On Seeley‐type universal extension operators for the upper half space

Modified from the standard half‐space extension via the reflection principle, we construct a linear extension operator for the upper half space R+n$\mathbb {R}^n_+$ that has the form Ef(x)=∑j=−∞∞ajf(x′,−bjxn)$Ef(x)=\sum _{j=-\infty }^\infty a_jf(x^{\prime },-b_jx_n)$ for xn<0$x_n<0$ . We prove that E$E$ is bounded in all Ck$C^k$ ‐spaces, Sobolev and Hölder spaces, Besov and Triebel–Lizorkin spaces, along with their Morrey generalizations. We also give an analogous construction on bounded smooth domains.


Introduction
Given a function f defined on the upper half space R n + = {(x ′ , x n ) : x n > 0}, the reflection principle gives a well-known construction to extend f to the total space R n while preserving its regularity property: we can define the extension Ef that depends linearly on f by (1) Ef (x ′ , x n ) = E a,b f (x ′ , x n ) := j a j f (x ′ , −b j x n ), for x n < 0, where b j > 0.
Here (a j , b j ) j are (finitely or infinitely many) real numbers that satisfy certain algebraic conditions (see below).This method traces back to Lichtenstein [Lic29] for C 1 -extensions and [Hes41] for C k -extensions.See also [Nik53] and [Bab53] for the corresponding Sobolev extensions.
In the spaces of distributions, a linear extension operator E is viewed as a right inverse of the natural restriction map D ′ (R n ) ↠ D ′ (R n + ).Triebel [Tri95] and Franke [Fra86] showed that for every ε > 0 there is a m = m(ε) > 0, such that, if the finite collection (a j , b j ) j satisfy j a j (−b j ) k = 1 for all integers −m ≤ k ≤ m, then for E in (1) have boundedness in Besov spaces E : B s pq (R n + ) → B s pq (R n ) and in Triebel-Lizorkin spaces E : F s pq (R n + ) → F s pq (R n ), for all ε < p, q ≤ ∞ and −ε −1 < s < ε −1 (p < ∞ in F -cases).In [Tri06,Section 1.11.5]such E is called a common extension operator which depends on ε(> 0).
In contrast, we call E a universal extension operator, if we have the boundedness E : A s pq (R n + ) → A s pq (R n ) for all A ∈ {B, F }, 0 < p, q ≤ ∞ and s ∈ R (p < ∞ in F -cases), simultaneously (i.e.we can take ε = 0).
Note that the collections (a j , b j ) j discussed above are all finite.In those cases, the range of spaces on which E is bounded would inevitably be finite and depends on ε > 0. Hence such E is never universal.
By taking some infinitely nonzero sequences (a j , b j ) j , Seeley [See64] constructed one such operator (1) such that E : C k (R n + ) → C k (R n ) for all k = 0, 1, 2, . . . .In particular Seeley's extension perverse C ∞ -smoothness.However, to the best of authors' knowledge, there was no proof of the boundedness of Seeley's operator on general Besov and Triebel-Lizorkin spaces.
In this paper we generalize Seeley's construction by extending the boundedness to the spaces of negative index, and we show that such operator is a universal extension operator.

Theorem 1.
(i) There exists an extension operator E : x n > 0 , where b j > 0 such that, (3) Moreover, when (3) is satisfied, then (2) always defines an extension operator E for functions on R n + , that has the following boundedness (simultaneously): (ii) (Sobolev, Hölder and C k ) E : W k,p (R n + ) → W k,p (R n ) and E : C k,s (R n + ) → C k,s (R n ) are defined and bounded for all k ∈ Z, 0 < p ≤ ∞ and 0 < s < 1.
(iii) (Besov and Triebel-Lizorkin) E : S ′ (R n + ) → S ′ (R n ) is continuous.Moreover we have boundedness in Besov spaces E : B s pq (R n + ) → B s pq (R n ) and in Triebel-Lizorkin spaces E : F s pq (R n + ) → F s pq (R n ) for all s ∈ R and 0 < p, q ≤ ∞. (iv) (Morrey-type) More generally E has boundedness on all Besov-type spaces B sτ pq , Triebel-Lizorkin-type spaces F sτ pq and Besov-Morrey spaces N sτ pq .That is, E : See Definitions 3, 6 and 8 for the spaces in the theorem.Here the Sobolev spaces W k,p for p < 1 can be defined and are discussed in [Pee75], see also Remark 4.
The analogue of Theorem 1 on smooth domains is also true; see Theorem 26.The operator is given by (39).
Remark 2.Here we allow δ k in (3) to tend to 0 as |k| → ∞.In practice we can choose (a j , b j ) j such that the sum j 2 δ|j| |a j |b k j < ∞ holds for all k and all arbitrarily large δ, see Proposition 14.It is not known to the authors whether the results still hold if we remove 2 δ k |j| in (3), i.e. if we only assume In the proof of Theorem 1 the terms 2 δ k |j| are used only when we consider p < 1 in (ii) and min(p, q) < 1 in (iii) and (iv).
We should remark that the universal extension operator exists not only on smooth domains but also on Lipschitz domains.This is done by Rychkov [Ryc99] using Littlewood-Paley decompositions.Rychkov's extension operator j=0 is a carefully chosen family of Schwartz functions.The shape of E R is totally different from (1) and the construction follows a different methodology.
Our result shows that the construction (1) can also be made to be universal extension.Cf. the comment below [JMHS15, Corollary 5.7].
In fact, our extension operator is "more universal" than the Rychkov extension E R [Ryc99].Although E R is bounded in all Besov and Triebel-Lizorkin spaces (see [Ryc99, Theorem 4.1]), it is not known whether E R is bounded on the endpoint Sobolev spaces W k,1 , W k,∞ and the C k -spaces.On the other hand, our extension operator is also defined on the measurable function space L p (R n + ) for 0 < p < 1, whose element may not be realized as a distribution on R n + .The range of the extension operator is important if we want to construct some operator that has the form (4) T f (x) = U K(x, y) Ef (y)dy, where E is an extension operator for the domain Ω ⊂⊂ U.
Here K(x, y) can be some kernel of integral operator or singular integral operator.This method has been used extensively in solving Cauchy-Riemann problem (the ∂-equation) on certain domains in C n .For example Wu [Wu98] used the Seeley's extension to construct a solution operator that is W k,p bounded for all k ≥ 0. If E is some common extension operator that is only bounded in a finite range of k, then T is only bounded in a finite range as well.
By replacing the Seeley's extension with our extension operator and doing more analysis on K(x, y), it might be possible that one can show that T in (4) is bounded in W k,p for k < 0 as well.
The estimates on negative Sobolev spaces are discussed in [SY24,Yao24], where we took E to be Rychkov's extension operator.In the case of smooth domains, if we replace Rychkov's extension by our Seeley-type extension, then it is possible not only to get some estimates related to L 1 -Sobolev spaces, but also to simplify some of the proofs in these papers (see for example [Yao24, version 1, Proposition 5.1 and Remark 5.2 (iii)]).

Function Spaces and Notations
In the paper we use the following definitions for Sobolev spaces and Hölder spaces, including both positive and negative indices.We also include the case p < 1 for Sobolev spaces.
We use C k (U ) = C k (U ) for the space of all continuous functions f : U → R such that ∂ α f are bounded and uniformly continuous for all |α| ≤ k.We use ∥f ∥ to be the space of bounded smooth functions.For 0 < s < 1, we define the Hölder space C k,s (U ) to be the space of all functions For 0 < p ≤ ∞, we define the Sobolev space W k,p (U ) by the following: For k > 0, 0 < s < 1 and 1 ≤ p ≤ ∞, we define to be the subset of distributions such that the above norms are finite, respectively.(iv) For 0 < p < 1, we define W −k,p (U ) to be the abstract completion of C ∞ c (U ) under the quasi-norm (6) with {g α } α ⊂ L p (U ) replacing by {g α } α ⊂ C ∞ c (U ) (where we only consider f ∈ C ∞ c (U ) on both sides).Remark 4. When 0 < p < 1 and k ≥ 1, the natural (continuous) mapping ι : W k,p (U ) → L p (U ) is not injective.The proof can be modified from [Pee75, Proposition 3.1].If one use W k,p (U )/ ker ι instead, then the elements can always be realized as measurable functions (in L p (U )).
One can check that Theorem 1 (ii) implies the boundedness When U satisfies nice boundary condition, particularly when U ∈ {R n + , R n } or when U is a bounded smooth domain, we have the following: (a) Definition 3 (i) and (ii) coincide, in the sense that For the proof see [Eva10, Section 5.3] or [Ada75, Theorem 3.17].Taking the distribution derivatives, we get the density of Theorem 3.9] for example.(c) For every k ∈ Z, we have equivalent norms ∥f ∥ W k+1,p (U ) ≈ ∥f ∥ W k,p (U ) + n j=1 ∥∂ j f ∥ W k,p (U ) for 1 < p < ∞, and ∥f ∥ C k+1,s (U ) ≈ ∥f ∥ C k,s (U ) + n j=1 ∥∂ j f ∥ C k,s (U ) for 0 < s < 1. However they fail for p = 1, ∞: in both cases we have ∥f ∥ L p (U ) ≳ ∥f ∥ W −1,p (U ) + n j=1 ∥∂ j f ∥ W −1,p (U ) but the converse inequalities are false.This can done by using (b) and [Pre97, Theorem 5].
We use the standard convention for spaces of tempered distributions.Definition 6.We use S (R n ) for the Schwartz space and S ′ (R n ) for the space of tempered distributions.
For an arbitrary open subset U ⊆ R n , we define ) to be the space of distributions in U that has tempered distributional extension.
We let S (U ) : to be the space of Schwartz functions supported in U .
is the quotient space.Since S (R n ) and S ′ (R n ) are dual to each other with respect to their standard topologies, we see that The Besov and Triebel-Lizorkin spaces, along with their Morrey analogies, can be defined using Littlewood-Paley decomposition.Note that we do not use these characterizations directly in the proof.
j=0 be a sequence of Schwartz functions satisfying: .
Here for q = ∞ we take natural modifications by replacing the ℓ q -sums with the supremums over j.
We define the corresponding spaces For arbitrary open U ⊆ R n , we define For the classical Besov and Triebel-Lizorkin spaces, we use q q,q (U ), for q ∈ (0, ∞] and s ∈ R. We shall see that A sτ pq (U ) is always a (quasi-)Banach space, and different choices of (λ j ) ∞ j=0 result in equivalent norms.See [Tri10, Proposition 2.3.2] and [Tri20, Propositions 1.3 and 1.8].
Remark 9.In the definition we only consider τ ≤ 1 p .If we extend them to τ > 1 p then by [YY13, Theorem 2] and [Sic12, Lemma 3.4] we have Our notation N sτ pq corresponds to the B sτ pq in [Sic12, Definition 5].For the usual conventions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces: we have For the notations in [Tri14,HT23], we have the correspondence For more discussions, we refer [YSY10, Sic12, Tri14, HT23] to readers.
In the following, we will use the notation x ≲ y to mean x ≤ Cy where C is a constant independent of x, y, and x ≈ y for "x ≲ y and y ≲ x".We use x ≲ ε y to emphasize the dependence of C on the parameter ε.
For r ̸ = 0 we use ϑ r as the dilation operator given by We use M ns pq and M ns pq for some fixed positive constants in Proposition 22 and Theorem 23, both of which depend on n ≥ 1, 0 < p, q ≤ ∞ and s ∈ R.
We let S = S n be the zero extension operator of R n + , that is Remark 10.We have the definedness and boundedness S n : [Tri20,Theorem 2.48].We will use the case n = 1 and the case p = ∞ in the proof of Theorem 1 (iii).
To clarify, the statement of [Tri20, Theorem 2.48] requires q ≥ 1 for the F s ∞q -case.We do not need this restriction as we do not use the unboundedness of S n for the case s / ∈ (max(n( 1 p − 1), 1 p − 1), 1 p ). See also [Tri20,Theorem 2.48,Proof Step 2].

The Construction of Universal Seeley's Extension: Proof of Theorem 1 (i) and (ii)
First we need to fulfill condition (3).In [See64] Seeley took b j := 2 j for j ≥ 0 and found (a j ) ∞ j=0 such that ∞ j=0 a j (−b j ) k = 1 for all k ≥ 0. This can be done by constructing an entire function (iii) When β > 2, we have the following quantitative estimate for l = 1, 2, 3, . . .: Proof.(i) is the result of the Weierstrass factorization theorem, see [Rud87, Theorem 15.9] for example.For (ii), one can see that for k ≥ 0, Since β > 1, we see that the sum ∞ k=0 u k /W ′ β (β k ) absolutely converges.Therefore the sum in (10) converges absolutely and locally uniformly in z.We conclude that F β u is indeed an entire function.For each j ≥ 0, we have as for each z = β k there is only one nonzero term in the sum (ii).For (iii), note that for each l ≥ 1, for each j ≥ 0. Therefore using (12) and (13) we see that for k, l ≥ 1, On the other hand Therefore taking sum over k ≥ 1 we obtain (11): . □ Remark 12.If one takes u k := (−1) k in Lemma 11 (ii), then the corresponding sequence a j := 1 j! (F β u ) (j) (0) (j = 0, 1, 2, . . . ) satisfies j a j β jk = (−1) k for all k ≥ 0. This is enough to prove the boundedness results in [See64] (C k for k ≥ 0), but in our case we also need the equality to be true for k < 0. However, one cannot construct an entire function F such that F (2 −k ) = (−1) −k for all k ≥ 0.
Let a j := ã|j| for j ̸ = 0 and a 0 := 2ã 0 , we see that ∞ j=−∞ a j 4 jk = (−1) k for all k ∈ Z with all summations converging absolutely.In particular j=−∞ satisfies the condition (3) and we prove Theorem 1 (i).□ One can see that Seeley-type extensions have the following structures: Lemma 15.
(i) Let (a, b) = (a j , b j ) ∞ j=−∞ be the collection satisfying the condition (3).Then for every b be the extension operator defined in (2).The formal adjoint E * has the following expression: Proof.The results follow from direct computations.(i): By assumption, Replacing y n by x n we get the expression (15).□ Remark 16.In the notations of (8) and (9) we can write E a,b = S + j a j ϑ −bj • S. For r < 0 we have where δ 0 is the Dirac measure at 0 ∈ R.
the domain where either side is defined) if and only if j a j = 1.Taking higher order derivatives we see that Lemma 17.Let (X, ∥ • ∥) be a quasi-Banach space such that (x, y) → ∥x − y∥ q is a metric for some 0 < q ≤ 1, i.e. ∥x + y∥ q ≤ ∥x∥ q + ∥y∥ q for all x, y ∈ X.Then for any δ > 0 there is a constant K q,δ > 0 such that ∞ j=−∞ x j ≤ K q,δ ∞ j=−∞ 2 δ|j| ∥x j ∥ for all sequence (x j ) j∈Z ⊂ X such that right hand summation converges.In this case the sum ∞ j=−∞ x j converges with respect to the quasi-norm topology of X. Remark 18.The couple (X, ∥ • ∥ q ) is also call a q-convex quasi Banach space (q-Banach space for short).
To prove the W k,p -boundedness we apply Lemma 17.For every When k ≥ 0, also for every By the assumption (3) we can take δ > 0 small such that j∈Z 2 δ|j| |a j |b k−⌈ 1 p ⌉ j < ∞ and j∈Z 2 δ|j| |a j |b k j < ∞.We now get the boundedness of W k,p for k ≥ 0.
When k < 0, applying Lemma 15 (ii) to (16) for Taking the infimum over all {g γ } |γ|≤−k for f , and taking δ from (3) such that j∈Z ) < ∞ as well, we get the W k,p -boundedness for k < 0, finishing the proof.□ Remark 19.The result can be extended to vector-valued functions.Indeed let 0 < q ≤ 1, 0 < p < ∞, and let (X, | • | X ) be a quasi Banach space such that |x + y| q X ≤ |x| q X + |y| q X .Then for every L p (strongly measurable) function f : R n + → X, by the same calculation to (16), Here K q,δ/2 and K p,δ/2 are the constants in Lemma 17.Also by Lemma 17 the sum of functions j∈Z a . In practice we will take X = B s pq (R n−1 ) in the proof of Theorem 23, Step A3.
4. Boundedness on Besov and Triebel-Lizorkin: Proof of Theorem 1 (iii) and (iv) Before we discuss the boundedness on Besov and Triebel-Lizorkin spaces, for completeness we prove that .
On the other hand, by the condition j a j (−b j ) −k = 1 for k = 1, 2, 3, . . . in (3), we have Therefore along with the zero extension to R n − , E * g defines a Schwartz function in R n which has support | with the implied constant being independent of g.
Since the seminorms sup We now turn to prove Theorem 1 (iii) and (iv).
To prove Theorem 1 (iii), it might be possible to use the mean oscillation approach in [Tri92, Chapter 4.5.5].However [Tri92, (4.5.2/9)] fails when (a j ) j has infinite nonzero terms.This is critical if one wants to prove the Besov and Triebel-Lizorkin boundedness of Seeley's operator (see Remark 12) using such method.The authors do not know what the correct modification of [Tri92, (4.5.2/9)] is.
for all k ∈ Z and 0 < s < 1, with equivalent norms.
Proposition 22.For any 0 < p, q ≤ ∞ and s ∈ R there is a M = M n,s p,q > 0 and C = C n,p,q,s,M > 0 such that Proof.We only need to prove the Triebel-Lizorkin case, i.e.A = F .Indeed suppose we get the case A = F , then by real interpolation (19), for each 0 < p, q ≤ ∞ and s ∈ R there is a C p,q,s > 0 such that (see [Tri95, Definition 1.2.2/2 and Theorem 1. ) is also bounded.Taking T = ϑ r and replacing M n,s p,q (obtained from the F -cases) with max(M n,s p,q , M n,s−1 p,p , M n,s+1 p,p ) we complete the proof.
Step 1: In particular we can take M 1,s p,q := max( 1 p , s − 1 p , 1 p − s) in this case.
Let m ≥ 0 be such that s + m > n max(0, In particular we can take M n,s p,q := M n,s+m p,q + m in this case. Step 4: The case p = ∞, 0 < q ≤ ∞ and s ∈ R. We use Proposition 21 (v).
Taking M n,s ∞,q := M n,s q,q + n/q, we complete the proof.□ We now start to prove Theorem 1 (iii).In order to handle Theorem 1 (iv) later, we consider a more quantitative version of Theorem 1 (iii).It will contain an estimate of the operator norm of E.
Theorem 23 (Quantitative common extensions).For every 0 < p, q ≤ ∞, s ∈ R, there is a constant M = M n,s p,q , such that the following holds: Let m 1 , m 2 ∈ Z ≥0 be two non-negative integers such that n max(0, ) < ∞ for some δ > 0.
Then E = E a,b given by (2) has the boundedness E : A s pq (R n + ) → A s pq (R n ) for A ∈ {B, F }. Moreover for the operator norms of E we have Here C n,p,q,s,δ, M does not depend on the choices of (a j ) j and (b j ) j .
Remark 24.By Proposition 14, the sequences (a j ) j and (b j ) j can be chosen independently of p, q, n, s, m 1 , m 2 .Note that δ in (25) is allowed to depend on m 1 , m 2 , M ns pq (cf. the δ k in (3) and Remark 2).Therefore, letting m 1 , m 2 → +∞ we obtain all boundedness properties of E in Theorem 1 (iii).Additionally, Theorem 23 also gives the following: • If one take m 1 = 0 and let m 2 → +∞, we see that Seeley's extension operator (see Remark 12) has boundedness E : A s pq (R n + ) → A s pq (R n ) for A ∈ {B, F }, p, q ∈ (0, ∞] and s > n max(0, 1 p − 1, 1 q − 1).It is possible that the range for s is not optimal.
• If one take (a j ) j to be finitely nonzero (depending on the upper bound of m 1 , m 2 ), then the qualitative result E : A s pq (R n + ) → A s pq (R n ) is well-known for common extension operators.See [Tri20, Remark 2.72].Proof of Theorem 23 (hence Theorem 1 (iii)).Similar to the proof of Proposition 22, using (23) we only need to prove the case A = F .
To make the notation clear, we use E a,b n for the extension operator of R n + given in (2) associated with sequences a = (a j ) ∞ j=−∞ and b = (b j ) ∞ j=−∞ .Recall the zero extension operator S n f (x) := 1 R n + (x)f (x) from (9).Case A: We consider p < ∞ and s suitably large.We sub-divide the discussion into three parts.