecular and wavelet decomposition

of Fs(·)p(·),q(·)(R

n) and Bs(·)p(·),q (and of 2-microlocal variants of these)

open for Bs(·)p(·),q(·)(R

n)


Outlook:

Spaces on domains:

- rather difficult due to the use of Fourier transform in thedefinition- usually defined by restriction- characterisation by differences- traces (at least for C∞-domains)- extension operators (at least for C∞-domains)- wavelets on domains- rough domains: Lipschitz?, or with boundary described again in

the terms of Fs(·)p(·),q(·)-spaces?

Duality

Mapping properties of classical operators


Approximation theory

Theoretical concepts of numerical analysis:

◮ widths

◮ entropy numbers

◮ best k-term approximation (through sequence spaces)

◮ linear vs. non-linear approximation

◮ greedy algorithms?


Literature

◮ L. Diening, P. Hasto, S. Roudenko: Function spaces of variable smoothness and

integrability, J. Funct. Anal. 256 (2009), (6), 1731–1768.

◮ A. Almeida, P. Hasto: Besov spaces with variable smoothness and integrability,J. Funct. Anal. 258 (2010), no. 5, 1628–1655.

◮ J. Vybıral: Sobolev and Jawerth embeddings for spaces with variable smoothness

and integrability, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 529–544.

◮ H. Kempka: Atomic, molecular and wavelet decomposition of 2-microlocal

Besov and Triebel-Lizorkin spaces with variable integrability, Funct. Approx. 43

(2010), (2), 171–208.

◮ H. Kempka, J. Vybıral: A note on the spaces of variable integrability and

summability of Almeida and Hasto, submitted.

◮ H. Kempka, J. Vybıral: Spaces of variable smoothness and integrability:

Characterizations by local means and ball means of differences, submitted.

◮ J.-S. Xu: Variable Besov and Triebel-Lizorkin spaces, Ann. Acad. Sci. Fenn.Math. 33 (2008), no. 2, 511–522.

◮ D. Drihem: Atomic decomposition of Besov spaces of variables smoothness and

integrability, preprint.

◮ T. Noi, Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with

variable exponents, preprint.

Documents

Besov and Triebel-Lizorkin spaces of variable smoothness ...msekce.karlin.mff.cuni.cz/~vybiral/Vort/Conf/Tabarz.pdf · Lebesgue spaces (Lp) Continuous and continuously diﬀerentiable