A note on the trace theorem for Besov-type spaces of generalized smoothness on $d$-sets
Vanja Wagner


Distributed private randomness distillation
    Andreas Winter
    30 August 2019
===========================================





The main goal of this paper is to give a complete proof of the trace theorem for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions on d-sets D⊂^n, d≤ n. The proof closely follows the classical approach by Jonsson, Wallin in <cit.> and the trace theorem for classical Besov spaces. Here, the trace space is defined by means of differences. When d=n, as an application of the trace theorem, we give a condition under which the test functions C_c^∞(D) are dense in the trace space on D.


Keywords: Function spaces of generalized smoothness, d-sets, trace space, Bernstein function

MSC[2010]: 46E35, 60J75, 60G51





§ INTRODUCTION


Besov-type spaces of generalized smoothness were introduced in the seventies by M.L. Goldman and G.A. Kalyabin as a generalization of the classical Sobolev and Besov spaces. Since then they have been studied by many authors from various points of view. Due to their close connection to the theory of stochastic processes and probability theory, these spaces continue to be of further interest. For a unified and general approach to function spaces of generalized smoothness and a review of results we refer the reader to <cit.> and <cit.>.

In this paper we consider traces on d-sets of spaces of generalized smoothness associated with complete Bernstein functions and their representation via differences. This approach allows us to easily associate these spaces with domains of Dirichlet forms corresponding to a certain class of purely discontinuous symmetric Markov processes. A trace theorem on h-sets for generalized Besov-type spaces associated with functions satisfying slightly different assumptions was presented by Knopova and Zähle in <cit.>. The authors consider the quarkonial representations of the trace spaces, as well as the representation via differences, and prove their equivalence. In the following paper <cit.>, Zähle proves a potential representation of the trace spaces and the corresponding trace Dirichlet forms. In order to make the theory of generalized Besov spaces more approachable in the context of application to the theory of stochastic processes, we present a complete and detailed proof of the trace theorem based on the characterization via differences, following the approach in <cit.>. 

Let j:(0,∞)→(0,∞) be a non-increasing function satisfying

    j(r) ≤ c_1 j(r + 1)  and ∫_0^∞ (1∧ s^2)j(s)ds≤∞.

for all r ≥ 1 and some constant c_1>0. Such a function j is a radial Lévy density of an isotropic unimodal Lévy process with characteristic exponent

    ψ(|ξ|)=∫_^n∖{0}(1-cos(x·ξ))j(|x|)dx, ξ∈^n.
  
Throughout the paper we will assume that ψ satisfies one or both of the following scaling conditions, 

(H1)(H1): There exist constants 0 < δ_1 ≤δ_2 < 1 and a_1,a_2 > 0 such that

    a_1 λ^2δ_1ψ(t) ≤ψ(λ t) ≤ a_2 λ^2δ_2ψ(t),  λ≥ 1,  t ≥1;

(H2)(H2): There exist constants 0 < δ_3 ≤δ_4 < 1 and a_3,a_4 > 0 such that

    a_3 λ^2δ_3ψ(t) ≤ψ(λ t) ≤ a_4 λ^2δ_4ψ(t),  λ≥ 1,  t < 1.

Under condition , by <cit.>, there exists a complete Bernstein function ϕ and a constant γ_2≥1 such that

    γ_2^-1ϕ(|ξ|^2)≤ψ(|ξ|)≤γ_2ϕ(|ξ|^2),  ξ∈^n,

and the radial Lévy density j enjoys the following property: for every R>0

    j(r)≍ϕ(r^-2)/r^n, r∈(0,R).

Due to the equivalence of norms, we will always assume that the constant γ_2 in (<ref>) is equal to 1. Furthermore, by <cit.> every Bernstein function ϕ satisfies the following useful inequality,

    1∧λ≤ϕ(λ r)/ϕ(r)≤ 1 ∨λ,  λ, r>0.

For further results on Bernstein functions, we refer the reader to <cit.>. Define by

    (^n)={u∈ L^2(^n): ||u||_ψ,1:=∫_^n(1+ψ(ξ))|ℱ u(ξ)|^2dξ<∞}

the ψ-Bessel-type potential space on ^n. This space naturally arises as the domain of the Dirichlet form associated with the Lévy process with the characteristic exponent ψ. In the first chapter, we give basic definitions and results regarding Besov-type spaces of generalized smoothness, connect them with ψ-Bessel-type potential spaces and provide several characterizations of the corresponding norms, based on differences. 

In the following two chapters, we give a complete proof of the trace theorem for spaces H^ψ,α(^n) (see Definition <ref>) on d-sets, where the trace space is defined by means of differences. 

Let D be a non-empty Borel subset of ^n and 0<d≤ n. A positive Borel measure μ on D is called a d-measure if there exist positive constants c_1 and c_2 such that for all x∈ D and r∈ (0, 1],
 
    c_1 r^d≤μ(B(x,r))≤ c_2 r^d.

A non-empty Borel set D is called a d-set if there exists a d-measure μ on D. 

Note that by definition all d-measures on D are equivalent to the restriction of the d-dimensional Hausdorff measure to D. For a d-set D in ^n with d-measure μ, we define the trace space as

    H^ψ,1(D,μ)   ={u∈ L^2(D,μ):||u||_(1),D,μ<∞},
    
    ||u||_(1),D,μ   =||u||_L^2(D,μ)+(  ∬_|x-y|<1|u(x)-u(y)|^2 ϕ(|x-y|^-2)/|x-y|^2d-nμ(dx)μ(dy))^1/2.

The proof of the trace theorem is divided into three parts; the restriction theorem, the extension theorem for d<n and finally, the extension theorem when d=n. We state the main result:

Let D be a d-set in ^n and ψ a radial function defined by (<ref>) such that and hold.
 
  
  * There exists a continuous restriction operators R:(^n)→ H^ψ,1(D,μ).
  
  * There exists a continuous extension operator E:H^ψ,1(D,μ)→ H^ψ,1(^n), such that R E u=u, μ-a.e. on D, for all u∈ H^ψ,1(D,μ).
 

This result is a direct consequence of Theorem <ref>, Theorem <ref> and Theorem <ref> for spaces H^ψ,α(^n). As a consequence of Theorem <ref> and <cit.>, we arrive to the following result on the correspondence of spaces H^ψ,1(D) and H^ψ,1_0(D), where the latter is the closure of C_c^∞(D) with respect to the norm ||·||_(1),D,λ_D. For an equivalent result in the case of the classical Bessel potential space see <cit.>.


Suppose that D⊂^n is an open n-set, ϕ is a complete Bernstein function such that and hold and ψ(ξ)=ϕ(|ξ|^2).

 
  * Suppose that 2δ_2 ≤ n and that ℋ_h (∂ D ∩ K_m ) < ∞ for an increasing sequence of Borel sets K_m such that ∪_m∈K_m ⊃∂ D, where h(r) = r
^n-2δ_2 if 2δ_2 < n and h(r) = max{log r , 0} when 2δ_2 = n=1, then H^ψ,1(D)=H^ψ,1_0(D).


  * If 2(δ_1∧δ_3)≥ n=1 or ℋ^d(∂ D)>0 for some d>n-2δ_1≥ 0 then H^ψ,1_0(D)⊊ H^ψ,1(D).

 

For easier notation, we write f≍ g if there exists a constant c>1 such that for all x, c^-1g(x)≤ f(x)≤ c g(x). Equivalently, f≲ g if there exists a constant c>0 such that for all x, f(x)≤ c g(x). By λ we denote the Lebesgue measure on ^n.




§ BESOV-TYPE SPACES OF GENERALIZED SMOOTHNESS AND EQUIVALENT NORMS



The aforementioned ψ-Bessel-type potential space is a type of a much more general class of function spaces called Besov-type spaces of generalized smoothness. First we recall these spaces in their most general form.     
 
 A sequence (γ_j)_j∈_0 of positive real numbers is called


  * almost increasing if there exists d_0>0 such that d_0γ_j≤γ_k, for all j≤ k;

  * strongly increasing if it is almost increasing and in addition there exists a κ_0∈ such that 2γ_j≤γ_k, for j≤ k-κ_0;

  * of bounded growth if there are positive constants d_1 and J_0∈_0 such that γ_j+1≤ d_1γ_j, for all j≥ J_0;
 
  * an admissible sequence if both (γ_j)_j∈_0 and (γ_j^-1)_j∈_0 are of bounded growth and J_0=0, i.e. there exist positive constants d_0 and d_1 such that d_0γ_j≤γ_j+1≤ d_1γ_j, for all j∈_0.



Let N=(N_j)_j∈_0 be a strongly increasing sequence and define

    Ω_0^N={x∈^n:|x|≤ N_0} and Ω_j^N={x∈^n:N_j-1≤|x|≤ N_j+1},   j∈.

Let Φ^N be a collection of all partitions of unity of C_c^∞(^n) functions associated with this decomposition. Let σ=(σ_j)_j∈_0 be an admissible sequence respectively and (φ_j^N)_j∈_0∈Φ^N. The Besov space of generalized smoothness associated with N and σ is defined by

    B_2^σ,N={g∈ S'(^n):||g||_B,σ,N:=||(σ_jφ^N_j(D)g)_j∈_0|l_2(L_2(^n))||<∞},

where φ(D)g(x)=ℱ^-1(φ(·)ℱ g)(x) and
||(f_j)_j∈_0|l^2(L^2(^n))||=(∑_j=0^∞ ||f_j||_L^2(^n)^2)^1/2.

By <cit.> the space B_2^σ,N is independent of the choice of system (φ_j^N)_j∈_0 in the sense of equivalent norms. We will restrict ourselves to a special subclass of spaces B_2^σ,N associated with an admissible symbol.

A non-negative function a∈ C^∞(^n) is an admissible symbol if


  * lim_|x|→∞a(x)=∞,

  * a is almost increasing in |x|, i.e. there exist constants δ_0≥ 1 and R>0 such that a(x)≤δ_0 a(y) if R≤ |x|≤ |y|,

  * there exists an m>0 such that x→a(x)/|x|^m is almost decreasing in |x|,

  * for every multi-index α∈_0^n there exist constants c_α>0 and R>0 such that

    |D^α a(x)|≤ c_αa(x)/(1+|x|^2)^|α|/2,   ∀ |x|≥ R.


The family of all admissible functions will be denoted by 𝒜.

Note that for every Bernstein function ϕ such that lim_r→∞ϕ(r)=∞, the function ξ→ϕ(|ξ|^2) is an admissible symbol, see <cit.>.

 By <cit.>, the sequence N_j^a,r=sup{|x|:a(x)≤ 2^jr}, j∈_0, where a∈𝒜 and r>0, is strongly increasing.


For a∈𝒜 we define the Besov space of generalized smoothness associated with a as B^σ,N^a,2_2(^n), where σ={2^j}_j∈_0 is an admissible sequence. These spaces have two useful representations in the sense of equivalent norms; one given by the Littlewood-Paley-type theorem and the other by means of differences.  

<cit.>

Let a∈𝒜, N=N^a,2 the strongly increasing sequence associated with a, α>0 and σ^α={2^α j}_j∈_0 an admissible sequence. Then the space (H^a,α(^n),||·||_a,α), defined by

    H^a,α(^n)={u∈ L^2(^n):||u||_a,α<∞},
       ||u||_a,α:=||(id+a(D))^α/2u||_L_2(^n)=(∫_^n (1+a(ξ))^α|û(ξ)|^2dξ)^1/2,

is equivalent to (B^σ^α,N^a,2_2(^n),||·||_B,σ^α,N^a,2).


Proposition <ref> implies that Besov-type spaces of generalized smoothness associated with ψ can be characterized as

    H^ψ,α(^n)={u∈ S'(^n):∃ f∈ L^2(^n) such that ℱ u=(1+ψ)^-α/2ℱ f}

 Since the function x↦(1+x)^-α/2 is completely monotone for every α>0, by <cit.> the function (1+ϕ)^-α/2 is also completely monotone. By Schoenberg's theorem, <cit.> it follows that the function (1+ψ)^-α/2 is a positive definite function and therefore a Fourier transforms of an integrable function, <cit.>, called the Bessel-type potential K_ψ,α. This means that the space H^ψ,α(^n) defined by (<ref>) can be characterized as a convolution space via the ψ-Bessel convolution kernel K_ψ,α i.e.

    H^ψ,α(^n)={K_ψ,α*f:f∈ L^2(^n)},   ||K_ψ,α*f||_ψ,α:=||f||_L^2(^n).

From now on we assume that the function ψ satisfies conditions and . The following estimates for the kernels K_ψ,α were obtained in <cit.> and <cit.>, for further results on Bessel-type potential spaces we refer the reader to the latter. 

Let α>0 and assume and hold. If α(δ_2∨δ_4)<n then there exist constants c_i=c_i(ϕ,α, n)>0, i=1,2, such that for all x∈^n and 0≤ j≤ n

    |K_ψ,α(x)|   ≤c_1/|x|^nϕ(|x|^-2)^α/2,
    
    |(K_ψ,α(x))'_x_j|   ≤c_2/|x|^n+1ϕ(|x|^-2)^α/2.



Next, we consider the characterization of spaces H^ψ,α(^n) via differences. First note that by applying (<ref>) and the Parseval's identity (see for example <cit.>), it easily follows that 

    ||u||_ψ,1≍ ||u||_L^2(^n)+(1/2∫_^n∫_^n (u(x+y)-u(x))^2 j(y)dy dx)^1/2

for all u∈(^n). We also introduce an equivalent norm on (^n), which we later use in the proof of the trace theorem,

    ||u||_(1)=||u||_L^2(^n)+(  ∬_|x-y|<1|u(x)-u(y)|^2 ϕ(|x-y|^-2)/|x-y|^n dx dy)^1/2,

with the equivalence of norms ||·||_(1) and ||·||_ψ,1 following from (<ref>), (<ref>) and the fact that j is a Lévy measure. In the remainder of this chapter we prove an equivalent result for spaces H^ψ,α(^n). 



For a function f on ^n, h∈^n and k∈ℕ, the k-th difference of function f is defined by (Δ_h^kf)(x):=Δ_h^1(Δ_h^k-1f)(x), x∈^n, where Δ_h^1f(x)=f(x+h)-f(x). Next, define the k-th modulus of continuity of a function f∈ L^2(^n) as ω_k(f,t)=sup_|h|<t||Δ_h^kf||_L^2(^n), t>0. Furthermore, for an admissible sequence (γ_j)_j∈_0 let γ_j=sup_kγ_j+k/γ_k and γ_j=inf_kγ_j+k/γ_k, and let s(γ):=lim_j→∞j^-1logγ_j and s(γ):=lim_j→∞j^-1logγ_j be the lower and upper Boyd index, respectively.

Since γ_j+i+k≤γ_jγ_i+k for all i,j,k∈_0 it follows that γ_j+i≤γ_jγ_i, so the sequence logγ_j is subadditive. By Fekete's subadditive lemma the sequence (logγ_j/j)_j converges to inf_jlogγ_j/j, so the upper index s(γ) is well defined. The analogous conclusion follows for the lower index s(γ), since logγ_j=-log(γ^-1_j). 

<cit.>

Let σ and N be admissible sequences and N_1=inf_k≥ 0N_k+1/N_k>1 and s(σ)/s(N)>0. Let k be an integer such that k>s(σ)/s(N). Then the norm ||·||_B,σ,N on B_2^σ,N is equivalent to 

    ||u||_L^2(^n)+(∑_j=0^∞σ_j^2ω_k(u,N_j^-1)^2)^1/2.




For a similar result see also <cit.>. Note that for every a∈𝒜 the sequence N^a,r satisfies the assumption N^a,r_1>1. Furthermore, recall that σ^α=(2^α j)_j∈_0 is an admissible sequence and by Remark <ref> the sequence N^ψ,2 is strongly increasing. One can easily show that N^ψ,2 is also admissible. Furthermore,

    s(σ^α)/s(N^a,2)≥α/2>0 s(σ)/s(N^ψ,2)≤αlog 2/1/δ_2log 2=αδ_2

so Theorem <ref> holds for k>αδ_2. Furthermore,

    ∑_j=0^∞ 2^α jsup_|H|<1/N_j^ψ,2||Δ_H^ku||_L^2(^n)^2   ≍∑_j=0^∞∫_2^-(j+1)≤ t<2^-j1/t^1+2αsup_|H|<1/N_j^ψ,2||Δ_H^ku||_L^2(^n)^2dt
       ≍∫_0^11/t^1+2αsup_|H|<1/ψ^-1(t^-2)||Δ_H^ku||_L^2(^n)^2dt,

since by 2^-(j+1)≤ t<2^-j implies ψ^-1(t^-2)≍ N_j^ψ,2. 
By change of variable t^-2=ψ(|h|^-1) it follows that (<ref>) is comparable to

    ∫_|h|<1(ψ^α)'(|h|^-1)/|h|^n+1sup_|H|<|h|||Δ_H^ku||_L^2(^n)^2dh.

Since (ψ^α)'(t)≍ψ^α(t)/t it follows that the last line is comparable to

    ∫_|h|<1ψ^α(|h|^-1)/|h|^nsup_|H|<|h|||Δ_H^ku||_L^2(^n)^2dh.




  * By applying a straightforward generalization of <cit.> to (<ref>), Theorem <ref>, Theorem <ref> and the calculation above imply that the norms

    ||u||_(1),α,k:=||u||_L^2(^n)+(∫_|h|<1ψ^α(|h|^-1)/|h|^n||Δ_h^k u||_L^2(^n)^2dh)^1/2

are equivalent to ||·||_a,1, for all k> αδ_2.

  * Since the function ψ^α(| · |^-1)/| · |^n is continuous and ||Δ_h^k u||_L^2(^n)≤ c(k)|| u||_L^2(^n) the norms ||·||_(1),α,k^h_0, 

    ||u||_(1),α,k^h_0:=||u||_L^2(^n)+(∫_|h|<h_0ψ^α(|h|^-1)/|h|^n||Δ_h^k u||_L^2(^n)^2dh)^1/2,

are equivalent for all h_0>0.

  * Let c>0, N∈ℤ and k> αδ_2. The norm ||·||_(2)^c,N on H^ψ,α(^n)
defined by 

    ||u||_(2),α,k^c,N=||u||_L^2(^n)+(  ∑_j=N^∞ψ^α(2^j) 2^nj∫_|h|<c2^-j||Δ_h^k u||_L^2(^n)^2dh)^1/2

is equivalent to the norm ||·||_(1). This follows by applying (<ref>) to the norm in (ii) for h_0=c2^-N,

    ∫_|h|<c2^-Nψ^α(|h|^-1)/|h|^n||Δ_h^k u||_L^2(^n)^2dh=∑_j=N^∞  ∫_c2^-j-1≤|h|<c2^-jψ^α(|h|^-1)/|h|^n||Δ_h^k u||_L^2(^n)^2dh
       ≍∑_j=N^∞ψ^α(2^j)2^nj∫_c2^-j-1≤|h|<c2^-j||Δ_h^k u||_L^2(^n)^2dh≍∑_j=N^∞∑_i=N^j ψ^α(2^i) 2^ni∫_c2^-j-1≤|h|<c2^-j||Δ_h^k u||_L^2(^n)^2dh
       =  ∑_i=N^∞ψ^α(2^i) 2^ni∫_|h|<c2^-j||Δ_h^k u||_L^2(^n)^2dh.

  




§ THE RESTRICTION THEOREM
 


In this section we provide a detailed proof of the continuity of the restriction operator, as a generalization of <cit.>. The same approach is used in <cit.>. Before we start with the proof, we show the following useful consequence of the estimates on Bessel-type potentials from Lemma <ref>. 


Let d≤ n, D a d-set in ^n and μ the d-measure on D. Let ψ be a function such that (<ref>), and hold and α>0 such that

    n-d/2<αδ_1≤α(δ_2∨δ_4)<n-d/2+1.

Then there exists a constant c>0 such that for all r≤1/3 and f∈ L^2(^n)

    ∬_|x-y|<r (K_ψ,α∗ f(x)-K_ψ,α∗ f(y))^2 μ(dx) μ(dy) ≤ c r^2d-n/ϕ^α(r^-2) ||f||_L^2(^n)^2.


Note that for every constant 0<a<1

    (K_ψ,α*f(x)-K_ψ,α*f(y))^2≤∫
    |K_ψ,α(x-t)-K_ψ,α(y-t)|^2af^2(t)dt·∫
    |K_ψ,α(x-t)-K_ψ,α(y-t)|^2(1-a)dt.

Let |x-y|<r. By Lemma <ref> it follows that

    ∫_|y-t|<2r|K_ψ,α(x-t)-K_ψ,α(y-t)|^2(1-a)
    dt≲∫_|z|<3r|K_ψ,α(z)|^2(1-a)dz≲∫_|z|<3r(1/|z|^nϕ^α/2(|z|^-2))^2(1-a)dz
       ≲r^-2αδ_1(1-a)/(ϕ^α/2(r^-2))^2(1-a)∫_|z|<3r(1/|z|^n-αδ_1)^2(1-a)dz = c̃_1 r^n/(r^nϕ^α/2(r^-2))^2(1-a),

for some c̃_>0 and a such that

    2(1-a)(n-αδ_1)<n.

Analogously, if

    2a(n-αδ_1)<d

then for all t∈^n there exists a constant c̃_2>0

    ∬_|x-y|<r
    |y-t|<2r
    |K_ψ,α(x-t)-K_ψ,α(y-t)|^2aμ(dx)μ(dy)≤c̃_2r^2d/(r^nϕ^α/2(r^-2))^2a.

Therefore, it follows that

    ∬_|x-y|<r(∫_|y-t|<2r (K_ψ,α(x-t)-K_ψ,α(y-t))f(t)dt)^2μ(dx)μ(dy)≤c̃_1c̃_2 r^2d-n/ϕ^α(r^-2)||f||_L^2(^n)^2.

For the second part, note that the mean value theorem and Lemma <ref> imply that

    ∬_|x-y|<r
    |y-t|>2r
    |K_ψ,α(x-t)-K_ψ,α(y-t)|^2aμ(dx)μ(dy)≲ r^2a ∬_|x-y|<r
    |y-t|>2r(1/|z_x,y|^n+1ϕ^α/2(|z_x,y|^-2))^2aμ(dx)μ(dy),

where z_x,y=y-t+θ_x,y(x-y) for some θ_x,y∈(0,1) and |z_x,y|≥|y-t|/2. Let δ=δ_2∨δ_4. By and there exists a constant c̃_3>0 such that the last line in (<ref>) is comparably less then
 
    r^2a+d/(r^αδϕ^α/2(r^-2))^2a∫_|z|>2r(1/|z|^n+1-αδ)^2aμ(dz)= c̃_3r^2d/(r^nϕ^α/2(r^-2))^2a

if

    2a(n+1-αδ )>d.

Similarly, if

    2(1-a)(n+1-αδ )>n

then there exists a constant c̃_4>0 such that |x-y|<r implies

    ∫_2r<|y-t||K_ψ,α(y-t)-K_ψ,α(x-t)|^2(1-a)dt≤c̃_4r^n/(r^nϕ^α/2(r^-2))^2(1-a).

Therefore, it follows that

    ∬_|x-y|<r(∫_|y-t|>2r (K_ψ,α(x-t)-K_ψ,α(y-t))f(t)dt)^2μ(dx)μ(dy)≤c̃_3c̃_4 r^2d-n/ϕ^α(r^-2)||f||_L^2(^n)^2.

Since n-d/2<αδ_1≤αδ<n-d/2+1 it follows that 

    (d/2(n+1-αδ), 1-n/2(n+1-αδ))∩(1-n/2(n-αδ_1),d/2(n-αδ_1))≠∅,
 
so we can choose a such that (<ref>), (<ref>), (<ref>) and (<ref>) hold.


For a d-set D in ^n with d-measure μ  and α>0 we define the trace space in terms of differences

    H^ψ,α(D,μ)={u∈ L^2(D,μ):||u||_(1),D,μ<∞},
       ||u||_(1),D,μ=||u||_L^2(D,μ)+(  ∬_|x-y|<1|u(x)-u(y)|^2 ψ^α(|x-y|^-1)/|x-y|^2d-nμ(dx)μ(dy))^1/2.

Note that, similarly as in Remark <ref>(iii), one can easily show that

    ||u||_(1),D,μ≍ ||u||_(2),D,μ^c,N=||u||_L^2(D,μ)+(  ∑_j=N^∞ψ^α(2^j) 2^(2d-n)j∬_|x-y|<c2^-j|u(x)-u(y)|^2
     μ(dx)μ(dy))^1/2.

Next we define the restriction operator. Let Ru be the pointwise restriction on D of the strictly defined function corresponding to u∈ H^ψ,α(^n), i.e. 

    Ru(x)=lim_r→01/λ(B(x,r))∫_B(x,r)u(y)dy,  x∈ D,

whenever the limit exists. Here, instead of u one can a take a quasi continuous modification ũ of u, see for example <cit.>. Recall that a function ũ is quasi continuous if for every ε>0 there exists an open set G_ε such that ũ_|G_ε^c is continuous, Cap_ψ,α(G_ε)<ε and ũ = u a.e. This means that outside of set N=∩_ε>0 G_ε function u can be strictly defined and therefore R̃u=u on N^c. To show that μ(N)=0, note that by the definition of capacity, for every ε>0 there exists a function v∈ H^ψ,α(^n) such that v≥ 1 on N and ||v||_a,1<ε, so

    μ(N)≤(∫_N |v(x)|^2μ(dy))^1/2≤c̃_1 ||v||_ψ,α<c̃_1ε

and therefore μ(N)=0. Here the second inequality follows from calculations analogous to ones in the proof of Lemma <ref>.



Let d≤ n, D a d-set in ^n and μ the d-measure on D. Let ψ be a function such that (<ref>), and hold and α>0 satisfying (<ref>). There exist a continuous restriction operators R:H^ψ,α(^n)→ H^ψ,α(D,μ).

Using the classical interpolation theorem for a special class of spaces associated with (^n) we will show that there exists a constant c̃_1>0 such that for all u∈ H^ψ,α(^n)
 
    ||Ru||_(2),D,μ^1,1≤c̃_1 ||u||_ψ,α.

Denote by a_j(x,y)=|Ru(x)-Ru(y)|1_|x-y|<2^-j, j∈_0 and let T be an operator on L^2(^n) such that Tu=(a_j)_j∈_0. Since Ru=u μ-a.e. on D, by Lemma <ref> there exists a constant c̃_2>0 such that

    sup_j∈_0(ϕ^α(2^2j) 2^j(2d-n)∬_|x-y|<2^-j|Ru(x)-Ru(y)|^2μ(dx)μ(dy))≤c̃_2 ||f||_L^2(^n)^2

for all u=K_ψ,α*f, f∈ L^2(^n). Let L=L^2(D× D,μ×μ). It follows that 

    (a_j)_j∈ l_∞^ϕ,α(L)={(ξ_j)_j∈_0: ξ_j∈ L,||ξ||_l_∞^ϕ,α(L)=sup_j∈_0ϕ^α(2^2j) 2^nj||ξ_j||_L^2<∞}

and that the operator T is bounded from H^ψ,α(^n) to l_∞^ϕ,α(L), i.e.  

    ||(a_j)_j∈_0||_l_∞^ϕ,α(L)≤c̃_2||K_ψ,α*f||_ψ,α=c̃_2||f||_L^2(^n).

Let (X_1,X_2)_θ,2:={a:  a∈ X_1+X_2,  ||a||_(X_1,X_2)_θ,2 <∞} be the K-interpolation space of Banach spaces X_1 and X_2 and ||a||_(X_1,X_2)_θ,2=(∫_0^∞ (t^-θK(t,a))^2dt/t)^1/2, where the Peetre K-functional is defined by K(t,a)=inf_a=a_1+a_2(||a_1||_X_1+t||a_2||_X_2), see <cit.>. Choose α_0<α<α_1 such that n-d/2<δ_1α_0≤(δ_2∨δ_4)α_1<n-d/2+1 and let θ=α_1-α/α_1-α_0∈(0,1). By <cit.> and <cit.>, since T is bounded from H^ψ,α_i(^n) to l^ϕ,α_i_∞(L), i=0,1 it is also bounded from (H^ψ,α_0(^n),H^ψ,α_1(^n))_θ,2 to (l^ϕ,α_0_∞(L),l^ϕ,α_1_∞(L))_θ,2. By a version of <cit.>, (l_∞^ϕ,α_0(L),l_∞^ϕ,α_1(L))_θ,2=l_2^ϕ,α(L), where

    l_2^ϕ,α(L)={(ξ_j)_j∈_0: ξ_j∈ L,||ξ||_l_2^ϕ(L)=(∑_j∈_0ϕ^α(2^2j) 2^nj||ξ_j||_L^2)^1/2<∞}.

Furthermore, (l_2^ϕ,α_0(L^2(^n)),l_2^ϕ,α_1(L^2(^n)))_θ,2=l_2^ϕ,α(L^2(^n)) and H^ψ,α(^n) is a retract of the space l_2^ϕ,α(L^2(^n)), <cit.>. Therefore, the interpolation identity
(H^ψ,α_0(^n),H^ψ,α_1(^n))_θ,2=H^ψ,α(^n) follows by <cit.>. Combining these results we get (<ref>).
 




§ EXTENSION THEOREM



The proof of the extension theorem is divided into two parts; case d<n and case d=n. In the first case we define the extension operator by using the Whitney decomposition of D^c and the approach as in <cit.>, which deals with the classical Besov spaces. The case d=n follows then by interpreting the n-set D in ^n as a n-subset of ^n+1. The extension operator is then defined as a composition of the extension operator to ^n+1 and restriction operator to ^n. 

Before proving the extension theorem for d<n, we recall the definition of the Whitney decomposition of a set.
 A Whitney decomposition of an open set A is a collection of closed cubes {Q_i}_i∈ with disjoint interiors and sides parallel to the axes such that A=⋃_i Q_i, where each cube Q_i has side length s_i=2^-M_i for some M_i∈ℤ and diameter l_i such that l_i≤ d(Q_i,A^c)≤ 4l_i.
 
Denote by x_i the center of the cube Q_i and let ε∈(0, 1/4). Denote by Q_i^* = (1 + ε)Q_i the cube with center x_i expanded by factor 1+ε. If x∈ Q_k∩ Q_i^* then  

    1/4 s_k≤ s_i≤ 4s_k
 
and Q_i and Q_k touch each other. This implies that every point in A is covered by at most N_0 cubes Q_i^*, where N_0∈ depends only on n. By <cit.> we can associate with decomposition {Q_i^*} a partition of unity
{φ_i}_i∈⊂ C_c^∞(^n), i.e. a family of non-negative functions with the following properties:

    φ_i⊂ Q_i^*,  ∑φ_i=1 on D̃^c,   |D^jφ_i|≤c̃ l_i^-|j| for some c̃>0.

If D is an d-set in ^n then by <cit.> the closure D of D is also a d-set and μ(D∖ D)=0 for every d-measure μ. Therefore, it is enough to prove the theorem for a closed d-set D. 

Next, we define the extension operator E from H^ψ,α(D,μ) to H^ψ,α(^n), when d<n. Let ω_i=μ(B(x_i,6l_i))^-1≍ l_i^-d and I={i∈:s_i≤ 1}. For H^ψ,α(D,μ) define 

    E u(x)= 
    	u(x),     x∈  D 
    ∑_i∈
        Iφ_i(x)ω_i∫_|y-x_i|<6l_iu(y)μ(dy),      x∉D.



 Let D be a closed d-set in ^n and d<n. Let ψ be a function such that (<ref>), and hold and α>0. There exist a continuous extension operators E:H^ψ,α(D,μ)→ H^ψ,α(^n) such that Eu=u μ-a.e. on D for all u∈ H^ψ,α(D,μ).

 We will show that for some c>0, N∈ℕ and k_0>αδ_2 there exists a constant c̃_1 such that 

    ||Ẽu||_(1),α,k_0^h_0≤c̃_1||u||_(2), D,μ^c,N,  ∀ u∈ H^ψ,α(D,μ).

Here we can choose the smallest k_0∈ satisfying the given condition, see Theorem <ref> and Remark <ref>. Since D is of Lebesgue measure zero in ^n it is enough to prove (<ref>) for E u 1_D^c. For every x∈ D^c there exists a k such that x∈ Q_k. If s_k>4 then by (<ref>) x∉Q_i^* for all i∈ I and Eu(x)=0. Therefore it is enough to consider the case when s_k≤ 4. Also note that ∑_i φ_i(x)=∑_i∈ Iφ_i(x) when s_k<1/4.

Let x∈ Q_k and let i∈ I be such that ϕ_i(x)≠ 0. Then for all y∈ B(x_i,6l_i) we have

    |y-x_k|≤|y-x_i|+|x_i-x|+|x-x_k|≤ 6l_i+l_i+l_k≤ 29 l_k,

which implies that

    |Eu(x)|   ≤∑_i∈ Iφ_i(x)ω_i∫_|y-x_i|<6l_i|u(y)|μ(dy)≲∑_i∈ Iφ_i(x)l_i^-d∫_|y-x_k|<29l_k|u(y)|μ(dy)
       (<ref>)≲∑_i∈ Iφ_i(x) ·
    l_k^-d∫_|y-x_k|<29l_k|u(y)|μ(dy)≲(
    l_k^-d∫_|y-x_k|<29l_ku^2(y)μ(dy))^1/2.

For j∈ define Δ_j:=⋃_{k:s_k=2^-j}Q_k. Note that there exists an integer N_1 depending only on n such that every point y∈^n is covered by at most N_1 balls B(x_k,29l_k) where Q_k⊂Δ_j. This follows from the fact
that |x_k-x_k'|≥ 2^-j and l_k=√(n) 2^-j, for all
Q_k,Q_k'⊂Δ_j. By the previous calculation it follows that 

    ∫_ D^c| Eu(x)|^2dx   =∑_j=-2^∞∑_Q_k⊂Δ_j∫_Q_k| Eu(x)|^2dx≲∑_j=-2^∞∑_Q_k⊂Δ_j∫_Q_k( l_k^-d∫_|y-x_k|<29l_ku^2(y)μ(dy))dx
       ≍∑_j=-2^∞ 2^-(n-d)j∑_Q_k⊂Δ_j∫_|y-x_k|<29l_ku^2(y)μ(dy) ≤ 2^2(n-d)∑_j=-2^∞∫_Δ_ju^2(y)μ(dy),
 
which implies that

    ||Eu||_L^2(^n)≲ ||u||_L^2( D,μ).

Next, for x∈Δ_i, y∈Δ_j and
|x-y|<2^-i/2 we have

    2^-j√(n)   ≤ d(Δ_j, D)≤ d(y,D)≤ |x-y|+d(x, D)≤ 6√(n) 2^-i,

so j≥ i-2. Analogously, √(n) 2^-i≤ 5√(n) 2^-j+2^-i-1 so j≤ i+2. Therefore,  

    x∈Δ_i, |x-y|<2^-i/2 ⇒  y∈⋃_j=i-2^i+2Δ_j.

Since Eu(x)=0 for for x∈Δ_i, i≤ -3 it follows that Eu(y)=0 when |x-y|<2^-i-1 for some x∈Δ_i, i≤ -5. Analogously, Δ_h^k_0 (Ẽu)(x)=0 if |h|<2^-5/k_0. Therefore, for h_i:=2^-i-1/k_0,

    ∫_D^c∫_|h|<h_0|   Δ_h^k_0 (Eu)(x)|^2ψ^α(|h|^-1)/|h|^n
    dh dx≤∑_i=-4^∞∬_x∈Δ_i
    |h|<h_i|Δ_h^k_0 (Eu)(x)|^2ψ^α(|h|^-1)/|h|^n
    dh dx
       +∑_i=5^∞∬_x∈Δ_i
    h_i≤|h|<2^-5/k_0|Δ_h^k_0 ( Eu)(x)|^2ψ^α(|h|^-1)/|h|^n
    dh dx=:A+B.

First we asses the term B. Let F_i:=⋃_j=i^∞Δ_j. Note that

    B   =∑_i=5^∞∑_m=4^i-1∫_h_m+1≤|h|<h_mψ^α(|h|^-1)/|h|^n∫_x∈Δ_i|Δ_h^k_0 (Eu)(x)|^2
    dx dh
       =∑_m=4^∞∫_h_m+1≤|h|<h_mψ^α(|h|^-1)/|h|^n∫_x∈ F_m+1|Δ_h^k_0 ( Eu)(x)|^2dx dh
       (<ref>)≲∑_m=4^∞ψ^α(2^m)2^mn∬_x∈ F_m+1
    h_m+1≤|h|<h_m |Δ_h^k_0 (Eu)(x)|^2 dx dh.

Similarly as in (<ref>), for x∈ F_i+1 and |h|<h_i it follows that x,x+h,..,x+k_0h∈ F_i-2. Since 

    |Δ_h^k_0 (Eu)(x)|^2≲ | Eu(x)- Eu(x+h)|^2+⋯ +|Eu(x+(k_0-1)h)-Eu(x+k_0h)|^2

it follows that

    B   ≲∑_m=4^∞ψ^α(2^m)2^mn∬_x,y∈ F_m-2
    |x-y|<h_m(Eu(x)-Eu(y))^2 dx dy.

For k,m≥ 2 and x∈Δ_k and y∈Δ_m it follows that 

    | Eu(x)-Eu(y)|≤∑_i∑_jφ_i(x)φ_j(y)ω_iω_j∬_|s-x_i|<6l_i
    |t-x_j|<6l_j|u(s)-u(t)|μ(ds)μ(dt)

From

    x∈Δ_k, φ_i(x)≠ 0⇒1/8 l_k≤ l_i≤ 64 l_k

it follows that 

    |Eu(x)-Eu(y)|   (<ref>)≲∑_i∑_jφ_i(x)φ_j(y)l_k^-dl_m^-d∬_|s-x_k|<29l_k
    |t-x_m|<29l_m|u(s)-u(t)|μ(ds)μ(dt)
       ≤(l_k^-dl_m^-d∬_|s-x_k|<29l_k
    |t-x_m|<29l_m(u(t)-u(s))^2μ(dt)μ(ds)
    )^1/2.

Here x_k,x_m are the centers and l_k,l_m diameters of cubes Q_p⊂Δ_k and Q_r⊂Δ_m containing x and y respectively. 
Now it follows that for i∈, y∈Δ_m and k,m≥ 2 

    ∫_x∈Δ_k,
    |x-y|<2^-i|Eu(x)-Eu(y)|^2 dx   ≲∫_x∈Δ_k
    |x-y|<2^-il_k^-dl_m^-d∬_|s-x_k|<29l_k
    |t-x_m|<29l_m(u(t)-u(s))^2μ(dt)μ(ds)dx
       ≤ N_0 l_k^-dl_m^-ds_k^-n∬_|s-y|<c2^-k+2^-i
    |t-x_m|<29l_m(u(t)-u(s))^2μ(dt)μ(ds),

where c=30√(n). Analogously, we get

    ∬_x∈Δ_k, y∈Δ_m
    |x-y|<2^-i|Eu(x)-Eu(y)|^2
    dx dy ≲ N_0^2 2^-(n-d)k2^-(n-d)m∬_|t-s|<2^-i+c2^-k+c2^-m(u(t)-u(s))^2dμ(s)dμ(t).

This implies that for i≥ 4 

    ∬_x,y∈ F_i-2
    |x-y|<2^-i| Eu(x)- Eu(y)|^2
    dx dy=∑_k,m=i-2^∞  ∬_x∈Δ_k, y∈Δ_m
    |x-y|<2^-i| Eu(x)- Eu(y)|^2
    dx dy
       ≲(∑_k,m=i-2^∞
    2^(d-n)k2^(d-n)m)∬_|t-s|<(8c+1)2^-i(u(t)-u(s))^2μ(ds)μ(dt)

and by applying this to (<ref>) we get that

    B   ≲∑_i=4^∞ψ^α(2^i) 2^i(2d-n)∬_|t-s|<(8c+1)2^-i(u(t)-u(s))^2μ(ds)μ(dt).

Next, by the mean value theorem 

    A    ≲∑_i=-4^∞ ∫_|h|<h_i(∑_|j|=k_0 ∫_Δ_i∫_0^1...∫_0^1
    |h|^2k_0 |D^j(Eu)(x+(t_1+...+t_k_0)h)|^2 dt_1 ...dt_k_0  dx)ψ^α(|h|^-1)/|h|^ndh
       (<ref>)≤∑_i=-4^∞ ∫_|h|<h_iψ^α(|h|^-1)/|h|^n-k_0dh·∑_|j|=k_0∫_F_i-2∖ F_i+3
    |D^j(Eu)(z)|^2 dz≲5∑_i=-2^∞ψ^α(2^i)/2^2k_0i∑_|j|=k_0∫_Δ_i |D^j(Eu)(z)|^2 dz.

In the last line we also used that D^j(Ẽu)(z)=0 if z∈Δ_i and i≤ -3. To find an upper bound for |D^j(Eu)(z)|, z∈Δ_i, we distinguish two cases; i≥ 2 and i<2. First, take z,y∈ Q_m⊂Δ_l, l≥ 2 and |j|=k_0. Since ∑_iD^jφ_i(z)=0, by similar calculations as before, we get

    |D^j( Eu)(z)|   =|∑_iD^jφ_i(z)ω_i∫_|s-x_i|<6l_i(u(s)-Ẽu(y))μ(ds)|
       ≤∑_i∑_k |D^jφ_i(z)|φ_k(y)(ω_iω_k∬_|s-x_i|<6l_i
    |t-x_k|<6l_k|u(s)-u(t)|^2μ(ds)μ(dt))^1/2.

Recall that there are at most N_0 indices i for which z∈ Q_i^* and D^jφ_i(z)≠ 0. By (<ref>) and (<ref>) z∈ Q_i^* implies ω_i≍ l_m^-d and |D^jφ_i(z)|≲ l_i^-|j|≲  l_m^-|j|. Also, by (<ref>) ω_k≍ l_m^-d for k such that φ_k(y)≠ 0. Therefore, 

    |D^j( Eu)(z)|   (<ref>)≲l_m^-k_0(l_m^-2d∬_|s-x_m|<29l_m
    |t-x_m|<29l_m|u(s)-u(t)|^2μ(ds)μ(dt))^1/2.

Applying this inequality to (<ref>) we arrive to 

    A   ≲∑_i=2^∞ψ^α(2^i)2^-2k_0i∑_|j|=k_0∑_Q_m⊂Δ_i∫_Q_m  2^2k_0i+2id∬_|s-x_m|<29l_m
    |t-x_m|<29l_m|u(s)-u(t)|^2μ(ds)μ(dt)dz

and since every s∈ D^c is covered by at most N_1 balls B(x_m,29l_m) it follows that

    A≲   ∑_i=2^∞ψ^α(2^i)2^i(2d-n)∬_|s-t|<60√(n)2^-i|u(s)-u(t)|^2μ(ds)μ(dt).

For the remaining part in A, take z∈Δ_k, k≥ -2. By the same arguments as before, 

    |D^j( Eu)(z)|   ≤∑_i|D^jφ_i(z)|ω_i∫_|s-x_i|<6l_i|u(s)|μ(ds)(<ref>)≲∑_φ_i(z)≠ 0l_i^-2(ω_i∫_|s-x_i|<6l_i|u(s)|^2μ(ds))^1/2
       ≲ 2^kk_0(2^kd∫_|s-x_k|<29l_k|u(s)|^2μ(ds))^1/2

and therefore ∑_i=-2^1∫_Δ_i|D(Eu)(z)|^2 dz≲||u||_L^2( D,μ). This inequality together with (<ref>), (<ref>) and (<ref>) implies (<ref>).

That E is truly the extension operator for R, i.e. that REu =u μ-a.e., follows from calculation similar to the calculation above. One first shows that for every t_0∈ D and r>0 small enough

    ∫_|x-t_0|≤ r(Eu(x)-u(t_0))^2dx ≲r^d/ψ^α(r^-1)∫_|t-t_0|<30 r(u(t)-u(t_0))^2ψ^α(|t-t_0|^-1)/|t-t_0|^2d-nμ(dt),

where the integral is finite for μ-almost all t_0 and decreasing as r goes to 0. Since lim_r→ 01/ψ^α(r^-1)r^n-d=0, it follows that for μ almost all t_0

    |R Eu(t_0)-u(t_0)|≲lim_r→ 0(r^-n∫_|x-t_0|≤ 
    r(Ẽu(x)-u(t_0))^2dx)^1/2= 0.




Let α∈(1/2,∞). Note that for a n-set D in ^n, D̃=D×{0} is a n-set in ^n+1 and that every function u∈ H^ψ,α(D,μ) can be represented as a function ũ in H^ψ̃,α̃(D̃,μ̃), where

    α̃:=2α, ψ̃(|ξ|):=ψ^1/2(|ξ|)|ξ|^1/α̃, μ̃(A×{0}):=μ(A)
       ũ (x,0):=u(x),  x∈ D  and   ||ũ||_(1),D̃,μ̃=|| u||_(1),D,μ,

By <cit.> the function ξ↦ϕ^1/2 (|ξ|)|ξ|^1/2α is a complete Bernstein function and ψ̃ satisfies conditions and with δ̃_i:=δ_i/2+1/4α. Analogously, the space H^ψ,α(^n) can be represented as H^ψ̃,α̃(^n×{0},μ̅), where μ̅ is the restriction of the n-dimensional Hausdorff measure in ^n+1 to ^n×{0}.




Theorem <ref> holds true as well in the case of d=n and α∈(1/2,∞). 


Take u∈ H^ψ,α(D,μ) and let ũ be the corresponding function in H^ψ̃,α̃(D̃,μ̃) from Remark <ref>. By Theorem <ref> function ũ can be extended to a function Ẽũ∈ H^ψ̃,α̃(^n+1), which can then be restricted to a function in H^ψ̃,α̃(^n×{0},μ̅) by applying the continuous restriction operator R̅ from Theorem <ref>. Again using the Remark <ref>, we can define the extension operator E as

    (Eu)(x)=(R̅Ẽũ)(x,0), x∈^n.

The continuity of E follows from the continuity of the extension and restriction operators Ẽ and R̅ and REu=u almost everywhere on D, where R is the restriction operator from H^ψ,α(^n) to H^ψ,α(D,μ).


Corollary <ref>:
By applying Theorem <ref>, the proof is an immediate consequence of <cit.>.


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