Smoothness spaces on Ahlfors regular sets

Lizaveta Ihnatsyeva

joint work with Antti Vahakangas and Riikka Korte

University of Helsinki

Toronto – August 29, 2012

1 / 17

Introduction Function spaces in Rn

Sobolev spaces

For a non-negative integer k and 1 ≤ p ≤ ∞, the Sobolev space W k,p(Rn)consists of all functions f ∈ Lp(Rn) having distributional derivatives Djf ,|j| ≤ k, in Lp(Rn).

2 / 17

Introduction Function spaces in Rn

Sobolev spaces

For a non-negative integer k and 1 ≤ p ≤ ∞, the Sobolev space W k,p(Rn)consists of all functions f ∈ Lp(Rn) having distributional derivatives Djf ,|j| ≤ k, in Lp(Rn).

Besov spaces

Let α > 0, 1 ≤ p, q ≤ ∞ and k be the integer such that 0 ≤ k < α ≤ k + 1.Then Bα

p,q(Rn) consists of functions f ∈ Lp(Rn) such that

∑

|j|≤k

‖Djf‖p +∑

|j|=k

(∫

Rn

‖Djf(·+ h)−Djf(·)‖qp|h|n+(α−k)q

dh

)1/q

< ∞,

if k < α < k + 1 and 1 ≤ p, q < ∞.If q = ∞, then the usual interpretation in the limiting way is used.If α = k + 1, the first difference of Djf is replaced by the second difference.

2 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

where Pk, k ≥ 0 is a space of polynomials on Rn of degree at most k

3 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

where Pk, k ≥ 0 is a space of polynomials on Rn of degree at most k

3 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

where Pk, k ≥ 0 is a space of polynomials on Rn of degree at most k

3 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

where Pk, k ≥ 0 is a space of polynomials on Rn of degree at most k

3 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

where Pk, k ≥ 0 is a space of polynomials on Rn of degree at most k

3 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

The sharp maximal function of f ∈ L1loc(R

n) is

f ♯k(x) := sup

t>0

1

tkEk(f,Q(x, t))L1(Rn), x ∈ R

n.

4 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

The sharp maximal function of f ∈ L1loc(R

n) is

f ♯k(x) := sup

t>0

1

tkEk(f,Q(x, t))L1(Rn), x ∈ R

n.

Th. (Calderon, 1972)

Let p > 1, k ∈ N. Then Sobolev spaces

W k,p(Rn) = f ∈ Lp(Rn) : f ♯k ∈ Lp(Rn).

4 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

The sharp maximal function of f ∈ L1loc(R

n) is

f ♯k(x) := sup

t>0

1

tkEk(f,Q(x, t))L1(Rn), x ∈ R

n.

Th. (Calderon, 1972)

Let p > 1, k ∈ N. Then Sobolev spaces

W k,p(Rn) = f ∈ Lp(Rn) : f ♯k ∈ Lp(Rn).

⊲ A.P. Calderon - Estimates for singular integral operators in terms of maximal functions, StudiaMath., 1972

4 / 17

Introduction Function spaces in Rn

Characterization in terms of local polynomial approximations

Let f ∈ Luloc(R

n) and 1 ≤ u ≤ ∞. The normalized local best approximation of fon a cube Q is

Ek(f,Q)Lu(Rn) := infP∈Pk−1

(

1

|Q|

∫

Q

|f(x)− P (x)|u dx

)1/u

,

The sharp maximal function of f ∈ L1loc(R

n) is

f ♯k(x) := sup

t>0

1

tkEk(f,Q(x, t))L1(Rn), x ∈ R

n.

Th. (Calderon, 1972)

Let p > 1, k ∈ N. Then Sobolev spaces

W k,p(Rn) = f ∈ Lp(Rn) : f ♯k ∈ Lp(Rn).

⊲ A.P. Calderon - Estimates for singular integral operators in terms of maximal functions, StudiaMath., 1972

⊲ A.P. Calderon, R. Scott - Sobolev type inequalities for p > 0, Studia Math., 1978.

4 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

5 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

Besov spaces

Bαp,q(R

n) consists of functions f ∈ Lp(Rn) such that

∫ 1

0

(

‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn)

tα

)qdt

t< ∞, if q < ∞,

and sup0<t≤1

t−α‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn) < ∞, if q = ∞.

5 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

Besov spaces

Bαp,q(R

n) consists of functions f ∈ Lp(Rn) such that

∫ 1

0

(

‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn)

tα

)qdt

t< ∞, if q < ∞,

and sup0<t≤1

t−α‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn) < ∞, if q = ∞.

Triebel-Lizorkin spaces

Fαp,q(R

n) consists of functions f ∈ Lp(Rn) such that g ∈ Lp(Rn), where

g(x) :=

(∫ 1

0

(

Ek(f,Q(x, t))L1(Rn)

tα

)qdt

t

)1/q

, if q < ∞,

and g(x) := supt−αEk(f,Q(x, t))L1(Rn) : 0 < t ≤ 1, if q = ∞.

5 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

Besov spaces

Bαp,q(R

n) consists of functions f ∈ Lp(Rn) such that

∫ 1

0

(

‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn)

tα

)qdt

t< ∞, if q < ∞,

and sup0<t≤1

t−α‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn) < ∞, if q = ∞.

Triebel-Lizorkin spaces

Fαp,q(R

n) consists of functions f ∈ Lp(Rn) such that g ∈ Lp(Rn), where

g(x) :=

(∫ 1

0

(

Ek(f,Q(x, t))L1(Rn)

tα

)qdt

t

)1/q

, if q < ∞,

and g(x) := supt−αEk(f,Q(x, t))L1(Rn) : 0 < t ≤ 1, if q = ∞.

5 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

Besov spaces

Bαp,q(R

n) consists of functions f ∈ Lp(Rn) such that

∫ 1

0

(

‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn)

tα

)qdt

t< ∞, if q < ∞,

and sup0<t≤1

t−α‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn) < ∞, if q = ∞.

Triebel-Lizorkin spaces

Fαp,q(R

n) consists of functions f ∈ Lp(Rn) such that g ∈ Lp(Rn), where

g(x) :=

(∫ 1

0

(

Ek(f,Q(x, t))L1(Rn)

tα

)qdt

t

)1/q

, if q < ∞,

and g(x) := supt−αEk(f,Q(x, t))L1(Rn) : 0 < t ≤ 1, if q = ∞.

⊲ Yu. Brudnyi - Piecewise polynomial approximation, embedding theorem and rational approximation,Lecture Notes in Math., 1976

5 / 17

Introduction Function spaces in Rn

Let α > 0, k be an integer: α < k, 1 < p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ u ≤ p.

Besov spaces

Bαp,q(R

n) consists of functions f ∈ Lp(Rn) such that

∫ 1

0

(

‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn)

tα

)qdt

t< ∞, if q < ∞,

and sup0<t≤1

t−α‖Ek(f,Q(·, t))Lu(Rn)‖Lp(Rn) < ∞, if q = ∞.

Triebel-Lizorkin spaces

Fαp,q(R

n) consists of functions f ∈ Lp(Rn) such that g ∈ Lp(Rn), where

g(x) :=

(∫ 1

0

(

Ek(f,Q(x, t))L1(Rn)

tα

)qdt

t

)1/q

, if q < ∞,

and g(x) := supt−αEk(f,Q(x, t))L1(Rn) : 0 < t ≤ 1, if q = ∞.

⊲ Yu. Brudnyi - Piecewise polynomial approximation, embedding theorem and rational approximation,Lecture Notes in Math., 1976

⊲ H. Triebel - Local approximation spaces, Z. Anal. Anwendungen, 1989

5 / 17

Besov spaces on d-sets

Ahlfors d-regular sets

Let Hd denote d-dimensional Hausdorff measure on Rn and

Q(x, r) = y ∈ Rn : ‖x− y‖∞ ≤ r.

A subset S ⊂ Rn is called an Ahlfors d-regular (or d-set) if there are c1, c2 > 0:

c1rd ≤ Hd(Q(x, r) ∩ S) ≤ c2r

d

for every x ∈ S and 0 < r ≤ diamS.

6 / 17

Besov spaces on d-sets

Ahlfors d-regular sets

Let Hd denote d-dimensional Hausdorff measure on Rn and

Q(x, r) = y ∈ Rn : ‖x− y‖∞ ≤ r.

A subset S ⊂ Rn is called an Ahlfors d-regular (or d-set) if there are c1, c2 > 0:

c1rd ≤ Hd(Q(x, r) ∩ S) ≤ c2r

d

for every x ∈ S and 0 < r ≤ diamS.

ExamplesCantor-type sets,

self-similar sets...

6 / 17

Besov spaces on d-sets

Let S ⊂ Rn be a d-set with n− 1 < d ≤ n

7 / 17

Besov spaces on d-sets

Let S ⊂ Rn be a d-set with n− 1 < d ≤ n

If f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q is a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

7 / 17

Besov spaces on d-sets

Let S ⊂ Rn be a d-set with n− 1 < d ≤ n

If f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q is a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

The Besov space Bαp,q(S), α > 0, 1 ≤ p, q ≤ ∞, is the set of those functions

f ∈ Lp(S) for which the norm ‖f‖Bαp,q(S) is finite.

‖f‖Bαp,q(S) := ‖f‖Lp(S) +

(∫ 1

0

(

‖Ek(f,Q(·, τ))Lu(S)‖Lp(S)

τα

)qdτ

τ

)1

q

,

where 1 ≤ u ≤ p and k is an integer such that α < k.

7 / 17

Besov spaces on d-sets

Let S ⊂ Rn be a d-set with n− 1 < d ≤ n

If f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q is a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

The Besov space Bαp,q(S), α > 0, 1 ≤ p, q ≤ ∞, is the set of those functions

f ∈ Lp(S) for which the norm ‖f‖Bαp,q(S) is finite.

‖f‖Bαp,q(S) := ‖f‖Lp(S) +

(∫ 1

0

(

‖Ek(f,Q(·, τ))Lu(S)‖Lp(S)

τα

)qdτ

τ

)1

q

,

where 1 ≤ u ≤ p and k is an integer such that α < k.

⊲ Yu. Brudnyi - Sobolev spaces and their relatives: local polynomial approximation approach, in:Sobolev spaces in mathematics II, 2009

7 / 17

Besov spaces on d-sets

Let S ⊂ Rn be a d-set with n− 1 < d ≤ n

If f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q is a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

The Besov space Bαp,q(S), α > 0, 1 ≤ p, q ≤ ∞, is the set of those functions

f ∈ Lp(S) for which the norm ‖f‖Bαp,q(S) is finite.

‖f‖Bαp,q(S) := ‖f‖Lp(S) +

(∫ 1

0

(

‖Ek(f,Q(·, τ))Lu(S)‖Lp(S)

τα

)qdτ

τ

)1

q

,

where 1 ≤ u ≤ p and k is an integer such that α < k.

⊲ Yu. Brudnyi - Sobolev spaces and their relatives: local polynomial approximation approach, in:Sobolev spaces in mathematics II, 2009

The first approach to Besov spaces on d-sets, 0 < d ≤ n, in terms of jet functionsA. Jonsson - The trace of potentials on general sets, Ark. Mat., 1979

7 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

If 1 < p < ∞, 1 ≤ q ≤ ∞

Fαp,q(R

n)|S = f ∈ Lp(S) :

(∫ 1

0

(

Ek(f,Q(·, t))L1(S)

tα

)qdt

t

)1/q

∈ Lp(S)

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

If 1 < p < ∞, 1 ≤ q ≤ ∞

Fαp,q(R

n)|S = f ∈ Lp(S) :

(∫ 1

0

(

Ek(f,Q(·, t))L1(S)

tα

)qdt

t

)1/q

∈ Lp(S)

Bαp,q(R

n)|S = Bαp,q(S), 1 ≤ p ≤ ∞, 0 < q ≤ ∞

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

If 1 < p < ∞, 1 ≤ q ≤ ∞

Fαp,q(R

n)|S = f ∈ Lp(S) :

(∫ 1

0

(

Ek(f,Q(·, t))L1(S)

tα

)qdt

t

)1/q

∈ Lp(S)

Bαp,q(R

n)|S = Bαp,q(S), 1 ≤ p ≤ ∞, 0 < q ≤ ∞

⊲ P. Shvartsman - Local approximations and intrinsic characterization of spaces of smooth functions

on regular subsets of Rn, Math. Nachr., 2006

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

If 1 < p < ∞, 1 ≤ q ≤ ∞

Fαp,q(R

n)|S = f ∈ Lp(S) :

(∫ 1

0

(

Ek(f,Q(·, t))L1(S)

tα

)qdt

t

)1/q

∈ Lp(S)

Bαp,q(R

n)|S = Bαp,q(S), 1 ≤ p ≤ ∞, 0 < q ≤ ∞

⊲ P. Shvartsman - Local approximations and intrinsic characterization of spaces of smooth functions

on regular subsets of Rn, Math. Nachr., 2006

8 / 17

Trace spaces on n-sets

Trace spaces on n-sets

Let S ⊂ Rn be an n-set, k ∈ N, 0 < α < k

Th. Shvartsman (2006)

W k,p(Rn)|S = f ∈ Lp(S) : supt>0

1tkEk(f,Q(·, t))L1(S) ∈ Lp(S), p > 1

If 1 < p < ∞, 1 ≤ q ≤ ∞

Fαp,q(R

n)|S = f ∈ Lp(S) :

(∫ 1

0

(

Ek(f,Q(·, t))L1(S)

tα

)qdt

t

)1/q

∈ Lp(S)

Bαp,q(R

n)|S = Bαp,q(S), 1 ≤ p ≤ ∞, 0 < q ≤ ∞

⊲ P. Shvartsman - Local approximations and intrinsic characterization of spaces of smooth functions

on regular subsets of Rn, Math. Nachr., 2006

If Ω is a W k,p-extension domain, 1 ≤ p < ∞, k ≥ 1, then Ω is an n-set.

⊲ P. Haj lasz, P.Koskela and H. Tuominen - Sobolev embeddings, extensions and measure density

condition, J. Funct. Anal., 2008.

8 / 17

Trace spaces on d-sets, n − 1 < d < n

Trace of a function on a subset

Suppose that f ∈ L1loc(R

n) and S ⊂ Rn. At those points x ∈ S where exists

f(x) = limr→0+

∫

Q(x,r)

f(y) dy = limr→0+

1

|Q(x, r)|

∫

Q(x,r)

f(y) dy,

define the restriction (trace) of f to S by

f |S(x) := f(x).

9 / 17

Trace spaces on d-sets, n − 1 < d < n

Trace of a function on a subset

Suppose that f ∈ L1loc(R

n) and S ⊂ Rn. At those points x ∈ S where exists

f(x) = limr→0+

∫

Q(x,r)

f(y) dy = limr→0+

1

|Q(x, r)|

∫

Q(x,r)

f(y) dy,

define the restriction (trace) of f to S by

f |S(x) := f(x).

By the Lebesgue differentiation theorem, f = f a.e in Rn.

9 / 17

Trace spaces on d-sets, n − 1 < d < n

Trace of a function on a subset

Suppose that f ∈ L1loc(R

n) and S ⊂ Rn. At those points x ∈ S where exists

f(x) = limr→0+

∫

Q(x,r)

f(y) dy = limr→0+

1

|Q(x, r)|

∫

Q(x,r)

f(y) dy,

define the restriction (trace) of f to S by

f |S(x) := f(x).

By the Lebesgue differentiation theorem, f = f a.e in Rn.

If f ∈ Bαp,q(R

n) or Fαp,q(R

n), then for a d-set S with

d > n− αp

the trace f |S is well defined.

9 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

10 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

10 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

i.e. for all functions f ∈ Bαp,q(R

n) the trace operator R : f 7→ f |S satisfies:

‖Rf‖B

α−n−dp

p,q (S)≤ c‖f‖Bα

p,q(Rn),

10 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

i.e. for all functions f ∈ Bαp,q(R

n) the trace operator R : f 7→ f |S satisfies:

‖Rf‖B

α−n−dp

p,q (S)≤ c‖f‖Bα

p,q(Rn),

and there exists an extension operator E : Bα−n−d

pp,q (S) → Bα

p,q(Rn), Eg|S = g:

‖Eg‖Bαp,q(R

n) ≤ c‖g‖B

α−n−dp

p,q (S).

for all g ∈ Bα−n−d

pp,q (S)

10 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

i.e. for all functions f ∈ Bαp,q(R

n) the trace operator R : f 7→ f |S satisfies:

‖Rf‖B

α−n−dp

p,q (S)≤ c‖f‖Bα

p,q(Rn),

and there exists an extension operator E : Bα−n−d

pp,q (S) → Bα

p,q(Rn), Eg|S = g:

‖Eg‖Bαp,q(R

n) ≤ c‖g‖B

α−n−dp

p,q (S).

for all g ∈ Bα−n−d

pp,q (S)

⊲ Yu. Brudnyi - Sobolev spaces and their relatives: local polynomial approximation approach, in:Sobolev spaces in mathematics II, 2009

10 / 17

Trace spaces on d-sets, n − 1 < d < n Besov spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S be an d-set, n− 1 < d < n, 1 ≤ p, q < ∞ and α > (n− d)/p. Then

Bαp,q(R

n)|S = Bα− n−d

pp,q (S),

i.e. for all functions f ∈ Bαp,q(R

n) the trace operator R : f 7→ f |S satisfies:

‖Rf‖B

α−n−dp

p,q (S)≤ c‖f‖Bα

p,q(Rn),

and there exists an extension operator E : Bα−n−d

pp,q (S) → Bα

p,q(Rn), Eg|S = g:

‖Eg‖Bαp,q(R

n) ≤ c‖g‖B

α−n−dp

p,q (S).

for all g ∈ Bα−n−d

pp,q (S)

⊲ Yu. Brudnyi - Sobolev spaces and their relatives: local polynomial approximation approach, in:Sobolev spaces in mathematics II, 2009

Trace theorems for Besov spaces and potential spaces on d-sets, 0 < d < n. A. Jonsson and H.Wallin Function spaces on subsets of Rn, 1984

10 / 17

Trace spaces on d-sets, n − 1 < d < n Triebel-Lizorkin spaces

Traces of Triebel-Lizorkin spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S ⊂ Rn be a d-set with n− 1 < d < n. Let 1 < p < ∞, 1 ≤ q ≤ ∞, and

α > (n− d)/p. Then

Fαp,q(R

n)|S = Bα−n−d

p

p,p (S).

11 / 17

Trace spaces on d-sets, n − 1 < d < n Triebel-Lizorkin spaces

Traces of Triebel-Lizorkin spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S ⊂ Rn be a d-set with n− 1 < d < n. Let 1 < p < ∞, 1 ≤ q ≤ ∞, and

α > (n− d)/p. Then

Fαp,q(R

n)|S = Bα−n−d

p

p,p (S).

Corollary

Since F kp,2(R

n) = W k,p(Rn), the space Bk− n−d

p

p,p (S) is the trace space for

W k,p(Rn).

11 / 17

Trace spaces on d-sets, n − 1 < d < n Triebel-Lizorkin spaces

Traces of Triebel-Lizorkin spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S ⊂ Rn be a d-set with n− 1 < d < n. Let 1 < p < ∞, 1 ≤ q ≤ ∞, and

α > (n− d)/p. Then

Fαp,q(R

n)|S = Bα−n−d

p

p,p (S).

Corollary

Since F kp,2(R

n) = W k,p(Rn), the space Bk− n−d

p

p,p (S) is the trace space for

W k,p(Rn).

⊲ G.A. Mamedov, Traces of functions in anisotropic Nikol’skii-Besov and Lizorkin-Triebel spaces onsubsets of the Euclidean space, 1993

11 / 17

Trace spaces on d-sets, n − 1 < d < n Triebel-Lizorkin spaces

Traces of Triebel-Lizorkin spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S ⊂ Rn be a d-set with n− 1 < d < n. Let 1 < p < ∞, 1 ≤ q ≤ ∞, and

α > (n− d)/p. Then

Fαp,q(R

n)|S = Bα−n−d

p

p,p (S).

Corollary

Since F kp,2(R

n) = W k,p(Rn), the space Bk− n−d

p

p,p (S) is the trace space for

W k,p(Rn).

⊲ G.A. Mamedov, Traces of functions in anisotropic Nikol’skii-Besov and Lizorkin-Triebel spaces onsubsets of the Euclidean space, 1993

⊲ K. Saka, The trace theorem for Triebel–Lizorkin spaces and Besov spaces on certain fractal sets. I.The restriction theorem, 1995

11 / 17

Trace spaces on d-sets, n − 1 < d < n Triebel-Lizorkin spaces

Traces of Triebel-Lizorkin spaces

Theorem∗(A. Vahakangas, L.I., 2011)

Let S ⊂ Rn be a d-set with n− 1 < d < n. Let 1 < p < ∞, 1 ≤ q ≤ ∞, and

α > (n− d)/p. Then

Fαp,q(R

n)|S = Bα−n−d

p

p,p (S).

Corollary

Since F kp,2(R

n) = W k,p(Rn), the space Bk− n−d

p

p,p (S) is the trace space for

W k,p(Rn).

⊲ G.A. Mamedov, Traces of functions in anisotropic Nikol’skii-Besov and Lizorkin-Triebel spaces onsubsets of the Euclidean space, 1993

⊲ K. Saka, The trace theorem for Triebel–Lizorkin spaces and Besov spaces on certain fractal sets. I.The restriction theorem, 1995

⊲ K. Saka, The trace theorem for Triebel–Lizorkin spaces and Besov spaces on certain fractal sets. II.The extension theorem, 1996

11 / 17

Trace spaces on d-sets, n − 1 < d < n Basic tools

Remez-type inequality

Let S be a d-set, n− 1 < d ≤ n.

Suppose that Q = Q(xQ, rQ) and Q′ = Q(xQ′ , rQ′ ) are cubes in Rn such that

xQ′ ∈ S, Q′ ⊂ Q, rQ ≤ RrQ′ and r′Q ≤ R for some R > 0.

Then, ∀ p ∈ Pk

(

1

|Q|

∫

Q

|p|r dx

)1/r

≤ C

(

1

Hd(Q′ ∩ S)

∫

Q′∩S

|p|u dHd

)1/u

,

where 1 ≤ u, r ≤ ∞ and C depends on S, R, n, u, r, k.

⊲ A. Brudnyi and Yu. Brudnyi - Remez type inequalities and Morrey-Campanato spaces on Ahlfors

regular sets, Contemp. Math., 2007

12 / 17

Trace spaces on d-sets, n − 1 < d < n Basic tools

The construction of the extension operator is based on a modification of theWhitney extension method

Let WS denote a Whitney decomposition of Rn \ S and

Φ := ϕQ : Q ∈ WS be a smooth partition of unity

To every cube Q = Q(xQ, rQ) ∈ WS assign the cube a(Q) := Q(aQ, rQ/2),where aQ ∈ S is such that ‖xQ − aQ‖∞ = dist(xQ, S)

13 / 17

Trace spaces on d-sets, n − 1 < d < n Basic tools

The construction of the extension operator is based on a modification of theWhitney extension method

Let WS denote a Whitney decomposition of Rn \ S and

Φ := ϕQ : Q ∈ WS be a smooth partition of unity

To every cube Q = Q(xQ, rQ) ∈ WS assign the cube a(Q) := Q(aQ, rQ/2),where aQ ∈ S is such that ‖xQ − aQ‖∞ = dist(xQ, S)

If f ∈ L1loc(S) and k ∈ N, then

Extk,Sf(x) :=

f(x), if x ∈ S;∑

Q∈WSϕQ(x)Pk−1,Qf(x), if x ∈ R

n \ S,

13 / 17

Trace spaces on d-sets, n − 1 < d < n Basic tools

The construction of the extension operator is based on a modification of theWhitney extension method

Let WS denote a Whitney decomposition of Rn \ S and

Φ := ϕQ : Q ∈ WS be a smooth partition of unity

To every cube Q = Q(xQ, rQ) ∈ WS assign the cube a(Q) := Q(aQ, rQ/2),where aQ ∈ S is such that ‖xQ − aQ‖∞ = dist(xQ, S)

If f ∈ L1loc(S) and k ∈ N, then

Extk,Sf(x) :=

f(x), if x ∈ S;∑

Q∈WSϕQ(x)Pk−1,Qf(x), if x ∈ R

n \ S,

13 / 17

Trace spaces on d-sets, n − 1 < d < n Basic tools

The construction of the extension operator is based on a modification of theWhitney extension method

Let WS denote a Whitney decomposition of Rn \ S and

Φ := ϕQ : Q ∈ WS be a smooth partition of unity

To every cube Q = Q(xQ, rQ) ∈ WS assign the cube a(Q) := Q(aQ, rQ/2),where aQ ∈ S is such that ‖xQ − aQ‖∞ = dist(xQ, S)

If f ∈ L1loc(S) and k ∈ N, then

Extk,Sf(x) :=

f(x), if x ∈ S;∑

Q∈WSϕQ(x)Pk−1,Qf(x), if x ∈ R

n \ S,

where the projection Pk,Q : L1(a(Q) ∩ S) → Pk are such that

(∫

a(Q)∩S

|f − Pk−1,Qf |u dHd

)1/u

≈ Ek(f, a(Q))Lu(S).

13 / 17

Relation between Besov spaces and Sobolev type spaces

Let f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q be a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

14 / 17

Relation between Besov spaces and Sobolev type spaces

Let f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q be a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

Fix α > 0 and set k = −[−α]. For f ∈ Luloc(S) define the fractional sharp

maximal function

f ♯α,S(x) := sup

t>0

1

tαEk(f,Q(x, t))L1(S), x ∈ S,

and corresponding smoothness spaces

Cαp (S) = f ∈ Lp(S) : ‖f‖Cα

p= ‖f‖p + ‖f ♯

α,S‖p < ∞, p ≥ 1.

14 / 17

Relation between Besov spaces and Sobolev type spaces

Let f ∈ Luloc(S), 1 ≤ u ≤ ∞, and Q be a cube centered at S ⊂ R

n. Thenthe normalized local best approximation of f on Q in Lu(S) norm is

Ek(f,Q)Lu(S) := infP∈Pk−1

(

1

Hd(Q ∩ S)

∫

Q∩S

|f − P |u dHd

)1/u

.

Fix α > 0 and set k = −[−α]. For f ∈ Luloc(S) define the fractional sharp

maximal function

f ♯α,S(x) := sup

t>0

1

tαEk(f,Q(x, t))L1(S), x ∈ S,

and corresponding smoothness spaces

Cαp (S) = f ∈ Lp(S) : ‖f‖Cα

p= ‖f‖p + ‖f ♯

α,S‖p < ∞, p ≥ 1.

⊲ (If S = Rn) R. DeVore, R. Sharpley - Maximal functions measuring local smoothness, Memoirs of

AMS, 1984

14 / 17

Relation between Besov spaces and Sobolev type spaces

Theorem (R. Korte, L.I., 2011)

Let S be an d-set with n− 1 < d ≤ n, 1 < p ≤ ∞ and α be a non-integer

positive number. Then

Bαp,p(S) ⊂ Cα

p (S) ⊂ Bαp,∞(S)

15 / 17

Relation between Besov spaces and Sobolev type spaces

Theorem (R. Korte, L.I., 2011)

Let S be an d-set with n− 1 < d ≤ n, 1 < p ≤ ∞ and α be a non-integer

positive number. Then

Bαp,p(S) ⊂ Cα

p (S) ⊂ Bαp,∞(S)

Corollary

Let S be an d-set, n− 1 < d < n, 1 < p < ∞, k ∈ N and α = k − (n− d)/p > 0.Then for any 0 < ε < (n− d)/p

Cα+εp (S) ⊂ W k,p(Rn)|S ⊂ Cα

p (S).

15 / 17

Relation between Besov spaces and Sobolev type spaces

Theorem (R. Korte, L.I., 2011)

Let S be an d-set with n− 1 < d ≤ n, 1 < p ≤ ∞ and α be a non-integer

positive number. Then

Bαp,p(S) ⊂ Cα

p (S) ⊂ Bαp,∞(S)

Corollary

Let S be an d-set, n− 1 < d < n, 1 < p < ∞, k ∈ N and α = k − (n− d)/p > 0.Then for any 0 < ε < (n− d)/p

Cα+εp (S) ⊂ W k,p(Rn)|S ⊂ Cα

p (S).

Relations between the trace spaces of first order Sobolev spaces and Haj lasz-Sobolev spacesP. Haj lasz and O. Martio - Traces of Sobolev functions on fractal type sets and characterization ofextension domains, J. Funct. Anal., 1997

15 / 17

Sobolev spaces on s-sets

Are Cpk (S) relevant analogs of classical Sobolev spaces in the setting of an s-set?

16 / 17

Sobolev spaces on s-sets

Are Cpk (S) relevant analogs of classical Sobolev spaces in the setting of an s-set?

If S = Rn or S is a W k,p-extension domain, then Cp

k(S) = W k,p(S)

16 / 17

Sobolev spaces on s-sets

Are Cpk (S) relevant analogs of classical Sobolev spaces in the setting of an s-set?

If S = Rn or S is a W k,p-extension domain, then Cp

k(S) = W k,p(S)

If k = 1, then Cp1 (S) coincides with Haj lasz-Sobolev spaces.

⊲ P. Haj lasz and J.Kinnunen, - Holder quasicontinuity of Sobolev functions on metric spaces, Rev.Mat. Iberoamericana, 1998.

16 / 17

Sobolev spaces on s-sets

Are Cpk (S) relevant analogs of classical Sobolev spaces in the setting of an s-set?

If S = Rn or S is a W k,p-extension domain, then Cp

k(S) = W k,p(S)

If k = 1, then Cp1 (S) coincides with Haj lasz-Sobolev spaces.

⊲ P. Haj lasz and J.Kinnunen, - Holder quasicontinuity of Sobolev functions on metric spaces, Rev.Mat. Iberoamericana, 1998.

Let S be an s-set, n− 1 < s < n, p ≥ 1, kp < s and q = sp/(s− kp). Then

‖f‖Lq(S) ≤ c(‖f ♯α,S‖Lp(S) + (diamS)−α‖f‖Lp(S))

16 / 17

Sobolev spaces on s-sets

Thank you for the attention

17 / 17