A truncated real interpolation method and characterizations of screened Sobolev spaces

In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We introduce a novel method of interpolation between seminormed spaces, called the truncated method, which generates this subfamily and more general scales that we call the screened Besov spaces. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a frequency space characterization in the spirit of the Littlewood-Paley decomposition.

1. Introduction 1.1. Background. The study of partial differential equations on unbounded domains is a catalyst for the development of new analytical tools and spaces of functions. One reason for this is that the classical scale of inhomogeneous Sobolev spaces fails to provide a suitable functional setting for PDEs on these domains. This is evident in the basic exterior Dirichlet problem in R 2 , where u (x) = log |x| solves −∆u = 0 in R 2 \ B (0, e) u = 1 on ∂B (0, e) . (1.1) While we have that u ∈Ẇ 1,p (R 2 \ B (0, e)) ∩Ẇ 2,q (R 2 \ B (0, e)) for all 2 < p ≤ ∞ and 1 < q ≤ ∞, u does not belong to L r (R 2 \ B (0, e)) for any 1 ≤ r ≤ ∞.
One approach for dealing with this problem is to switch to weighted inhomogeneous Sobolev spaces: see, for instance [BF79,BS72,EE73,Kuf85,MMS10,Tha02]. An alternative approach is to directly utilize homogeneous Sobolev spaces, but for this to be fruitful in the study of boundary value problems it is essential to know their associated trace spaces. The trace spaces associated to homogeneous Sobolev spaces on infinite strip-like domains of the form {x ∈ R n+1 : η − (x ′ ) < x n+1 < η + (x ′ )}, for η ± : R n → R Lipschitz with η − < η + , were recently characterized by Leoni and Tice [LT19]. They used this to characterize the solvability of certain quasilinear elliptic boundary value problems in these domains. This trace theory has also been used in recent studies of the Muskat problem by Nguyen and Pausader [NP20], Nguyen [Ngu19], and Flynn and Nguyen [FN20].
A curious feature of this trace theory is that regularity of the trace function is measured with a fractional Sobolev seminorm involving a screening effect. These screened Sobolev seminorms were first studied by Strichartz [Str16], who proved that the fractional regularity associated to traces ofḢ 1 (R × (0, 1)) is characterized by the seminorm Comparing the expression in (1.2) to the seminorm on a homogeneous Sobolev-Slobldeckij spaceḢ 1 2 (R), one sees that the moniker 'screening' is justified since in the former only small difference quotients are allowed, and larger ones are screened away.
The generalization of this result in [LT19] required the introduction more general seminorms. For an open set ∅ = U ⊆ R n , a lower semicontinuous function σ : U → (0, ∞], s ∈ (0, 1) (called the screening function), and 1 ≤ p < ∞, [LT19] defines the screened Sobolev spaceW s,p (σ) (U ) as the collection of locally integrable functions f , defined on U , for which (1.3) Variants of these screened spaces have appeared in recent work on fractional Sobolev-type seminorms [BBM01,BBM02,BN18,Pon04,PS17] and in weak formulations of nonlocal elliptic equations [DGLZ12,FKV15,ZD10]. However, the spaceW s,p (σ) above did not appear in previous literature, so [LT19] established its basic properties: completeness, strict inclusion ofẆ 1,p , and a partial frequency space characterization in the case that p = 2, σ = 1: for f ∈ S (R n ; R) we have the equivalence (1.4) Deeper questions related to density, embeddings, traces, a more robust frequency space characterization, and interpolation were left open in [LT19], and a central goal of this paper is to fill that gap.
The key to unlocking these deeper properties is the characterization of the screened spaces in terms of interpolation theory, specifically the real method of abstract interpolation (see [BL76,Lun18,Pee68]). One expects such a characterization, as this is the case for the Sobolev-Slobodeckij and Besov spaces. We refer to the works [AF03, BL76, BIN78, BIN79, Bur98, DNPV12, Gri11, Leo17, Maz11, Neč12, Pee76,Tri78] and their references for a thorough study of these spaces and their interpolation properties. However, the standard methods of abstract interpolation only generate Banach interpolation spaces intermediate to an appropriately compatible pair of Banach spaces. The screened Sobolev spaces are only seminormed spaces with non-Hausdorff topologies and thus, without appropriate modification, interpolation methods cannot generate these spaces. Literature regarding the theory of interpolation of seminormed spaces appears to be sparse, aside from a technical report of Gustavsson [Gus70], which is difficult to find in print.
1.2. Primary results and discussion. We survey the principle new results regarding the screened spaces obtained in this paper. For brevity's sake, we do not provide fully detailed statements and only record their abbreviated forms. The proper statements can be found later in the indicated theorems.
In order to study the screened Sobolev spaces, we actually introduce a more general scale of screened Besov spaces,B s,p q (R n ), for s ∈ (0, 1) and 1 ≤ p, q ≤ ∞. See Definition 4.1 for the precise definition. Our first result finds sufficient conditions that identify the screened Sobolev spaces within the screened Besov scale.
Theorem 1 (Proved in Corollary 4.5). If σ : R n → R + is a lower semicontinuous screening function bounded above and below by positive constants, s ∈ (0, 1), and 1 ≤ p < ∞, then the screened Sobolev spacẽ W s,p (σ) (R n ) is equivalent to the screened Besov spaceB s,p p (R n ).
With the established connection between the scales of screened Sobolev and screened Besov spaces, we move to structurally and topologically characterize the latter. We find that there is a method of interpolation of seminormed spaces that generates the screened Besov spaces.
Theorem 2 (Proved in Theorem 4.4). There is a method of interpolation of seminormed spaces, called the truncated real-method, that generates the screened Besov spaces as interpolation spaces with respect to a Lebesgue space and a homogeneous Sobolev space.
The interpolation characterization of the screened Besov spaces leads to the following characterization.
Theorem 3 (Proved in Theorem 4.9). For s ∈ (0, 1) and 1 ≤ p, q ≤ ∞, the screened Besov spaceB s,p q (R n ) is equivalent to the sum of the inhomogeneous Besov space B s,p q (R n ) and the homogeneous Sobolev spacė W 1,p (R n ).
The sum characterization from the previous theorem allows us to characterize when the subspace of compactly supported smooth functions is dense.
Theorem 4 (Proved in Corollaries 4.6 and 4.10). For 1 ≤ p, q < ∞ and s ∈ (0, 1) the set C ∞ c (R n ) is dense in the screened Besov spaceB s,p q (R n ) if and only if 1 < p or 2 ≤ n.
We next generalize (1.4) by giving a Littlewood-Paley characterization of the screened spaces.
Theorem 5 (Proved in Corollary 4.14). Let 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ (0, 1). For all functions f ∈B s,p q (R n ) we have that f is a tempered distribution and that the following equivalence holds: [f ]Bs,p q ≍ j∈Z\N 2 j |π j f | 2 1/2 L p + 2 sj π j f L p j∈N ℓ q (N) , (1.5) where {π j } j∈Z are a family of dyadic localization operators, as in Definition 3.2.
Theorem 5 follows from a somewhat more general result that provides a Littlewood-Paley characterization of the interpolant between two Riesz potential spaces,Ḣ r i ,p (R n ) for i ∈ {1, 2} (see Definition 3.11).
Theorem 6 (Proved in Theorem 4.12). Let 1 < p < ∞, 1 ≤ q ≤ ∞, α ∈ (0, 1), σ ∈ R + , and r, s ∈ R with r < s. Set t = (1 − α) r + αs. Then the interpolation space Ḣ r,p (R n ) ,Ḣ s,p (R n ) (σ) α,q is characterized by the Littlewood-Paley seminorm f → j∈Z\N 2 sj |π j f | 2 1/2 L p + 2 tj π j f L p j∈N ℓ q (N;R) . (1.6) The interesting feature of Theorems 5 and 6 is that the Littlewood-Paley characterization changes form between low frequencies and high frequencies. For low frequencies, a Triebel-Lizorkin type of seminorm arises, but for high frequencies it is of Besov type. Note also that the power 2 sj in the low frequencies is inherited from the second factor in the interpolation. This explains why the low frequency Fourier multiplier in (1.4), |ξ| 2 , matches that associated toḢ 1 (R n ).
Our final result concerns embedding and restriction (trace) results for these spaces.
Broadly speaking, our strategy for proving the above results is to take the analytical high road: these results are consequences of our development of a more general abstract theory. We begin the paper, in Section 2, with the development of interpolation methods of seminormed spaces in the abstract. Recall that the K-method for Banach spaces takes a pair of spaces X 0 and X 1 and constructs their intermediate s, q-interpolation space, (X 0 , X 1 ) s,q , as the collection of all elements x belonging to the sum of X 0 and X 1 for which the map R + ∋ t → t −s K (t, x) ∈ R belongs to L q (R + ; µ), were µ is the Haar measure associated to the multiplicative group on R + (see Section 1.3). We find that this K-method is not quite right to produce the screened Spaces as interpolation spaces. However, it is nearly correct. We need only consider a slight generalization to seminormed spaces and allow for a larger family of domains of integration.
For a parameter σ ∈ (0, ∞] we study the 'truncated' interpolation space (Y 0 , Y 1 ) (σ) s,q with respect to seminormed spaces Y 0 and Y 1 . The truncated spaces are characterized as the set of y belonging to the sum of Y 0 and Y 1 for which the map (0, σ) ∋ t → t −s K (t, y) ∈ R belongs to L q ((0, σ) , µ). We find that for σ = ∞ the seminormed interpolation mirrors that of the interpolation theory of Banach spaces, with only a few more subtleties regarding notions of compatibility. On the other hand, when σ < ∞ the truncated method does give interpolation spaces; however it is interestingly asymmetric in the roles of Y 0 and Y 1 . The upshot of studying these methods of abstract seminormed interpolation is that we obtain a general relationship between the methods for σ < ∞ and σ = ∞. More precisely, in Theorem 2.24 we find an abstract sum characterization: for σ < ∞ the truncated interpolation space (Y 0 , Y 1 ) (σ) s,q is equivalent to the sum of (Y 0 , Y 1 ) (∞) s,q and the second factor, Y 1 . Section 3 is a three-fold development of vital analytical tools utilized in the later study of screened Sobolev and screened Besov spaces. The inspiration for this section is the Littlewood-Paley theory in Chapter 5 of [Gra14a], the applications of harmonic analysis to study smoothness in Chapter 6 of [Gra14b], and the interpolation of Sobolev and Besov spaces in Chapter 6 of [BL76]. First, we define the homogeneous Sobolev spaces and the Riesz potential spaces. The latter is a two parameter space,Ḣ s,p (R n ), for s ∈ R and 1 < p < ∞ (see Definition 3.11) where, roughly speaking, a tempered distribution f belongs toḢ s,p (R n ) if [|·| sf ] ∨ defines a function in L p (R n ). Note that this scale is intimately related to the homogeneous Sobolev spaces; however, we work directly with seminorms rather than quotient by polynomials to obtain a normed space. We prove a frequency space characterization ofẆ 1,p (R n ) that says that the former space is essentially equivalent to the Riesz potential spaceḢ 1,p (R n ). We then pair this with a Littlewood-Paley decomposition of the Riesz potential spaces to deduce a Littlewood-Paley characterization of the homogeneous Sobolev spaceẆ 1,p (R n ).
Next, we study the L p -modulus of continuity and its relationship with the K-functional on the sum of L p (R n ) andẆ 1,p (R n ). Having established this, we use the interpolation of seminormed spaces developed in Section 2 to show that the homogeneous Besov spaces (see Definition 3.15),Ḃ s,p q (R n ), are generated via s,q for s ∈ (0, 1) and 1 ≤ q ≤ ∞. As a final development in Section 3, we explore the homogeneous Besov-Lipschitz scale of spaces, ∧Ḃ s,p q (R n ) with parameters s ∈ R, 1 < p < ∞, and 1 ≤ q ≤ ∞ (see Definition 3.19). With the Littlewood-Paley decomposition of the Riesz-Potential spaces, we see that the theory of seminorm interpolation realizes the equivalence: ∧Ḃ α,q for r = (1 − α) s + αt. This interpolation result, supplemented with the interpolation characterization of the homogeneous Besov spaces, reveals a Littlewood-Paley characterization of the latter scale.
Section 4 synthesizes the abstract seminormed space interpolation of Section 2 and the analysis of Section 3 to obtain a deeper understanding of the screened Sobolev spaces. First we generalize the scale of screened Sobolev spaces by defining the screened Besov spaces in Definition 4.1. Having already developed the homogeneous Besov spaces and the connection between the L p modulus of continuity and the K-functional associated to the sum of L p (R n ) andẆ 1,p (R n ), the claims in Theorem 2 above are now immediate. Then the abstract sum characterization of the truncated interpolation method gives, with a little more work, Theorems 3 and 4. We next apply the truncated interpolation method to general pairs of Riesz potential spaces and from this analysis we obtain the claims of Theorem 5. Finally, we use the sum characterization of the screened Besov spaces to quickly read off some results on embeddings and traces.
1.3. Conventions of notation. We record our conventions of notation used throughout this paper. The number sets N, Z, R, and C are the natural numbers, integers, reals, and complex numbers, respectively. We assume that 0 ∈ N and write N + = N \ {0} and R + = (0, ∞). In writing R n we always assume n ∈ N + .
Throughout the paper we denote the field K ∈ {R, C}. The spaces of rapidly decreasing and analytic functions taking values in K are denoted by S (R n ; K) and C ω (R n ; K), respectively. The space of tempered distributions valued in K is denoted S * (R n ; K). The Fourier transform is denoted either as· or F . For 0 < α < 1, the homogeneous Hölder spaceĊ 0,α (R n ; K) is the space of functions f : We let µ denote the standard Haar measure with respect to the multiplicative structure on R + , i.e. µ (E) = R + χ E (t) t −1 dt for Lebesgue measurable sets E ⊆ R + . The n-dimensional Lebesgue measures and s-dimensional Hausdorff measures are denoted L n and H s , respectively. Moreover we choose the normalization of H s so that when s = n we have H n = L n .
Finally, whenever the expression a b appears in a proof of a result, it means that there is a constant C ∈ R + , depending only on the parameters quantified in the statement of the result such that a ≤ Cb. We may sharpen this by occasionally writing the explicit dependence of the constant C as a subscript on , i.e. a s,p,q b. We write a ≍ b to mean a b and b a.

Interpolation of seminormed spaces
In this section we present two distinct methods of generating interpolation spaces intermediate to a given pair of seminormed spaces satisfying certain compatibility conditions. The first method is a seminorm generalization of the well-known 'real method of interpolation' (see for instance, the paper [Pee68], Chapter 3 in [BL76], or Chapter 1 in [Tri78,Lun18]), and as such we refer to this method as the real method of interpolation of seminormed spaces. The real seminorm method has its origins in the work of Gustavsson [Gus70], and here we essentially follow his approach, with a few embellishments.
The second method is, to the best of our knowledge, a new method of generating interpolation spaces. This method, which we call the truncated real method, generates spaces in a seemingly similar way to the real method; however, it is bizarrely asymmetric and generates larger spaces than the non-truncated method. As mentioned in the introduction, the impetus for studying the truncated method of interpolation stems from trace theory of homogeneous Sobolev spaces on certain unbounded domains.
The spaces obtained from the non-truncated method appear crucially at a few points in the truncated theory, so it is important for us to have a careful enumeration of their properties. The technical report [Gus70] is not available in journals or online, so we have recorded a number of its results below and indicated how to obtain the proofs from the arguments used in the second method.
A concise review of relevant topological notions in seminormed spaces is presented in Appendix A. Throughout the following section all generic seminormed spaces are over a fixed common field -either real or complex.
2.1. Topology of compatible couples. We begin by exploring notions of compatibility of seminormed spaces. In this first subsection we consider what happens when two seminormed spaces are simultaneously contained within some larger vector space. We can then consider their sum and intersection and give each of those a seminorm in a natural way.
(1) We say that they are a strongly compatible pair if there exists a topological vector space where the annihilator A is defined in Definition A.1. Note that, due to Proposition A.4, the second condition implies that either Note that every strongly compatible pair is automatically a weakly compatible pair. (3) In the case that X 0 and X 1 are either a strong or weak compatible pair of seminormed spaces we form their sum and their intersection in the usual way: We endow these spaces with the seminorms [·] Σ : Σ (X 0 , X 1 ) → R and [·] ∆ : ∆ (X 0 , X 1 ) → R defined by Observe that we have the continuous embeddings ∆ (X 0 , X 1 ) ֒→ X i ֒→ Σ (X 0 , X 1 ) for each i ∈ {0, 1}.
(1) We say that a seminormed space and (Y 1 , [·] 1 ) are another weakly compatible couple of seminormed spaces, and that (X, [·] X ) and (Y, [·] Y ) are another pair seminormed spaces, with X ⊆ Σ (X 0 , X 1 ) and Y ⊆ Σ (Y 0 , Y 1 ). We say that X and Y are a pair of interpolation spaces if for every linear map The definition of strong compatibility that we give ensures that the intersection of a compatible couple behaves well with respect to completeness. Proposition 2.3. Suppose that (X 0 , [·] 0 ) and (X 1 , [·] 1 ) are a weakly compatible pair of semi-Banach spaces. Then the space (Σ (X 0 , X 1 ) , [·] Σ ) is semi-Banach. If we assume that the pair is strongly compatible, then Proof. If {x k } ∞ k=0 ⊂ ∆ (X 0 , X 1 ) is Cauchy, then the continuous embedding ∆ (X 0 , X 1 ) ֒→ X i for i ∈ {0, 1} paired with completeness of X i implies that there are (a, b) ∈ X 0 × X 1 such that By definition of strongly admissible pair, there is some topological vector space Y such that Let's handle the case that a − b ∈ A (X 0 ) (the other case is similar). Then, b = a + (b − a) ∈ X 0 + A (X 0 ) ⊆ X 0 , and hence b ∈ ∆ (X 0 , X 1 ). Moreover for any k ∈ N we have the bound Thus (2.4) paired with (2.3) shows that x k → b in ∆ (X 0 , X 1 ), showing this space to be complete. To prove completeness of Σ (X 0 , X 1 ), we use Lemma A.7. Suppose that For each k ∈ N, we can find, by the definition of the seminorm on Σ (X 0 , X 1 ), a pair (a k , b k ) ∈ X 0 × X 1 for which As K → ∞, the previous right hand side vanishes, completing the proof.
We can also say something about the annihilators of the sum and intersection.
Proof. The first assertion is trivial, so we only prove the second. If x ∈ Σ (A (X 0 ) , A (X 1 )), then there are Suppose first that the pair of seminormed spaces are strongly admissible. Thus we may find (Y, τ ) a topological vector space such that ∀ i ∈ {0, 1} we have X i ֒→ Y , and A (Y ) = A (X 0 ) ∪ A (X 1 ). Thus, if x ∈ A (Σ (X 0 , X 1 )), then we may find sequences {y m } m∈N ⊂ X 0 and {z m } m∈N ⊂ X 1 such that x = y m + z m for all m ∈ N, and [y m ] 0 + [z m ] 1 ≤ 2 −m for m ∈ N. The continuous embeddings X 0 , X 1 ֒→ Y imply that y m , z m → 0 in Y as m → ∞, and hence x ∈ A (Y ) ⊆ Σ (A (X 0 ) , A (X 1 )).
This result tells us that one of the downsides to having a weak, but not strong, compatible pair is that the annihilator of the sum may grow larger than one desires.
2.2. The K-methods. We define the K-functional in the same was as the normed space theory.
Proof. These three items are immediate from the definition of K .
Using the K-functional, we can define the following families of extended seminorms on the sum.
2.3. Basic properties. We now study basic properties of the K-methods of interpolation. In particular, we will prove that they are intermediate and interpolation spaces in the sense of Definition 2.2, then we study various inclusion, embedding, and completeness properties, and finally we exhibit equivalent discrete seminorms. Along the way, we will see that for fixed s and q, the K-methods' interpolation spaces are only topologically distinct for σ finite and σ infinite.
Next, we show that the K-methods' interpolation spaces preserve completeness.
Proof. We again only prove the case for σ < ∞, as the case for σ = ∞ follows with a similar argument. We verify completeness through the series characterization in Lemma A.7 be a sequence such that ∞ k=0 [x k ] (σ) s,q < ∞. By Proposition 2.8 it follows that ∞ k=0 [x k ] Σ < ∞. Then, Proposition 2.3 implies that there exists x ∈ Σ (X 0 , X 1 ) such that lim K→∞ −x + K k=0 x k Σ = 0. For K, M ∈ N with M > K we may use Proposition 2.6 to bound and since K (t, ·) is an equivalent seminorm on Σ (X 0 , X 1 ) we may send M → ∞ in (2.13) and then multiply by x k ), for all K ∈ N and all t ∈ (0, σ). In the case that q = ∞ we deduce immediately that s,∞ . Since the right-hand-side tends to zero as K → ∞, completeness is established in this case.
We now examine the inclusion relations among the interpolation and truncated interpolation spaces.
Proposition 2.10 (Inclusions and embeddings of K-methods' spaces). Suppose that (X 0 , [·] 0 ) and (X 0 , [·] 1 ) are a weakly compatible pair of seminormed spaces, s ∈ (0, 1), 1 ≤ p, q ≤ ∞, and σ ∈ (0, ∞], ρ ∈ R + . The following hold: (1) We have the continuous embedding (2) If σ < ∞, then we have the equality of spaces, (3) If σ < ∞ and s < t, then we have the continuous embedding (X 0 , X 1 ) s,q , with the following estimate for all x ∈ (X 0 , X 1 ) s,q , with the following estimate for all x ∈ X 1 : (5) If p < q, then we have the continuous embedding (X 0 , X 1 ) (σ) s,p ֒→ (X 0 , X 1 ) (σ) s,q . Proof. For the first four items we only prove the case for 1 ≤ q < ∞, as the case for q = ∞ is proved analogously. The first item follows trivially from the definitions. Given x ∈ (X 0 , X 1 ) (σ) s,q , we estimate via a change of variables and Proposition 2.6: This proves the second item. Next, for x ∈ (X 0 , X 1 ) , and the fourth item is proved. We will only prove the fifth item in the case that σ < ∞, as the case σ = ∞ follows similarly. Let s,q . We first consider q = ∞. For t, τ ∈ (0, σ) we may use Proposition 2.6 to bound The fifth item is proved.
The next theorem shows that the spaces (X 0 , X 1 ) (σ) s,q and (X 0 , X 1 ) s,q are interpolation spaces in the sense of Definition 2.2.
Proof. We only present the proof for σ < ∞, as the proof with σ = ∞ follows similarly. Let Taking the infimium over all such decompositions and multiplying by t −s yields the bound In the case q = ∞ we take the supremum over t ∈ (0, σ), and in the case q < ∞ we take the q th power, integrate, and employ a change of variables; in either case we arrive at the bound: We can also quantify the annihilators of the K-methods' interpolation spaces.
Proof. The first item follows from Propositions 2.6, 2.8, and 2.10. The second item follows from Proposition 2.4.
Finally, we can characterize these spaces with discrete seminorms.

Integration into seminormed spaces.
To study the J-method for interpolation of seminormed spaces, we develop the following variant of the Bochner integral for functions valued in seminormed spaces. Simple functions and their integrals are defined as usual.
Definition 2.14 (Simple functions). Let (Y, M, µ) be a measure space and (X, [·]) a seminormed space. We say that a function s : Y → X is a simple function if: (2) card (s (Y )) is finite, and so there exist n ∈ N + , {a j } n j=1 ⊆ X, and a pairwise disjoint collection The collection of X-valued simple functions over Y is denoted simp (Y ; X). We define the functional With the functional I in hand, we can define the integral as a set-valued map.
Definition 2.15 (Strongly measurable and X-integrable). Let (X, [·]) be a seminormed space and (Y, M, ν) be a measure space. We say that a function f : X → Y is strongly measurable if: (1) f is measurable in the sense that f −1 (U ) ∈ M for all U ⊆ X open.
We say that a strongly measurable function f : (2.24) One of the benefits of defining the integral as a set-valued map is that it allows us to avoid invoking completeness to guarantee {I (s n )} n∈N converges. The trade-off is that it can be the case that Y f dν = ∅. Note, though, that in the event that X is a Banach space, the integral is the singleton containing the usual Bochner integral of f . We next record some simple properties of the mapping Y (·) dν.
Proposition 2.16. Let (Y, M, ν) be a measure space and (X, [·]) be a seminormed space over K ∈ {R, C}. Then, the following hold for all f, g ∈ L 1 (Y, ν; X) and α ∈ K: n → 0 as n → ∞, and hence ℓ 0 −ℓ 1 ∈ A (X), which proves the second item. For the third item consider ℓ ∈ Y f dν and pick the approximation sequence {s n } n∈N as above. Then we may estimate and send n → ∞ to arrive at the bound [ℓ] ≤ Y [f ] dν. This proves the third item. The fourth item is immediate from the second item and continuity of vector operators in a seminorm space.
2.5. J-method and equivalence with K-method. With a notion of seminorm integration in hand, we now turn our attention to the development of the J−method of interpolation for seminormed spaces.
Definition 2.17 (J-functional). Suppose that (X 0 , [·] 0 ) and (X 1 , [·] 1 ) are a pair of weakly compatible seminormed spaces. We define the following functional on their intersection: J : Some simple properties of the J-functional are recorded in the next proposition.
Proof. These are immediate from the definition J .
We now define the J-method of interpolation.
We will now show that the K-method and the J-method give the same interpolation spaces. We need a seminormed space version of the so called 'fundamental lemma of interpolation theory' (for the normed space version, see for instance Lemma 3.3.2 in [BL76]). The case for seminormed spaces is marginally more subtle, since [x] Σ = 0 need not imply that there is a decomposition of Our proof of the lemma is a slight generalization of the ideas in [Gus70]. (2.28) Let ε ∈ R + , 1 < r < ∞, and suppose that ϕ : R + → R + is Lebesgue measurable and satisfies the following: Then there exists a strongly measurable u : R + → ∆ (X 0 , X 1 ) with the following properties: Proof. Given k ∈ Z, by the definition of the K functional we can find a decomposition x = y k + z k with (y k , z k ) ∈ X 0 × X 1 and (2.30) The assumptions (2.28) and (2.29) imply that (2.31) Note that for each k ∈ Z we have that ζ k+1 = y k+1 − y k = −z k+1 + z k ∈ ∆ (X 0 , X 1 ). This leads us to define v : . It is clear that v is strongly measurable, as it is a step function with countable image. For k ∈ N we have that v restricted to r −k , r k is a simple function. Hence (2.32) Notice that (2.32) paired with item (2) of Proposition 2.16 reveal that (2.34) Thus u = v/ log (r) satisfies items (1) and (2). To prove (3), we take t ∈ R + with r k−1 ≤ t < r k , for some k ∈ Z and estimate We now give important sufficient conditions for the satisfaction of the hypotheses of Lemma 2.20.
Proof. We appeal to item (5) of Proposition 2.10; whence: With the lemmas in hand, we are now ready to prove the equivalence theorem for the non-truncated interpolation spaces.
We first show that u ∈ D (x), which amounts to proving that u ∈ L 1 (R + , µ; X) and x ∈ R + u dµ. Since Lemma (2.20) tells us that we may take u a step function, there is a natural choice of simple functions to attempt to satisfy Definition 2.15. For k ∈ Z and t ∈ [2 k−1 , 2 k ), there is some ξ k ∈ ∆ (X 0 , X 1 ) such that u (t) = ξ k . Then for each n ∈ N + we define s n = n k=−n ξ k χ [2 k−1 ,2 k ) ∈ simp (R + ; Σ (X 0 , X 1 )). It is clear that s n → u everywhere as n → ∞. Also, according to Proposition 2.18 and item (3) from Lemma 2.20, we may bound (2.36) To show that the right hand side of (2.36) tends to zero as n → ∞, it suffices to show that both integrands are integrable over R + . This is clear for the latter term involving ϕ. To handle the former, we use Hölder's inequality to bound where p = q (q − 1) −1 . Hence, u is Σ (X 0 , X 1 )-integrable, with x ∈ R + u dµ; indeed, item (2) in Lemma 2.20 implies the limit: [x − I (s n )] Σ → 0 as n → ∞ holds. Finally, we check that x ∈ (X 0 , X 1 ) s,q,J . If 1 ≤ q < ∞, then again we use Lemma 2.20 to see that The integral on the right hand side is finite. As ε ∈ R + was chosen arbitrarily, we can let ε → 0 + and see that (X 0 , X 1 ) s,q ֒→ (X 0 , X 1 ) s,q,J . The case for q = ∞ is proved in the same way.
On the other hand, let x ∈ (X 0 , X 1 ) s,q,J . Then, there is some u ∈ D (x) by hypothesis. Then for t ∈ R + we use Proposition 2.18 to bound (2.39) In the case that 1 ≤ q < ∞, we use (2.39) and Hardy's inequalities (see Lemma B.2) to estimate Taking the infimum over all u ∈ D (x) gives the case for 1 ≤ q < ∞. For the case q = ∞, we consider some t ∈ R + and use (2.39): Taking the supremum over t ∈ R + , and then the infimum over all Finally, we show that we have a discrete characterization of the seminorm on (X 0 , X 1 ) s,q,J .
Proposition 2.23 (Discrete characterization of the J-method). Let (X 0 , [·] 0 ) and (X 1 , [·] 1 ) be a pair of weakly compatible seminormed spaces. For x ∈ Σ (X 0 , X 1 ) we define the discrete decomposition set of x as Then for each r ∈ (1, ∞) there is a constant c ∈ R + such that for all x ∈ Σ (X 0 , X 1 ), Proof. Let x ∈ (X 0 , X 1 ) s,q,J and ε ∈ R + . Again, we take ϕ : R + → R + to be defined as in the proof of Theorem 2.22. By Theorem 2.22 we have that x ∈ (X 0 , X 1 ) s,q with [x] s,q ≤ c [x] s,qJ for some c depending only on s and q. Thus, we can apply Lemma 2.20 and then Theorem 2.22 again to find a step function u : where C is a constant depending on ϕ, s, and q. Moreover, there is a sequence Finally, item (2) in Lemma 2.20 implies that lim K→∞ [−x+ log (r) K k=−K ξ k ] Σ = 0. Thus {log (r) ξ k } k∈Z ∈ D (x). Proposition 2.18 provides a constantc, depending on s and q, such that (2.46) Together, (2.44), (2.45), and (2.46) imply for all ε ∈ R + . Hence, the second inequality of (2.43) is proved. Now suppose that x ∈ Σ (X 0 , X 1 ) is such that ∅ =D (x) and choose {ξ k } k∈Z ∈D (x). We define u : R + → ∆ (X 0 , X 1 ) via u (t) = log (r) −1 k∈Z ξ k χ [r k−1 ,r k ) (t). For n ∈ N + we take s n = u| (r −n ,r n ) . Then s n ∈ simp (R + ; Σ (X 0 , X 1 )) and s n → u everywhere as n → ∞, which shows u to be strongly measurable. The condition k∈Z [ξ k ] Σ < ∞ easily implies that R + [s n − u] Σ dµ → 0 as n → ∞. Then u is Σ (X 0 , X 1 )-integrable. Moreover, x ∈ R + u dµ, as the condition lim K→∞ [−x + K k=−K ξ k ] Σ = 0 implies that lim n→∞ [−x + I (s n )] = 0. Hence, D (x) = ∅. Using Proposition 2.18 once more, we see that for some constant c depending on r, s, q. This implies the first inequality in (2.43), and the proof is complete.
2.6. Sum characterization of the truncated K-method. The following theorem shows that the truncated spaces are the sum of the second factor and the K-method with σ = ∞ space between the two factors.
Theorem 2.24 (Sum characterization for truncated method). Let (X 0 , [·] 0 ) and (X 1 , [·] 1 ) be a pair of weakly compatible seminormed spaces. Suppose that s ∈ (0, 1), σ ∈ R + , and 1 ≤ q ≤ ∞. Then we have the following equality of spaces and equivalence of seminorms: Proof. We begin by defining the functionalK : Note that for all t ∈ R + , the mapK (t, ·) is an equivalent seminorm on Σ (X 0 , X 1 ) s,q , X 1 . First suppose that x ∈ Σ (X 0 , X 1 ) s,q , X 1 ⊆ Σ (X 0 , X 1 ), where the latter inclusion follows from Proposition 2.8. Pick y ∈ (X 0 , X 1 ) s,q and z ∈ X 1 such that x = y + z. By Proposition 2.10 we have the bound (2.50) Thus, upon taking the infimum over all such decompositions of x, we arrive at the estimate (2.51) In particular, this implies that Σ (X 0 , X 1 ) s,q , X 1 ⊆ (X 0 , X 1 ) (σ) s,q . On the other hand, suppose x ∈ (X 0 , X 1 ) (σ) s,q . Let ε ∈ R + . For each k ∈ N we may then find (a k , b k ) ∈ X 0 × X 1 with the following properties for each k ∈ N: (2.52) Set η ∈ X 1 via η = b 0 . Proposition 2.8 gives the bound for some c depending on s, q, and σ. Note that (2.52) implies that for all k ∈ N we have ξ k = a k − a k+1 = b k+1 − b k ∈ ∆ (X 0 , X 1 ). Hence, for m ∈ N, we may use telescoping sums to compute η + m k=0 ξ k = η + a 0 − a m+1 = x − a m+1 . Proposition 2.13 provides a constant c ∈ R + such that 2 sk K (σ2 −k , x) k∈N ℓ q (N) ≤ s,q < ∞, which means that lim k→∞ K σ2 −k , x = 0. This and (2.52) imply that [a m+1 ] 0 → 0 as m → ∞, and hence lim For k ∈ Z \ N we set ξ k = 0. Then (2.52) implies that where finiteness follows from the inclusion x ∈ (X 0 , X 1 ) s,q thanks to Proposition 2.13 and Hölder's inequality (see the proof of Theorem 2.22). We deduce from this and (2.54) that {ξ k } k∈Z ∈D (x − η), where the latter set is defined in Proposition 2.23. Next we again use (2.52) and the fact that K (·, x) is increasing to bound for k ∈ N. Since ξ k = 0 for k ∈ Z\N we have J σ2 −k , ξ k = 0 in this case. Combining these and using Proposition 2.23, we arrive at the bound for a constant c ∈ R + depending on s, q and σ, and possibly increasing from line to line. Combining (2.53) and (2.57) then yields the estimateK (1, s,q + 2ε for every ε ∈ R + and some C ∈ R + depending only on s, q, and σ. Letting ε → 0 + , we find that (X 0 , X 1 ) (σ) s,q ֒→ Σ (X 0 , X 1 ) s,q , X 1 . As a corollary, we have the following density result.

Homogeneous Sobolev and homogeneous Besov spaces
We now use the interpolation theory developed in the previous section to realize the homogeneous Besov spaces as intermediate interpolation spaces with respect to members of the scale of homogeneous Sobolev spaces. Along the way we will also develop frequency space characterizations used later in the paper. Many of the results we present in this section are essentially already known in the literature, and we have attempted to omit as many proofs as possible. The proofs we have included are meant to highlight the direct use of seminorms rather techniques employing spaces of distributions modulo polynomials. The precise statements of the results in our notation will also be essential in the following section, where we develop the theory of screened Sobolev and screened Besov spaces. The reader already fluent in analysis of homogeneous function spaces could skip to Section 4.
3.1. Dyadic localization. Here, for convenience of the reader, we recall the essentials of dyadic localization and Littlewood-Paley theory. We refer the reader to Appendix B.2 for the relevant notions of real valued tempered distributions and multipliers.
This dyadic partition of unity leads to the creation of 'projection-like' operators that localize a given distribution at a certain dyadic annulus of frequencies.
The following lemmas record some basic properties of these operators. (1) Suppose that ϕ ∈ S (R n ; K) is such that 0 ∈ suppφ. Then m j=−m π j ϕ → ϕ in S (R n ; K) as m → ∞.
(2) If f ∈ S * (R n ; K), then for each ℓ ∈ Z the sequence m j=0 π j+ℓ f m∈N converges in S * (R n ; C) to g ∈ S * (R n ; K) with the property that for all ϕ ∈ S (R n ; C) with 0 ∈ suppφ where the right-hand-side is well defined since π j f, ϕ = 0 for all but finitely many j ∈ Z, j < ℓ.
(3) Suppose that f, g ∈ S * (R n , K) satisfy π j f = π j g for all j ∈ Z. Then there exists a polynomial Q : R n → K such that f + Q = g.
Proof. The first item follows from standard properties of the Schwartz class, and the second item follows from the first. We now prove the third item. If ϕ ∈ S (R n ; K) is such that 0 ∈ suppφ, by the first item ϕ = j∈Z π j ϕ, with convergence in S (R n ; K). Consequently: Thenĝ −f is a tempered distribution supported at the origin, and hence g − f is a K-valued polynomial by, for instance, Proposition 2.4.1 in [Gra14a].
The next lemma shows that the operators are almost idempotent and almost orthogonal.
Lemma 3.4 (Almost idempotence and almost orthogonality of dyadic localization). The operators {π j } j∈Z from Definition 3.2 are 'almost idempotent': if j ∈ Z and f ∈ S * (R n ; K), then for all m, k ∈ N + we have that They are also 'almost orthogonal': if j, k ∈ Z and |j − k| > 1, then π j π k f = 0.
Proof. These follow immediately from the properties of ψ from Lemma 3.1.
Next we recall the Littlewood-Paley characterizations of L p .
Theorem 3.5 (Littlewood-Paley inequalities in L p ). Let 1 < p < ∞. The following hold: (1) Frequency characterization of L p (R n ; K): For f ∈ S * (R n ; K) write There exists a constant c ∈ R + , depending only on n and p, such that the following hold: (3.5) For f ∈ L p (R n ; C) and j ∈ Z we write π φ j f = (δ 2 j φ) ∨ * f . Let 1 < r < ∞. There is a constant c ∈ R + , depending only on n, p, r, and φ, such that for any sequence {f k } k∈Z ⊂ L p (R n ; C) we have the bound j∈Z k∈Z π φ k f j 2 r/2 1/r L p ≤ c j∈Z |f j | r 1/r L p . 3.2. Homogeneous Sobolev spaces. Our primary goal in this subsection is to develop frequency-space characterizations of the homogeneous Sobolev spaces.

(3.7)
This vector space is endowed with the seminorm [·]Ẇ 1,p :Ẇ 1,p (R n ; K) → R given by [f ]Ẇ 1,p = n j=1 ∂ j f L p . Next we recall some useful facts about homogeneous Sobolev spaces. The first fact is a density result.
The second shows that functions inẆ 1,p define tempered distributions.
Lemma 3.8 (Members of homogeneous Sobolev spaces are tempered). Let 1 ≤ p ≤ ∞. Then the inclusioṅ W 1,p (R n ; K) ⊂ S * (R n ; K) holds. More precisely, if f ∈Ẇ 1,p (R n ; K), then the mapping is well defined, continuous on S (R n ; C), and defines a K-valued distribution.
Proof. If 1 ≤ p < n, then the Gagliardo-Nirenberg-Sobolev embedding (see, for instance, Theorem 12.9 in [Leo17]) implies that each member ofẆ 1,p (R n ; K) is the sum of a constant function and an L q -integrable function with q = np n−p , and thus defines a tempered distribution. If p = n, thenẆ 1,n (R n ; K) ֒→ BMO (R n ; K) by, for instance, Theorem 12.3 in [Leo17]. The fact that functions of bounded mean oscillation are tempered is a consequence of item (ii) in Proposition 7.1.5 in [Gra14b]. Finally if n < p ≤ ∞, thenẆ 1,p (R n ; K) ֒→Ċ 0,1−n/p (R n ; K) thanks to Morrey's embedding (see, for instance, Lemma 12.47 in [Leo17]). The latter space is tempered since its members grow at most linearly.
The third result concerns the completeness of this space.
Proof. This follows from the completeness of the Lebesgue spaces paired with Poincaré inequalities on cubes.
We now prove a strong compatibility result.
Lemma 3.10 (Strong compatibility). For 1 ≤ p ≤ ∞, the seminormed spaces L p (R n ; K) andẆ 1,p (R n ; K) are strongly compatible in the sense of definition 2.1.
Proof. This result is an easy consequence of Proposition 2.4. We view L p (R n ; K) andẆ 1,p (R n ; K) as simultaneously belonging to L 1 loc (R n ; K). Let X denote the vector subspace consisting of their sum. Notice that ∆ L p (R n ; K) ,Ẇ 1,p (R n ; K) = W 1,p (R n ; K) is a Banach space. Hence the annihilator of X, A (X), is the sum of the annihilators of each factor -this is exactly the collection of constant functions. Therefore L p (R n ; K) ,Ẇ 1,p (R n ; K) ֒→ X, and A (X) = A (L p (R n ; K)) ∪ A Ẇ 1,p (R n ; K) . This shows that the pair L p (R n ; K) andẆ 1,p (R n ; K) are strongly compatible. Now we explore the precise relation between the scales of homogeneous Sobolev spaces and the Riesz potential spaces. This yields a Fourier characterization of the former.
Definition 3.11 (Riesz potentials and spaces). Let s ∈ R. If f ∈ S * (R n ; K) is such that 0 ∈ suppf , then we define Λ s f ∈ S * (R n ; K) via: Λ s f, ϕ = f , ̺ |·| sφ ∈ C, where ̺ ∈ C ∞ (R n ) is any radial function satisfying ̺ = 1 on suppf , and ̺ = 0 on B (0, κ), κ = min 1, dist(suppf , 0) ∈ R + . It is easy to see that ̺ |·| sφ ∈ S (R n ; C), and it can be shown that this definition is independent of ̺ and preserves the property of being K-valued. For 1 < p < ∞ we define the Riesz potential spacė H s,p (R n ; K) = f ∈ S * (R n ; K) : j k=−j Λ s π k f j∈N ⊂ L p (R n ; K) is convergent . (3.9) We equip this space with the seminorm [·]Ḣ s,p → [0, ∞) defined by (3.10) We first present a Littlewood-Paley characterization ofḢ s,p that gives a more useful seminorm to work with. The proof is similar to that of Theorem 6.2.7 in [Gra14a], but here we work directly with the seminorms and avoid the technique of quotienting by polynomials.
Theorem 3.12 (Littlewood-Paley characterization of the Riesz potential spaces). Let 1 < p < ∞ and s ∈ R. Define the extended seminorm Then there exists c ∈ R + , depending on s, n, p, such that following hold: ( Suppose first that f ∈Ḣ s,p (R n ; K). By hypothesis, there exists f s ∈ L p (R n ; K) such that m j=−m Λ s π j f → f s as m → ∞ in L p (R n ; K). Consider φ ∈ C ∞ c (R n ; R) ⊂ S (R n ; C) defined via φ (ξ) = |ξ| −s ψ (ξ). Observe this function is radial and that for j ∈ Z it holds that Hence by item (2) from Theorem 3.5 we obtain the bound On the other hand, suppose that f ∈ S * (R n ; K) satisfies [f ] ∼ H s,p < ∞. We again use Theorem 3.5 to show that the sequence m j=−m Λ s π j f j∈N is L p (R n ; K)-Cauchy (note that this sequence is indeed K-valued the results in Appendix B.2). Let m, k ∈ N with m < k. Using Lemma 3.4 shows that for j ∈ Z we have (3.14) Hence, by item (1) from Theorem 3.5 we may bound (3.15) By Theorem B.6 applied to ψ and then Lemma B.5, there is a constant c ∈ R + , depending only on ψ, n, and p, such that for all j ∈ Z it holds that δ 2 j ψ Mp = c. Therefore the bound (3.15) implies that Now let ν ∈ C ∞ c (R n ; R) ⊂ S (R n ; C) be the radial function defined via ν (ξ) = |ξ| s ψ (ξ). Arguing as in (3.12) shows that Λ s π j f = 2 sj π ν j f ; moreover, Lemma 3.4 implies that π ν j = π ν j 1 ℓ=−1 π j+ℓ for each j ∈ Z. Hence, (3.16) paired with item (2) of Theorem 3.5 yield the bounds we can now show that as m → ∞ the final expression in (3.17) tends to zero. Indeed, for a.e. x ∈ R n the sum r∈Z (2 sr |π r f (x)|) 2 is finite. For such x we have that m−1<|r|≤k+1 (2 sr |π r f (x)|) 2 1/2 → 0 as m < k → ∞. (3.18) The limit in L p (R n ; K) follows now from the dominated convergence theorem. We deduce then that the sequence m j=−m Λ s π j f j∈N is Cauchy in L p (R n ; K), and hence f ∈Ḣ s,p (R n ; K). We can now argue exactly as above to deduce that for each m ∈ N it holds that m j=−m Λ s π j f L p s,n,p,ψ Using the Littlewood-Paley characterization of the Riesz potential spaces, we are now able to see that the spacesḢ 1,p andẆ 1,p essentially coincide. The proof of the following result is technical refinement of Theorem 6.3.1 in [BL76] in the sense that we do not require the Fourier transform of f to vanish near the origin.
Theorem 3.13 (Frequency space characterization ofẆ 1,p ). Let 1 < p < ∞. There exists a constant c ∈ R + , depending only on n, p, and ψ, such that the following hold: (1) If f ∈Ẇ 1,p (R n ; K), then c −1 [f ]Ḣ 1,p ≤ [f ]Ẇ 1,p (2) If f ∈Ḣ 1,p (R n ; K), then there exists a K-valued polynomial Q with the property that f − Q can be identified with anẆ 1,p -function and [f − Q]Ẇ 1,p ≤ c [f ]Ḣ 1,p . Moreover, the coefficients of terms of degree 1 and higher of Q are uniquely determined.
Proof. Suppose first that f ∈Ḣ 1,p (R n ; K). Let us first show that the sequence m j=−m π j f m∈N is Cauchy inẆ 1,p (R n ; K). This sequence of tempered distributions is identified with a sequence of locally integrable functions since each member has compactly supported Fourier transform (see the Paley-Wiener-Schwartz theorem in, for instance, Chapter 6, Section 4 of [Yos95]). If k ∈ {1, . . . , n} the mapping R n ∋ ξ → iξ k |ξ| −1 (a scalar multiple of the usual Riesz transform) belongs to M p (R n ; K) by Theorem B.6 and Lemma B.7; therefore, since m j=−m Λ 1 π j f ∈ L p (R n ; K) it then holds that (3.20) The above argument, supplemented with ideas from the latter half of the proof of Theorem 3.12, yields for m, k ∈ N with m < k: (3.21) This estimate paired with Theorem 3.12 shows that the sequence in question is indeed Cauchy in the spacė W 1,p . As this seminormed space is semi-Banach thanks to Lemma 3.9, we are assured of the existence of g ∈Ẇ 1,p (R n ; K) with the property that for each ℓ ∈ {1, . . . , n} (3.22) Theorem B.6 assures us that for all j ∈ Z, π j ∈ L (L p (R n ; K) ; L p (R n ; K)). This fact, paired with Lemma 3.4, shows that π j ∂ ℓ g = π j ∂ ℓ f for all j ∈ Z and for all ℓ ∈ {1, . . . , n} .
(3.23) Hence Lemma 3.3 implies that ∇f = ∇g + P for a K n -valued polynomial P . By Poincaré's lemma there is K−valued polynomial Q such that ∇Q = P . We are free to adjust the constant term of Q so that f = g + Q. IfQ were another polynomial with the property that f −Q ∈Ẇ 1,p (R n ; K). ThenQ − Q would also belong to the spaceẆ 1,p (R n ; K). Hence ∇(Q − Q) is necessarily zero.
3.3. Homogeneous Besov spaces. We now turn our attention to the scale of homogeneous Besov spaces.
(1) For h ∈ R n we define the h-translation operator, τ h , and the h−forward difference operator, ∆ h , as follows. Given f : (2) We define the L p -modulus of continuity as the functional ω p : Definition 3.15 (Homogeneous Besov spaces). Let s ∈ (0, 1) and 1 ≤ p, q ≤ ∞. We define the homogeneous Besov spaceḂ s,p q (R n ; K) = f ∈ L 1 loc (R n ; K) : (3.34) The following equivalent seminorm is occasionally useful. (3.35) Proposition 17.21 in [Leo17] shows that [·] ∼ B s,p q is equivalent to [·]Ḃs,p q . The proof of the following lemma is a slightly modified excerpt from the proof of Theorem 17.24 in [Leo17]. We include it to emphasize the connection between the K-functional on the sum of L p andẆ 1,p and the L p modulus of continuity on L 1 loc . Lemma 3.17 (Relation between K-functional and moduli of continuity). Fix 1 ≤ p ≤ ∞. Let K denote the K-functional, from Definition 2.5, corresponding to the space L p (R n ; K) andẆ 1,p (R n ; K). Then for all (t, u) ∈ R + × L 1 loc (R n ; K) we have the equivalence: Proof. First, we prove the left inequality in (3.36). We may reduce to proving this under the extra assumption that u ∈ Σ L p (R n ; K) ;Ẇ 1,p (R n ; K) since otherwise the right-hand-side is infinite, and there is nothing to prove. Assume this and let t ∈ R + . Suppose that (v, w) ∈ L p (R n ; K) ×Ẇ 1,p (R n ; K) are a decomposition of u, that is:  (B (0, 1)) be a standard mollifier. Then for ε ∈ (0, 1) we can use the fundamental theorem of calculus and Minkowski's integral inequality to bound: Letting ε → 0 + and using Fatou's lemma gives the claim. Now we simply take the supremum over We then take the infimum over all such decompositions of u to see that ω p (t, u) ≤ 2K (t, u). Next, we prove the first inequality in (3.36) in the case that 1 ≤ p < ∞. Suppose that t ∈ R + , and u ∈ L 1 loc (R n ; K) is such that ω p (u, t) < ∞ (if this is infinite, then there is, again, nothing to prove). Let Q 0, n − 1 2 t denote the cube centered at the origin with sides of length n − 1 2 t which are parallel to the coordinate axes. Consider v, w : R n → K given by By construction we have that u = v + w. We estimate v with the Minkowski integral inequality: where in the last inequality we used that Q 0, n − 1 2 t ⊆ B (0, t). Next we estimate w inẆ 1,p (R n ; K) (see also Lemma B.1). Let j ∈ {1, . . . , n}. For z ∈ R n we adopt the following notation: the canonical basis of R n is the set {e 1 , . . . , e n } and z = z ′ j , z j , z j ∈ R, z ′ j ∈ R n−1 , and z ′ j = (z 1 , . . . , z j−1 , z j+1 , . . . , z n ). By a change of coordinates we have that Then for a.e. x ∈ R n we have that Thus, upon differentiating under the integral and applying Fubini's theorem, we find that for a.e. x ∈ R n (3.42) WhereQ 0, n − 1 2 t ⊂ R n−1 is the cube centered at zero with sides parallel to the coordinate axes of length n − 1 2 t. Again Minkowski's integral inequality and the fact that ω p (·, u) is increasing show that (3.43) Synthesizing (3.39) and (3.43), we deduce that With the first bound and the estimate (3.44) in hand, the proof is complete when p < ∞.
On the other hand, if p = ∞ we again decompose u = v + w as in equation (3.38). In this case it is straightforward to see that v L ∞ ≤ ω ∞ (t, u) and for all j ∈ {1, . . . , n}, ∂ j w L ∞ ≤ t −1 ω ∞ (t, u).
From this equivalence we can characterize the homogeneous Besov spaces as seminorm interpolation spaces.
Corollary 3.18 (Interpolation characterization of homogeneous Besov spaces). For all s ∈ (0, 1) and 1 ≤ p, q ≤ ∞ we have the equality of seminormed spaces with equivalence of seminorms: Consequently, the following hold:Ḃ s,p q (R n ; K) is semi-Banach and A Ḃ s,p q (R n ; K) = {constants}; if p, q < ∞, then C ∞ c (R n ; K) is dense inḂ s,p q (R n ; K); and we have the inclusionḂ s,p q (R n ; K) ⊂ S * (R n ; K). Proof. Equation 3.45 is an immediate consequence of Lemma 3.17 and the definition of the seminorm onḂ s,p q . The three consequences follow from the results on interpolation of seminormed spaces from Section 2.
We now explore frequency space characterizations of the homogeneous Besov spaces.
Definition 3.19 (Homogeneous Besov-Lipschitz spaces). Let 1 < p < ∞ and 1 ≤ q ≤ ∞. Then we define the space ∧Ḃ . (3.46) The following theorem shows that the theory of seminormed space interpolation applied to pairs of Riesz potential spaces yields the homogeneous Besov-Lipschitz spaces. We note that the following result appears as Theorem 6.3.1 in [BL76], where the proof is abbreviated.
Theorem 3.20. Let 1 < p < ∞ and s 0 , s 1 ∈ R with s 0 < s 1 . Then for α ∈ (0, 1) and 1 ≤ q ≤ ∞ we have the equality of seminormed spaces with equivalence of seminorms: Ḣ s 0 ,p (R n ; K) ,Ḣ s 1 ,p (R n ; K) α,q = ∧Ḃ s,p q (R n ; K) , where s = (1 − α) s 0 + αs 1 . (3.47) Proof. The pair of seminormed spacesḢ s 0 ,p (R n ; K) andḢ s 1 ,p (R n ; K) are weakly compatible as witnessed by the space of tempered distributions. Suppose that f ∈ Ḣ s 0 ,p (R n ; K) ,Ḣ s 1 ,p (R n ; K) α,q . Let f = f 0 +f 1 be a decomposition with f ℓ ∈Ḣ s ℓ ,p (R n ; K) for ℓ ∈ {0, 1}. We first claim that we have the universal bound π j f ℓ L p n,p,ψ 2 −s ℓ j [f ℓ ]Ḣs ℓ ,p , for j ∈ Z and ℓ ∈ {0, 1} . (3.48) This follows from the Littlewood-Paley characterization of the Riesz potential spaces from Theorem 3.12. Thus, Therefore by Proposition 2.6 and Proposition 2.13, On the other hand, suppose that f ∈ ∧Ḃ s,p q (R n ; K). Then, we will see that for all j ∈ Z it holds that π j f ∈ ∆ Ḣ s 0 ,p (R n ; K) ,Ḣ s 1 ,p (R n ; K) . In fact, we claim that the sequence {π j f } j∈Z belongs to the discrete decomposition set of f ,D (f ). For j ∈ Z and k ∈ {0, 1} we estimate via Lemma 3.4 and Theorem 3.12: π j f Ḣs k ,p n,p,ψ 1 ℓ=−1 2 s k (j+ℓ) π j+ℓ π j f L p n,p,ψ,s k 2 s k j π j f L p . (3.51) In turn, we have that Then (3.52) and Proposition 2.18 imply that the following series converges absolutely: with the series converging in S * (R n ; C) by virtue of Lemma 3.3. Both factors in this decomposition are K-valued, thanks to Lemma B.3. Then for m ∈ N we have the bound We prove that lim m→∞ I m = 0. The argument that II m → 0 as m → ∞ follows similarly. With the aide of Lemma 3.3 and Lemma 3.4, we compute the action of the family {π k } k∈Z on the expression appearing in I m : (3.55) Thus by Theorem 3.12, Theorem B.6, and Lemma B.5 (3.56) To obtain good bounds on the last expression in (3.56) we break to cases on the size of p. Suppose first that 1 < p ≤ 2. In this case the mapping The finiteness follows from the hypothesis that f ∈ ∧Ḃ s,p q (R n ; K). Notice also that the final expression in (3.57) tends to zero as m → ∞. Thus I m → 0 as m → ∞ in the case 1 < p ≤ 2.
On the other hand, in the case that 2 < p < ∞, we bound via Minkowski's integral inequality: (3.58) This bound again implies that I m → 0 as m → ∞.
Thus, we learn that {π j f } j∈Z ∈D (f ). Using the discrete characterization of the J-method in Proposition 2.23 and equation (3.52) we obtain the estimate that completes the proof: [f ] α,q α,q 2 α(s 1 −s 0 )j J (2 −j(s 1 −s 0 ) , π j f ) j∈Z ℓ q (Z) n,p,s 0 ,s 1 ,ψ 2 sj π j f L p j∈Z ℓ q (Z) = [f ] ∧Ḃ s,p q . (3.59) We can now relateḂ s,p q (R n ; K) and ∧Ḃ s,p q (R n ; K). Corollary 3.21. For each s ∈ (0, 1), 1 < p < ∞, 1 ≤ q ≤ ∞, there exists c ∈ R + with the following properties: ( (2) On the other hand if f ∈ ∧Ḃ s,p q (R n ; K), then there exists a K-valued polynomial Q such that f −Q is identifiable with a member ofḂ s,p q (R n ; K) and ∧Ḃ s,p q . Moreover, the coefficients of Q, aside from the constant term, are uniquely determined.
Proof. The first item follows at once from the embeddings L p (R n ; K) ֒→Ḣ 0,p (R n ; K) andẆ 1,p (R n ; K) ֒→ H 1,p (R n ; K) (see Theorems 3.5 and 3.13) and the interpolation characterizations ofḂ s,p q (R n ; K) (Theorem 3.18) and ∧Ḃ s,p q (R n ; K) (Theorem 3.20). For the second item, we let f ∈ ∧Ḃ s,p q (R n ; K). The finiteness of [f ] ∧Ḃ s,p q implies that for each j ∈ Z we have π j f ∈ L p (R n ; K). Using the Littlewood-Paley characterization ofẆ 1,p (R n ; K) and the almost orthogonality of {π j } j∈Z (see Lemma 3.4) we learn that π j f ∈Ẇ 1,p (R n ; K) and [π j f ]Ẇ 1,p n,p,ψ k∈Z 2 k |π k π j f | 2 1/2 L p n,p,ψ 2 j π j f L p . (3.60) Consequently we have the absolute convergence: Proposition 2.3 ensures that Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) is semi-Banach. Hence, there existsf belonging to this sum such that m j=−m π j f −f Σ(L p ,Ẇ 1,p ) → 0 as m → ∞. Moreover, {π j f } j∈Z belongs to the discrete decomposition set off , so we are free to estimate the seminorm off in the interpolation spaceḂ s,p q (R n ; K) = L p (R n ; K) ,Ẇ 1,p (R n ; K) s,q via the discrete characterization of the J-method (see Proposition 2.23) and (3.60): (3.62) It remains to show that f andf differ by a polynomial. As the family of operators {π j } j∈Z are continuous on both L p (R n ; K) andẆ 1,p (R n ; K) (and hence their sum) we find that π j m k=−m π j f − π jf Σ(L p ,Ẇ 1,p ) → 0 as m → ∞ for each j ∈ Z. But if m > |j| then Lemma 3.4 tells us that π j m k=−m π k f = π j f . Therefore π j f −f ∈ A Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) = {constant functions} for all j ∈ Z. (3.63) Since supp F π j f −f ⊂ R n \ B (0, 2 j−2 ) and constant functions are supported at the origin on the Fourier side, we must have π j f = π jf for all j ∈ Z. Thus Lemma 3.3 provides us a K-valued polynomial Q such that f − Q =f ∈Ḃ s,p q (R n ; K). If P, Q are two polynomials such that f − Q, f − P ∈Ḃ s,p q (R n ; K), then P − Q ∈Ḃ s,p q (R n ; K) is a polynomial which implies that P − Q ∈Ḃ s,p q (R n ; K) is a constant.

Screened Sobolev and screened Besov spaces
Recall from the introduction that [LT19] defines the screened Sobolev spaceW s,p (σ) (U ) as the collection of locally integrable functions f : U → R for which (1.3) holds. In this section we introduce a generalized scale of spaces, the screened Besov spaces, and use our previous seminorm interpolation theory to study their properties.
4.1. Motivation, definitions, and basic properties. In an effort to better understand the screened Sobolev spaces, we introduce the following scale of screened Besov spaces with constant screening function.
We begin by providing an equivalent seminorm that utilizes the L p −modulus of continuity.
Proposition 4.2. Let 1 ≤ p, q ≤ ∞ and ω p be the L p -modulus of continuity from Definition 3.14.
Then for all f ∈ L 1 loc (R n ; K) and all s ∈ (0, 1), σ ∈ R + we have the equivalence Proof. The result is trivial when q = ∞, so we only prove the case 1 ≤ q < ∞.
We then average over ξ ∈ B (h/2, |h| /2) and use a change of variables to arrive at the bounds ∆ h f L p ≤ 2 n L n (B (0, 1)) |h| −n B(h/2,|h|/2) In this expression we take the supremum over h ∈ B (0, t), raise the result to the q th power, then multiply by t −1−sq , and finally integrate over (0, σ); this results in the following chain of inequalities, in which we also employ Lemma B.2, Hölder's inequality, and (4.3): Proposition 4.2 leads us to define the following equivalent extended seminorm.

4.2.
Interpolation characterization of screened Besov spaces. Using the equivalent seminorm on the space (σ)B s,p q (R n ; K) from Definition 4.3, we can realize that the s, q, σ-truncated interpolation space between L p (R n ; K) andẆ 1,p (R n ; K) is equal to the screened space (σ)B s,p q (R n ; K). Precisely, we have the following theorem.
Theorem 4.4 (Interpolation characterization of screened spaces). Let 1 ≤ p, q ≤ ∞, s ∈ (0, 1), and σ ∈ R + . Then we have the equality of vector spaces with equivalent seminorms: s,q . (4.8) In fact, for all f ∈ Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) we have the equivalence (4.9) Proof. Let K denote the K-functional on the sum of L p (R n ; K) andẆ 1,p (R n ; K) and note that the strong compatibility of these spaces is shown in Lemma 3.10. It is sufficient to observe that for all t ∈ (0, σ) and all f ∈ Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) we have the equivalence This is a consequence of Lemma 3.17.
This interpolation characterization has numerous important and useful corollaries that we can read off from the abstract theory of seminorm interpolation presented previously. The first is that we can now can build an explicit bridge to well-studied function spaces.
Corollary 4.5 (Sum characterization of screened spaces). Let s ∈ (0, 1) and 1 ≤ p, q ≤ ∞. The following hold: (1) If σ, τ ∈ R + , then we have the equality of vector spaces with equivalence of seminorms: (σ)B s,p q (R n ; K) = Σ Ḃ s,p q (R n ; K) ;Ẇ 1,p (R n ; K) = (τ )B s,p q (R n ; K) . (4.11) (2) If p < ∞ and σ : R n → R + is a screening function with log σ a bounded function, then we have the equality of spaces with equivalence of seminorms: Proof. Given Corollary 4.4, the first item is immediate from Theorem 2.24. For the second item we set σ + = sup σ and σ − = inf σ. By hypothesis, these are both positive. It is a simple matter to observe that: Thus the second item follows from the first.
The next corollary shows us when we have density of smooth and compactly supported functions in the screened spaces. This result is, in fact, sharp, as we will see in the next section.
We also learn that the screened spaces are semi-Banach and their annihilator is nothing more than the space of constant functions.
Proof. This follows from Theorem 4.4, Propositions 2.9 and 2.12, and finally Lemma 3.10.
We note that Corollary 4.7 appears in [LT19] for the scale of screened Sobolev spaces with general screening functions.

4.3.
A concrete decomposition. The previous subsection shows that the screened Besov spaces coincide with the sum of a homogeneous Sobolev and a homogeneous Besov space. In either case the seminorms are, at best, tedious to work with. The purpose of this subsection is to show that we achieve a nearly optimal decomposition into the summands in a simple way. We then use this decomposition to show that compactly supported smooth functions are not dense in the spaceB s,1 q (R; K) for any s ∈ (0, 1), 1 ≤ q ≤ ∞.
Definition 4.8. Let Q = (−1/2, 1/2) n ⊂ R n and define the operators H, L : L 1 loc (R n ; K) → L 1 loc (R n ; K) via (4.14) Notice that the sum of H and L is the identity.
The following theorem utilizes these to arrive at another equivalent seminorm.
There exists a constant c ∈ R + such that for all f ∈ L 1 loc (R n ; K) (4.15) Proof. By the sum characterization of Corollary 4.5 and the embedding B s,p q (R n ; K) ֒→Ḃ s,p q (R n ; K), it is sufficient to prove the second inequality in (4.15). Suppose that f ∈B s,p q (R n ; K). By Lemma B.1, we have that Lf ∈ W 1,1 loc (R n ; K) and for j ∈ {1, . . . , n} and a.e. x ∈ R n it holds that where Q (x) = n k=1 (−1/2 + x k , 1/2 + x k ) and Π j = (I − e j ⊗ e j ). Then when 1 ≤ p < ∞ an application of Minkowski's integral inequality, Proposition 4.2, and Proposition 2.8 show that (4.17) Note that K is the K-functional associated to the sum Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) and that [·] (1) s,q is the seminorm on the truncated interpolation space L p (R n ; K) ,Ẇ 1,p (R n ; K) (1) (4.20) The same estimates work when q = ∞ as well.
As a corollary, we show that the density of compactly supported continuous functions fails in the cases not covered by Corollary 4.6.

4.4.
Frequency space characterizations. Our goal in this subsection is to synthesize the sum characterization of the screened Besov spaces and the frequency characterizations of the Riesz potential and Besov-Lipschitz spaces. We find that the 'low mode' part of the function behaves no worse than a general W 1,p function while the 'high mode' part behaves like a general B s,p q function. To achieve this we will generalize yet again and characterize the frequency behavior of truncated interpolation spaces between certain pairs of Riesz potential space pairs. We then read off the specifics for the screened Besov spaces.
The following theorem characterizes these spaces as interpolation spaces.
t,p q (R n ; K) ×Ḣ s,p (R n ; K) . (4.27) The sum characterization of the truncated interpolation spaces, Theorem 2.24, and the interpolation theorem of Riesz potential spaces, Theorem 3.20, ensure the equality of seminormed spaces with equivalence of seminorms: Ḣ r,p (R n ; K) ;Ḣ s,p (R n ; K) (4.28) Therefore, [·] Σ is an equivalent seminorm on the truncated interpolation space on the left side of (4.28). Now let f ∈ Ḣ r,p (R n ; K),Ḣ s,p (R n ; K) α,q and decompose f = g + h with g ∈ ∧Ḃ t,p q (R n ; K) and h ∈ H s,p (R n ; K). We estimate each factor in the spaceB t,s p,q (R n ; K), beginning with g: [g]Bt,s p,q = j∈Z\N 2 sj |π j g| 2 1/2 ∧Ḃ t,p q . (4.29) To handle the remaining term controlling the low modes, we can break into cases on the size of p. First, suppose that 1 < p ≤ 2. Then the mapping R + ∪ {0} ∋ η → η p/2 ∈ R + ∪ {0} is subadditive, and hence t < s implies that j∈Z\N 2 sj |π j g| 2 1/2 j∈Z\N ℓ u (Z\N) 2 jt π j g L p j∈Z\N ℓ q (Z\N) p < q, 1/p = 1/q + 1/u [g] On the other hand, in the case that 2 < p < ∞, we can apply Minkowski's integral inequality to switch the sum and integral: j∈Z\N ℓ u (Z\N) 2 jt π j g L p j∈Z\N ℓ q (Z\N) 2 < q, 1/2 = 1/q + 1/u [g] ∧Ḃ t,p q . (4.31) Next, let us show the estimates of h.
[h]Bt,s p,q = j∈Z\N 2 sj |π j h| 2 1/2 (4.32) Again, we break into cases based on the size of p to control the high mode term. Let w ∈ R satisfy t < w < s. If 1 < p < 2, then 2 tj π j h L p j∈Z ℓ q (N) ≤ j∈N 2 tj π j h L p ≤ j∈N 2 In the case that 2 ≤ p < ∞ we bound 2 tj π j h L p j∈Z ℓ q (N) ≤ j∈N 2 tj π j h L p ≤ j∈N 2 On the other hand, let f ∈B t,s p,q (R n ; K). Set h = j∈N π j f and g = f − h. We will prove that h ∈Ḃ t,p q (R n ; K) and g ∈Ḣ s,p (R n ; K). Note first that the series defining h is a well-defined tempered distribution, thanks to Lemma 3.3. We now compute the action of the family {π j } j∈Z on h using almost orthogonality, see Lemma 3.4. For j ∈ N + it holds π j h = π j f , π 0 h = π 0 2 f + π 0 π 1 f , π −1 h = π −1 π 0 f , and finally if Z ∋ j < −1 then π j h = 0. This allows us to then estimate (4.37) We apply Theorem B.6 and Lemma B.5 to the first two terms on the right hand side to find a constant c, depending only on n, p, and ψ, such that for ℓ ∈ {−1, 0, 1}, π ℓ π 0 f L p ≤ c π 0 f L p . Plugging this into (4.37) yields the bound (4.38) We now handle the estimates of g. Again we use Lemma 3.3 to see that for j ∈ N + we have π j g = 0, π 0 g = π −1 π 0 f , π −1 g = π −1 2 f + π −2 π −1 f , while for j ∈ Z \ (N ∪ {−1}) we have π j g = π j f . Thus with c as before, we find that [g]Ḣ s,p = j∈Z 2 js |π j g| 2 1/2 Together, estimates (4.38) and (4.39) prove the other embedding:B t,s p,q (R n ; K) ֒→ Ḣ r,p (R n ; K) ,Ḣ s,p (σ) α,q .
The following result should be contrasted with Theorem 4.9 Corollary 4.13 (Fundamental decomposition of generalized screened Besov and Riesz potential spaces). Let r, s ∈ R, 1 < p < ∞, and 1 ≤ q ≤ ∞. Consider the high and low pass filters P + , P − : S * (R n ; K) → S * (R n ; K) defined by P + f = j∈N π j f and P − f = f − P + f = (I − P + )f = I − j∈N π j f . These are well defined thanks to Lemmas 3.3 and B.7. The following hold: (1) If r < s, then for all f ∈B r,s p,q (R n ; K) we have P + f ∈ ∧Ḃ r,p q (R n ; K), P − f ∈Ḣ s,p (R n ; K), and [f ]Br,s p,q ≍ P + f ∧Ḃ r,p q + P − f Ḣs,p . (4.40) (2) If s < r, then for all f ∈H r,s p,q (R n ; K) we have P + f ∈Ḣ s,p (R n ; K) and P − f ∈ ∧Ḃ r,p q (R n ; K), and [f ]Hr,s p,q ≍ P + f Ḣs,p + P − f ∧Ḃ r,p q (4.41) Proof. Again we only prove the first item, as the second item follows from similar arguments. A consequence of Theorem 4.12 is the sum characterization: Σ ∧Ḃ r,p q (R n ; K) ,Ḣ s,p (R n ; K) =B r,s p,q (R n ; K). Therefore the ' ' inequality in (4.40) is handled by the Theorem 4.11. As for the ' ' inequality, we see that this is covered in the latter half of the proof of Theorem 4.12. There we showed that for f ∈B r,s p,q (R n ; K) we can decompose f = P + f + P − f , and the seminorms of the factors in ∧Ḃ r,p q (R n ; K) andḢ s,p (R n ; K), respectively, can be bounded above by a universal constant times [f ]Br,s p,q .
Next we obtain another characterization of the screened Besov spaces.
(2) If f ∈B s,1 p,q (R n ; K), then f is identified with a locally integrable function and there exists a polynomial Q whose coefficients, aside from the constant term, are uniquely determined, with the property that f − Q ∈B s,p q (R n ; K). Moreover, there exists a constant c ∈ R + depending only on s, p, q, and n such that: p,q . Proof. Corollary 4.5 and Theorem 4.12 gave us the identities: L p (R n ; K) ;Ẇ 1,p (R n ; K) (1) s,q =B s,p q (R n ; K) and Ḣ 0,p (R n ; K) ;Ḣ 1,p (R n ; K) (1) s,q =B s,1 p,q (R n ; K) . (4.42) Theorem 3.5 shows that L p (R n ; K) ֒→Ḣ 0,p (R n ; K), and Theorem 3.13 showsẆ 1,p (R n ; K) ֒→Ḣ 1,p (R n ; K). These combine to prove the first item.
On the other hand, if f ∈B s,1 p,q (R n ; K), then Corollary 4.13 tells us that P + f ∈ ∧Ḃ s,p q (R n ; K) and P − f ∈Ḣ 1,p (R n ; K). The former has Fourier transform supported away from the origin and hence (by Corollary 3.21) P + f ∈Ḃ s,p q (R n ; K). The second conclusion of Theorem 3.13 gives us a polynomial Q such that P − f − Q ∈Ẇ 1,p (R n ; K), and hence f − Q ∈B s,p q (R n ; K). Moreover, from Corollary 3.21, Theorem 3.13, and Corollary 4.13 we obtain the universal bound (4.43) Finally, if P , Q are both polynomials such that f − Q, f − P ∈B s,p q (R n ; K), then P − Q ∈B s,p q (R n ; K), which then implies that P − Q is a constant. 4.5. Embeddings. In this subsection we shed some light on the nature of the Sobolev embeddings for the screened Besov spaces. We start in the subcritical case.
Proposition 4.15 (Subcritical embedding). Suppose that n ∈ N \ {0, 1}, 1 ≤ p < n, 1 ≤ u ≤ ∞, and s ∈ (0, 1). Set q = np n−p and r = np n−sp . Then there is a constant c ∈ R + such that for all f ∈B s,p u (R n ; K) there exists a ∈ K such that f − a ∈ Σ (L q (R n ; K) , L r,u (R n ; K)), with the estimate f − a Σ(L q ,L r,u ) ≤ c [f ]Bs,p q . Proof. First, we claim that if f ∈ Σ L p (R n ; K) ;Ẇ 1,p (R n ; K) there is a unique a (f ) ∈ K such that f − a (f ) ∈ Σ (L p (R n ; K) , L q (R n ; K)). Uniqueness is clear since if f − a, f − b ∈ Σ (L p (R n ; K) , L q (R n ; K)), for a, b ∈ K, then a − b ∈ Σ (L p (R n ; K) , L q (R n ; K)), which can only happen if a = b. Existence is a consequence of the Gagliardo-Nirenberg-Sobolev inequality (see, for instance, Theorem 12.9 in [Leo17]): there is a constant c ∈ R + such that for all w ∈Ẇ 1,p (R n ; K) there exists a (w) ∈ K such that w − a (w) ∈ L q (R n ; K) and w − a (w) L q ≤ c [w]Ẇ 1,p . Thus, if u ∈ Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) , then we can take a (u) = a (w) for any decomposition u = v + w, with v ∈ L p (R n ; K) and w ∈Ẇ 1,p (R n ; K).
Next we define ι : Σ L p (R n ; K) ,Ẇ 1,p (R n ; K) → Σ L p (R n ; K) , L q (R n ; K) via ιf = f − a (f ). The dependence of a (f ) on f is linear, so ι is linear. It is also the case that ι is continuous. Indeed, for any f ∈ Σ L p (R n ; K) ;Ẇ 1,p (R n ; K) and any decomposition f = v + w for v ∈ L p (R n ; K) andẆ 1,p (R n ; K), we may estimate ιf Σ(L p ,L q ) ≤ v L p + w − a (w) L q ≤ (1 + c) [f ] Σ(L p ,Ẇ 1,p ) . Similar arguments show that ι continuously maps L p (R n ; K) to itself (and, in fact, equals the identity mapping) and continuously mapsẆ 1,p (R n ; K) to L q (R n ; K). Now we use the fact that the screened spaces are interpolation spaces (see Theorem 2.11). This implies, by the abstract sum characterization in Theorem 2.24, that ι :B s,p u (R n ; K) → (L p (R n ; K) , L q (R n ; K)) (1) s,u = Σ (L q (R n ; K) , L r,u (R n ; K)) (4.45) is a continuous linear mapping. Here we have used the fact thatB s,p q (R n K) = L p (R n ; K) ,Ẇ 1,p (R n ; K) (1) s,u by Corollary 4.5 and Theorem 4.4, and that (L p (R n ; K) , L q (R n ; K)) (1) s,p = Σ (L q (R n ; K) , L r,p (R n ; K)) by Theorem 2.24 and Example 2.27.
Next we consider a first mixed case.
The next mixed case follows.
[αx] = |α| [x]. We say that the pair (X, [·]) is a seminormed space. This space is endowed with the topology τ = [·] −1 (U ) : U ⊆ R is open . Note that this is the smallest topology in which [·] is a continuous mapping.
Seminormed spaces are topological vector spaces and we have the following realization of their annihilators.
Proposition A.4. If (X, [·]) is a seminormed space, then the topology τ from Definition A.3 makes (X, τ ) a topological vector space in the sense of Definition A.1. Moreover, A (X) = {x ∈ X : [x] = 0} is a closed vector subspace.
Note that seminormed spaces are, in particular, semimetric spaces. Hence, we can quotient out be the annihilator of the seminorm and the resulting structure is, at the least, a metric space; but actually, this quotient space results in a normed vector space.
Proposition A.5 (Quotient by annihilator). Let (X, [·]) be a seminormed space. We make the following definitions: (1) For x, y ∈ X we say that x ∼ y if x − y ∈ A (X). This obviously defines an equivalence relation on X. Let X/A (X) denote the resulting set of equivalence classes. This leads us to a natural notion of completeness in seminormed spaces.
Definition A.6 (Semi-Banach spaces). We say that a seminormed space (X, [·]) is semi-Banach or complete if the normed quotient space (X/A (X) , |[·]|) is complete or Banach as a normed vector space.
The following characterization of completeness in seminormed spaces spaces is often useful.
x ∈ X such that x − M k=0 x k → 0 as M → ∞. We now turn our attention to linear mappings between seminormed spaces.
Proposition A.8 (Properties of linear mappings). Let (X 0 , [·] 0 ) and (X 1 , [·] 1 ) be seminormed spaces, and T : X 0 → X 1 be a linear mapping. Then T is continuous if and only if there is a constant c ∈ R + such that for all x ∈ X 0 we have the bound [T x] 1 ≤ c [x] 0 . If either condition holds, then T A (X 0 ) ⊆ A (X 1 ).
A particularly common linear mapping between seminormed spaces is an embedding. Definition A.9 (Continuous embeddings and equivalent seminormed spaces). Let (X 0 , [·] 0 ) and (X 1 , [·] 1 ) be seminormed spaces related via the inclusion X 0 ⊆ X 1 . We say that X 0 is continuously embedded into X 1 if the inclusion mapping ι : X 0 → X 1 is continuous; we write X 0 ֒→ X 1 in this case. If, in addition, we assume that X 1 ⊆ X 0 and the opposite inclusion X 1 → X 0 is continuous, we say that X 0 and X 1 are equivalent as seminormed spaces.
Note that Proposition A.8 implies that if X 0 ֒→ X 1 , then the annihilator of X 1 must be at least as large as the annihilator of X 0 ; in particular, a non-Hausdorff space cannot be continuously embedded into a Hausdorff space.
Lastly, we note that seminormed spaces are equivalent if and only if their seminormed are uniformly comparable.

Appendix B. Miscellaneous facts from analysis
This appendix serves to collect some analysis results used in this paper.