A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.


Introduction
To give an overview, we first comment on standard Lizorkin-Triebel spaces (i.e. isotropic, inhomogeneous spaces with unmixed norms). These are throughout denoted by F s p,q .
Since around 1977 the question of Sobolev embeddings of Lizorkin-Triebel spaces has been answered affirmatively, with a unified proof of the following Proposition 1. Let s, s 0 ∈ R and p, p 0 ∈ ]0, ∞[ be given such that s − n p = s 0 − n p 0 , p > p 0 . (1) There is then a continuous embedding F s 0 p 0 ,∞ (R n ) ֒→ F s p,q (R n ) for every q ∈ ]0, ∞].
Specifically this means that there exists a number C > 0, depending on the parameters, such that the following inequality is valid for every u ∈ F s 0 p 0 ,∞ (R n ): (2) To explain the notation, note that a Littlewood-Paley decomposition 1 = ∞ j=0 Φ j (ξ) can be obtained by letting Φ j = Ψ j − Ψ j−1 for j > 0 when Ψ j (ξ) = ψ(2 −j |ξ|) for some ψ ∈ C ∞ (R, R) fulfilling ψ(t) = 1 for t < 1 and ψ(t) = 0 for t > 2. By definition, a tempered distribution u ∈ S ′ (R n ) is in the Lizorkin-Triebel space F s p,q (R n ) when the quantity on the left hand side of (2) is finite; hereby u j = F −1 (Φ jû ). As usual f p denotes the norm of a function f in L p (R n ) with respect to the Lebesgue measure λ on R n .
Jawerth's original proof [Jaw77] of the Sobolev inequalities (2) was given for the homogeneous spacesḞ s p,q , utilising the rewriting of L pnorms via the distribution function (0 < p < ∞) ( However, readers familiar with the subject may recall that for the inhomogeneous F s p,q , it is not possible for every t > 0 to find j ∈ N 0 such that 2 jn/p < t (unlikeḞ s p,q , where j ∈ Z), so Jawerth's proof needs an adaptation to this case. Triebel [Tri78,Tri83] introduced a splitting into the t-intervals ]0, K[ and ]K, ∞[ , where K essentially was the q th root of u |F s p,q . Since then this proof has widely been considered 'best possible'. Perhaps this is because the strategy of Jawerth and Triebel covers all p ∈ ]0, ∞[ in an elegant way (at least forḞ s p,q ), whilst previous attempts did not cover all cases.
In comparison, the Besov spaces B s p,q (R n ) have corresponding embeddings with a well-known one-line proof based on the Nikol ′ skij-Plancherel-Polya inequality that we now recall.
For 0 < p ≤ r ≤ ∞ there exists a c > 0 such that for every f ∈ S ′ (R n ) ∩ L p with supp F f compact, say contained in the closed ball B(0, R) = { ξ ∈ R n | |ξ| ≤ R }, Applying this to each u j (R = 2 j+1 ) in u |B s p,q := ( 2 sjq u j q p ) 1 q , one finds at once that B s 0 p 0 ,q ֒→ B s p,q when (1) holds. Generally F s p,q -spaces are rather more complicated to treat than B s p,qspaces, so it would seem plausible that even the basic Sobolev inequalities in (2) are substantially more technical to achieve for the F s p,q . But also (2) has a short proof based on the Nikol ′ skij-Plancherel-Polya inequality (4). The trick is to handle the infinite sum by means of the following result on the weighted sequence spaces ℓ s q , that (for q < ∞) is normed by Lemma 2. Let real numbers s 1 < s 0 be given, and θ ∈ ]0, 1[ . For 0 < q ≤ ∞ there is c > 0 such that holds for all complex sequences (a j ) j∈N 0 in ℓ s 0 ∞ . This result was to our knowledge first crystallised in the works of Brezis and Mironescu [BM01]; cf. the elementary proof there.
In what follows we shall discuss certain generalizations of Proposition 1, and of (4), to mixed L p -norms and anisotropic smoothness. For the reader's convenience we shall first give the details of the proof in the isotropic (unmixed) case. Hence we prove Proposition 1; as it will become clear in Sections 2 and 3, the method carries over straightforwardly to the more general situation.
Proof of Proposition 1 (based on (4), (6)). For u ∈ F s 0 p 0 ,∞ the claim is obtained by interpolating its F s p,q -norm between those of B s 1 ∞,∞ with s 1 := s 0 − n p 0 = s − n p and the given space Clearly q < ∞ suffices in (2), and for each x ∈ R n , Lemma 2 gives Here u j ∞ ≤ c ′ 2 j n p 0 u j p 0 follows from (4) since u j is in L p 0 with supp F u j ⊂ B(0, 2 j+1 ). The definition of s 1 therefore yields Hence the fact that θp = p 0 gives, by taking L p -norms in (8), That is, (2) is proved.
Remark 3. For simplicity we have taken the sum-exponent q 0 = ∞ on the right hand side of (2), but the other Sobolev inequalities can be recovered via the simple embedding F s 0 p 0 ,q 0 ֒→ F s 0 p 0 ,∞ . The Nikol ′ skij-Plancherel-Polya inequality (4) will be generalised to mixed L p -norms in Section 2 below. In the course of the proof given there, we utilise (4) in the above unmixed form; so to provide a complete overview we insert an argument for this (known) standard version.
Proof of inequality (4). It suffices to establish (4) when in addition to the stated assumptions u ∈ S(R n ), where S(R n ) is the Schwartz space of rapidly decreasing C ∞ -functions. To see this we takeψ ∈ C ∞ (R n ) such thatψ(t) = 1 for |t| ≤ 1,ψ(t) = 0 for |t| ≥ 2, while ψ(0) = 1. Defining u j by then u j has its spectrum in suppû+B(0, 2 j ), and u j ∈ S(R n ) for u(x) is O((1 + |x|) N ) for some N ≥ 0 by the Paley-Wiener-Schwartz theorem (û ∈ E ′ ). There is only something to show for p < r, so p < ∞ and hence So if (4) can be proved for all Schwartz functions with compact spectra, it will follow that (u j ) is a fundamental sequence in L r , hence converging to some g ∈ L r with g(x) = u(x) a.e. Since · p is continuous L p → R, the above would entail that (c > 0 will be independent of u and the size of its spectrum.) In the smooth case we proceed in the spirit of [Tri83, 1.3.2]. When For 1 ≤ p ≤ ∞ = r and p + p * = pp * , the above identity gives hence (4). In case 0 < p < 1 and r = ∞, so the x-independence of the right hand side entails where Ψ 1 p ∞ = R n p ψ ∞ . This proves (4) for r = ∞. If r < ∞ and p < 1, insertion of (17) into u r gives For 1 ≤ p ≤ r < ∞ one can insert (15) instead. This completes the proof of (4), with c equal to a power of a norm of ψ.

The mixed norm case
One advantage of the above proof method is that it extends to spaces with mixed norms. In this set-up L p (R n ) is replaced by L p (R n ) where p = (p 1 , . . . , p n ) for 0 < p j ≤ ∞, j = 1, . . . , n, and It is well known that such spaces frequently enter the analysis of evolution problems for partial differential equations.
The corresponding Lizorkin- The purpose is not to go into the general theory of such spaces here. Instead we want to show that the Sobolev embeddings follow directly from first principles, namely the definition (20) and a mixed norm version of the Nikol ′ skij-Plancherel-Polya inequality.
To prepare for the mixed norm version of the Nikol ′ skij-Plancherel-Polya inequality, we recall the Paley-Wiener-Schwartz Theorem in a precise version, cf. [Hör85, Ch. 7]: when K ⋐ R n , a distribution u ∈ S ′ (R n ) fulfils suppû ⊂ K if and only if u extends to a function u(x+i y) on C n which is entire analytic and fulfils for some N ≥ 0 (the order ofû); here H(y) = sup{ y · ξ | ξ ∈ K } is the supporting function of the compact set K .
We also note that for a rectangle [−R 1 , R 1 ] × · · · × [R n , R n ], Indeed, H(y) can be estimated by the triangle inequality, and equality is attained in one of the corners (±R 1 , . . . , ±R n ). Whenû is supported in this rectangle, and (x ′ , x ′′ ), (y ′ , y ′′ ) ∈ R n ′ × R n ′′ , n = n ′ + n ′′ , it follows when x ′′ , y ′′ are kept as parameters that U(x ′ + i y ′ ) := u((x ′ , x ′′ ) + i(y ′ , y ′′ )) is analytic on C n ′ and Therefore , as one would expect. These facts are convenient for the proof of it also holds that f ∈ L r (R n ) and This result was established by Unin ′ skij [Uni66,Uni70] for exponents p k ≥ 1; Schmeisser and Triebel [ST87, 1.6.2] covered the case n = 2. We give a direct proof where the treatment of 0 < p j < 1 is inspired by a paper of Stöckert [Stö78, Satz 2.1], who proved (26) for subclasses of S(R n ) with exponential decay.
The strategy of the proof is perhaps best described as a succession of embeddings, which means that (26) is realised as a composition of the following n embeddings, that each only affect a single coordinate direction: L p ֒→ L (r 1 ,p 2 ,...,pn) ֒→ L (r 1 ,r 2 ,p 3 ,...,pn) ֒→ . . . ֒→ L r .
For r k = p k the k th map is just the identity map from the k th space to itself. This gives a convenient reduction to the case in which r and p differ in only one component, say r m > p m whilst r j = p j for j = m. Thereby some technicalities are circumvented. (Already (12) is troublesome to carry over to the mixed case, for if some p k = ∞ we cannot obtain convergence in the norm topology.) Proof. 1 • . We prove (26) for an arbitrary u ∈ L p (R n ) by means of a succession of embeddings, as explained above. So it suffices to assume that r m > p m only holds for one value of m, and we can assume that this is m = n. Indeed, seeing u as a function of x ′ = (x 1 , . . . , x m ), it was found in (24) ff. that its spectrum lies in [−R 1 , R 1 ] × · · · × [−R m , R m ]; and by (22) it is in S ′ ∩ L (p 1 ,...,pm) on R m , so once the case m = n is covered, integration with respect to x ′′ yields (26) in general when 1 ≤ m ≤ n.
But this means that and since 1 − s q = s( 1 pn − 1 rn ) the claim follows by taking roots. Using Lemma 2 and ideas from the proof of Proposition 1, the above theorem can now relatively easily be extended to a sequence version of the Nikol ′ skij-Plancherel-Polya inequality.
This version deals with sequences (f j ) in S ′ (R n ) that fulfill the following spectral condition, that we could describe as a geometric rectangle condition, Here A > 0 is a constant, while the fixed numbers R 1 ,. . . ,R n > 1 define the rectangles. The next inequality will give the Sobolev embeddings in Theorem 6 below at once, but the inequality is also interesting for other purposes; it was obtained in [ST87, Prop. 2.4.1] for n = 2.
Theorem 5. When p = r and 0 < p k ≤ r k < ∞ for k = 1, . . . , n, then there is for 0 < q ≤ ∞ a number c > 0 such that for all sequences (f j ) in S ′ (R n ) fulfilling (38).
Proof. Using a succession of embeddings as in the proof of Proposition 4, it is enough to cover the case in which r m = p m only holds for m = n. Furthermore it can be assumed that q < min(r 1 , . . . , r n ). Then since r n < ∞, it is a consequence of Minkowski's inequality that the left hand side of (39) is less than To proceed we note that Lemma 2 also holds if the base of the exponential is shifted from 2 to R n > 1 (since R n is a power of 2). We use this version below with θ := pn rn ∈ ]0, 1[ , and setting s 0 = 1 pn − 1 rn , Applying this to the ℓ 0 q -norm, the above integral is majorised by Using Proposition 4 on the L (r ′ ,∞) -norm, one has for the last factor (43) The first factor (. . . ) θ/pn can be treated similarly, and it follows that (39) holds.
Using Theorem 5 we arrive at the following Sobolev embedding.
Proof. Since the ball B(0, A · 2 j ) is contained in the rectangle in (38) with R k = 2 for all k, it suffices to take a Littlewood-Paley decomposition of an arbitrary u in F s p,∞ (R n ) and then insert f j := 2 tj u j into (39).
In the following section, the above embedding will be extended to a set-up with an additional anisotropy in s.

Spaces with mixed norms and quasi-homogeneous smoothness.
As is well known, it is important, say for parabolic differential equations to consider spaces that are anisotropic in the quasi-homogeneous sense concerning the smoothness index s. This may be combined with the mixed Lebesgue norms in the way we now describe briefly.
Each coordinate x j in R n is given a weight a j ≥ 1, and a = (a 1 , . . . , a n ); i.e. a = (1, . . . , 1) is the case previously treated in this paper. With the anisotropic dilation t a x := (t a 1 x 1 , . . . , t an x n ) for t ≥ 0, and t s a x := (t s ) a x for s ∈ R, and in particular t − a x = (t −1 ) a x, the anisotropic distance function |x| a is introduced as the unique t > 0 such that t − a x ∈ S n−1 (for x = 0; |0| a = 0); i.e.
For the reader's convenience we recall that | · | a is C ∞ on R n \ {0} by the Implicit Function Theorem; the formula |t a x| a = t|x| a is seen directly, and this implies the triangle inequality: |x + y| a ≤ |x| a + |y| a .
The inequality to the right follows from (46), since |(x 1 , 0, . . . )| a = |x 1 | 1/a 1 etc. By taking t equal to the above maximum, the left hand side of (45) would be ≥ 1 (the maximum is attained), so that t ≤ |ξ| a . Along with | · | a , the Littlewood-Paley decomposition is chosen with the modification that Ψ j (ξ) = ψ(2 −j |ξ| a ). Then Φ j is supported in the anisotropic corona 2 j−1 ≤ |ξ| a ≤ 2 j+1 . As usual u j := F −1 (Φ jû ), but here it is understood that it is the anisotropic distance function | · | a that goes into the construction.
Using this, the anisotropic Lizorkin-Triebel space F s, a p,q (R n ), with s ∈ R, p ∈ ]0, ∞[ n and 0 < q ≤ ∞, consists of the u in S ′ (R n ) having finite quasi-norms The corresponding Besov space B s, a p,∞ (R n ), now with 0 < p k ≤ ∞ for all k, is given by the quasi-norm These are more precisely anisotropic spaces of the quasi-homogeneous and mixed norm type. In the case of usual Lebesgue norms, i.e. p 1 = · · · = p n = p, such spaces have been used in general investigations, for example of microlocal properties of non-linear differential equations in [Yam88] and of pointwise multiplication in [Joh95]. For 1 < p k < ∞, k = 1, . . . , n and positive s, anisotropic Triebel-Lizorkin spaces based on mixed L p -norms have been investigated in the second edition of the famous book of Besov, Il'in and Nikol ′ skij [BIN96]. There they are introduced by means of differences. Cf. Remark 10 below.
Let us first note that · |F s, a p,q and · |B s, a p,q are quasi-norms. Indeed, (21) implies that for λ = min(1, q, p 1 , . . . , p n ) their powers · λ are subadditive ; i.e. an analogue of (21) holds for them. So by taking ℓ 1/λ and ℓ (1/λ) * -norms, ( 1 λ ) * = 1 1−λ , the quasi-triangle inequality results: When both q ≥ 1 and all p k ≥ 1, the spaces are therefore Banach spaces (in view of the completeness shown in Proposition 9 below). As usual the simple embeddings F s, a p,q ֒→ F s−ε, a p,q for ε > 0, respectively F s, a p,q ֒→ F s, a p,r for q < r are easy to see. Similarly for B s, a p,q . A direct argument and the generalised Minkowski inequality gives that B s, a p,min(p 1 ,...,pn) ֒→ F s, a p,q ֒→ B s, a p,max(p 1 ,...,pn) .
Next we show that the Sobolev embedding for these spaces is a direct consequence of the previous inequality for distributions fulfilling the geometric rectangle condition (38).
Theorem 7. Let s, t ∈ R, s > t, and p k , For any q ∈ ]0, ∞] there are then continuous embeddings F s, a p,∞ ֒→ F t, a r,q (all r k < ∞) and B s, a p,q ֒→ B t, a r,q between these spaces over R n . Proof. Since supp Φ j is contained in the set where |ξ| a ≤ 2 j+1 , the left inequality in (48) gives for all k = 1, . . . , n that |ξ k | ≤ 2 a k +ja k . Hence supp Φ j is a subset of the rectangle in (38) for R k = 2 a k and A = 2 max(a 1 ,...,an) .
Applying Theorem 5 to f j := 2 tj u j for an arbitrary u in F s, a p,∞ (R n ) therefore yields that u |F t, a r,q ≤ c u |F s, a p,∞ under the assumption in (53). The B-case follows from (26).
Remark 8. In [BIN96,Thm. 29.10] embeddings of F -spaces into L r and into Besov spaces are studied. Our Theorem 7 supplements and improves (at least partly) the assertions stated there.
Finally, as a supplement to the existing literature, we add the next result. It is another application of the mixed-norm version of the Nikol ′ skij-Plancherel-Polya inequality, cf. Proposition 4 above.
Proposition 9. For s ∈ R, 0 < p k < ∞ (1 ≤ k ≤ n) and 0 < q ≤ ∞ the Lizorkin-Triebel space F s, a p,q (R n ) is a quasi-Banach space with continuous embeddings Analogous results hold for the B s, a p,q spaces with p k ∈ ]0, ∞] for all k. Proof. By (52), it suffices to show (54) in the Besov case; and q = ∞ is also enough. That S(R n ) ֒→ B s, a p,∞ (R n ) is a direct consequence of the definition, with a proof similar to the isotropic one in [Tri83,2.3.3].
The continuity of B s, a p,∞ (R n ) ֒→ S ′ (R n ) can also be carried over from [Tri83,2.3.3]. In so doing, the anisotropies can be handled by Proposition 4 (like in the proof of Theorem 7), which for any u ∈ B s, a p,∞ (R n ) gives Then one can setΦ j = Φ j−1 + Φ j + Φ j+1 , so thatΦ j ≡ 1 on supp Φ j ; this gives for u ∈ B s, a p,∞ and ψ ∈ S(R n ) Formally the last infinite series has the structure of a B a 1 p 1 +···+ an pn −s 1,1 norm on ψ. However, the fact that the family (Φ j ) appears instead of the decomposition (Φ j ) is inconsequential, for the proof of the continuity of S ֒→ B s p,q also gives an estimate in this situation. So for some seminorm p N on S it holds that | u, ψ | ≤ c ′′ u |B s, a p,∞ p N (ψ), as desired.
Completeness is shown for the F -case ([Tri83, 2.3.3] is without details on this). Given a fundamental sequence (u l ) in F s, a p,q , the just shown continuity gives that u l − u m belongs eventually to any given neighbourhood of 0 in S ′ (R n ); hence (u l ) converges in S ′ to some u. Sǒ Φ j * u l →Φ j * u in S ′ for l → ∞ (since the index l refers to the sequence, we shall writeΦ j * u l for the frequency modulations by means of the Littlewood-Paley decomposition). HenceΦ j * u l (x) = u l ,Φ j (x − ·) converges pointwisely toΦ j * u(x).
It remains to prove that u ∈ F s, a p,q (R n ) and u l → u in the topology of this space. For q < ∞ the sum over j ∈ N 0 can be seen as an integration w.r.t. the counting measure, and then n + 1 applications of Fatou's lemma give u |F s, a p,q ≤ lim inf u l |F s, a p,q < ∞ (it is convenient that positive measurable functions always have integrals in [0, ∞] and that Tonelli's theorem implies the measurability during the successive integrations). Given ε > 0 and N so that u m − u l |F s, a p,q < ε for m, l > N , similar applications of Fatou's lemma gives This shows the convergence in F s, a p,q . For q = ∞ it is seen directly that sup j |Φ j * u(x)| ≤ lim inf l (sup j |Φ j * u l (x)|), and thence the conclusions follow as above. In the B-cases the ingredients are the same, only with the j-integration carried out last.
Remark 10. To conclude we comment on the background.
Anisotropic Sobolev (or Bessel potential) spaces H s, a p and Besov spaces B s, a p,q (with 1 < p < ∞ respectively 1 ≤ p, q ≤ ∞, partly with s > 0) and in particular embedding relations between them have been investigated e.g. in the monographs of Nikol ′ skij [Nik75] and Besov, Il'in and Nikol ′ skij [BIN79], [BIN96]. Nikol ′ skij and his co-authors departed from a definition based on derivatives and differences. For a characterization of anisotropic Lizorkin-Triebel spaces F s, a p,q by differences we refer to Yamazaki [Yam86b,Thm. 4.1] and Seeger [Se89] (for p = (p, . . . , p), but general a). Since Sobolev spaces H s, a p represent particular cases of the Lizorkin-Triebel scale, there is some overlap between our work and these quoted books. In connection with anisotropic Lizorkin-Triebel spaces we would like to mention also Triebel [Tri77] and Stöckert, Triebel [StT79] for the Fourier-analytic characterization; and concerning the ϕ-transform and characterization by atoms, we refer to Dintelmann [Di96] and Farkas [Fa00].
As others before us, we have preferred to define the anisotropy in terms of the function | · | a . This procedure is well known and goes back at least to the 1960's. A list of some basic properties of | · | a can be found in [Yam86a], together with further historical remarks.
One advantage of using | · | a is that it gives an efficient formalism where the powerful tools from Fourier analysis and distribution theory are easy to invoke. This is clearly illustrated by the rather manageable proofs of e.g. Theorems 5 and 7. In general the Fourier-analytic approach gives streamlined, if not simpler proofs of some basic properties of the spaces.
Finally we note that this paper has been partly motivated by some works of Weidemaier [Wei98,Wei02] and Denk, Hieber and Prüess [DHP], whose results on traces in connection with parabolic problems can be roughly summarised as follows: taking traces by setting x 1 = 0 in the anisotropic F s, a p,q -spaces, with s = 2 and a = (1, . . . , 1, 2), leads to trace spaces in the same scale, and only to Besov spaces if p 1 = · · · = p n .
Concerning the trace problem for the full scale F s, a p,q , i.e. with general s, a, p and q, we have corroborated this conclusion in another joint work [JS]. With the present article our intention was to extract some preparations that should be of independent interest.