Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators

Abstract:We consider the general framework of ametricmeasure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.


Introduction
Function spaces play a leading role in analysis and its applications for almost a century. For a plenty of scopes the researcher may need to "count" the behavior of a function regarding its continuity, di erentiability, integrability and so and so forth. The functions that enjoy a similar level of the preceding properties, build the corresponding function spaces.
Smoothness spaces have been initially introduced because of the straightforward need to measure the integrability of a function and its derivatives and have been further developed under the leverage of the Fourier transform.
Two of the most general scales, which are involved in a large number of applications, are Triebel-Lizorkin spaces and the companion class of Besov spaces. For the historical path of the development of function spaces, we refer to [30].
On the other hand, the problem under study may be governed by a geometry which is not the Euclidean one. For such a purpose signi cant progress in function spaces on several geometric settings -spheres, balls, manifolds and more-has been obtained the last decades [3, 4, 6, 19-21, 26, 31, 32].
Today the above function spaces have been introduced and explore in a very broad set-up that covers all the aforementioned cases [23].
Embeddings between various spaces of distributions play an important role in Function Theory with applications in PDE's, Approximation Theory and Statistics. Our goal in this article is to extend known results regarding the (Sobolev-type) embeddings of Triebel-Lizorkin and Besov spaces of di erent smoothness on R d to the corresponding Classical and Nonclassical spaces associated to a non-negative self-adjoint operator L that allows as to deal with spaces with di erent geometries, compact and non-compact spaces and spaces with non-trivial weights.
The organization of the paper is as follows: In §2 we present the basic framework that will be needed as well as the de nitions and the main assumptions of the various spaces. §3 contains the main result of the paper Theorem 3.1 which states that for s , s ∈ R, p , p ∈ ( , ∞) and q, r ∈ ( , ∞] such that s > s and p < p , then the Triebel Lizorkin space F s p ,q (L) is continuously embedded in F s p ,r (L) provided s −s where d is the homogeneous dimension of the metric measure space. In §4 this result is extended to nonclassical Triebel-Lizorkin spaces in order to deal with anisotropic geometries. In section §5 we discuss similar results concerning classical and non classical Besov spaces. We note that the nonclassical spaces are of great interest and appear naturally in nonlinear approximation of su ciently smooth disitributions by various decompositions systems such as frames (see [23] and the references therein for details). Finally in §6 we present a typical example that illustrates our results.

Background
This section provides the geometric setting we work on, some illustrative remarks and the necessary machinery for our study.

. The framework
We are ready to present the main assumptions needed for our study.
Assumption I. We assume that (M, ρ, µ) is a metric measure space such that (M, ρ) is locally compact with distance ρ(·, ·), and µ is a positive Radon measure satisfying: Assumption II. The geometry of the space (M, ρ, µ) is related to a self-adjoint non-negative operator L on L (M, dµ), mapping real-valued to real-valued functions, such that the associated semi-group e −tL , t > , consists of integral operators with (heat) kernel p t (x, y) obeying the conditions: (c) Gaussian localization: There exist constants c , c > such that Hölder continuity: There exists a constant α > such that The setting considered in this paper has been put forward in [7,23] and has been studied extensively in recent years [1, 2, 5, 9, 10, 15-18, 22, 24, 25].
Note further that the doubling volume property is very classical in the literature starting by the celebrated work of Coifmann and Weiss [6]. Also the non-collapsing condition holds automatically on the Euclidean space and whenever the measure of the space is nite; µ(M) < ∞, therefore it is restrictive only when µ(M) = ∞.
From (2.1) it follows that there exist constants c ≥ and d > such that (2.6)

De nition 2.1
The minimum value d > of the above quantity d will be called homogeneous or geometrical dimension of (M, ρ, µ).
The setting (M, ρ, µ, L) is very general. A realization of it appears in the general framework of strictly local regular Dirichlet spaces with a complete intrinsic metric, see [7]. Some more examples of settings satisfying our assumptions are Riemannian manifolds with non-negative Ricci curvature, associated with the Laplace-Beltrami operator, Lie groups of polynomial volume growth with sub-laplacians, the Euclidean space R d , with uniformly elliptic divergence form operators, the weighted ball or the sphere with the corresponding Laplacians, the interval with Jacobi operators, the upper hemisphere and the simplex.

. Distributions
For the de nition of Besov and Triebel-Lizorkin spaces we follow Kerkyacharian and Petrushev [23] and introduce the notion of test functions and the corresponding distributions associated with L.
In the present setup, one has to distinguish between the two cases µ(M) < ∞ and µ(M) = ∞. We have the following.
Here x ∈ M is selected arbitrarily and xed from now on. Note that the particular selection of x in the above de nition is not important.
Following a standard approach, the space S = S (L) of distributions on M is de ned as the set of all continuous linear functionals on S and the pairing of f ∈ S and ϕ ∈ S will be denoted by f , ϕ := f (ϕ).
For further details on distributions in the present setting we refer the reader to [23].

. Spectral multipliers
We recall that according to the spectral theorem, since L is a non-negative self-adjoint operator, there exists a unique spectral resolution associated with L, consisted of orthoprojections on L (M) such that Moreover, L maps real-valued to real-valued functions and for any real-valued, measurable and bounded function g on R+ the operator (spectral multiplier) g(L), de ned by is bounded on L (M), self-adjoint, and maps real-valued functions to real-valued functions as well.

. Besov and Triebel-Lizorkin spaces
In order to de ne Besov and Triebel-Lizorkin spaces, we need to consider the following auxiliary functions with the following properties.
The Triebel-Lizorkin space F s p,q = F s p,q (L) is de ned as the set of all f ∈ S such that

14)
with the q -norm replaced by the sup-norm when q = ∞.
It can be veri ed that the Triebel-Lizorkin spaces are independent from the admissible couple (φ , φ), see [23]. Furthermore the preceding de nition generalizes the standard de nition of Triebel-Lizorkin spaces on R d , the sphere S d , the ball B d and other settings.

. A Nikol'skii's type inequality
A fundamental tool for the establishment of embedding theorems is Nikol'skii's inequality which, in its classical form on R d , relates di erent Lp-norms of band-limited distributions. In our setting the role of band-limited functions will be played by the spectral spaces: Let λ ≥ and Y a space of measurable function in M. We denote by Note that for every k ≥ and every f ∈ S it holds that φ k ( √ L)f ∈ Σ k+ . The following Nikol'skii's type inequality was proven in [24]: Proposition 2.3 Let < p ≤ q ≤ ∞ and β ∈ R. Then there exists a constant c > such that for any f ∈ Σ λ , λ ≥ , We note that (2.15) holds even without assuming (2.2). In addition, if we apply (2.7) to (2.15) we easily get

Embeddings between Triebel-Lizorkin spaces
We recall that in the case of Triebel-Lizorkin spaces de ned on R d , it holds that when −∞ < s < s < ∞, < p < p < ∞ and < q, r ≤ ∞, then F s p ,q → F s p ,r , provided that s −s d = p − p . Here we generalize this Sobolev embedding to the broad framework of this article.
Furthermore our results are new for many geometric settings of independent interest, such as the ball, the interval and Riemannian manifolds with nonnegative Ricci curvature and many more.
We are now ready to state and prove the main result of this paper.
Proof. We need to prove that there exists c > such that for every f ∈ F s p ,q , Using the trivial embedding between the sequence spaces q → ∞ , it su ces to assume that q = ∞.
Let now f ∈ F s p ,∞ be such that f F s p ,∞ = . Using that for every p > and g ∈ L p we can write In order to estimate the summation in (3.4) from inequality (2.16) (p < p ) we get that for every k ∈ N , Let now K ∈ N , employing (3.5) we nd a constant C * := C * (p , p , r) > , such that where we also used assumption (3.2). On the other hand to estimate the upper sum of the series since s − s < , we have that for any integer K ≥ − ∞ k=K+ ks φ k ( Using (3.4) we write (3.8) We rst bound I . Applying (3.7) with K = − we derive where we used the facts that p < p , t ≤ c and we changed the variable in the integral.
We next bound I . Let t > ( C * ) /r . We denote by K the unique integer such that It follows that where we used the right hand inequality in (3.10) and (3.2). We replace this expression in I and by changing to the variable τ = t p /p we obtain ≠ . This completes the proof of the theorem.
A class of independent interest that fall under the scale of Triebel-Lizorkin spaces, is the one of (fractional) Sobolev spaces. Let s ∈ R and p ∈ ( , ∞). A distribution f ∈ S belongs to the (generalized) sobolev space H s p (L) when f H s p := (I + L) s/ f p < ∞. It has been proved in [23] that H s p = F s p, , for every s ∈ R and p ∈ ( , ∞). Theorem 3.1 takes the following form: Corollary 3.2 Let s , s ∈ R and p , p ∈ ( , ∞) be such that s > s and p < p . Then

Embeddings between non-classical Triebel-Lizorkin spaces
Nonclassical spaces, allow us to deal with anisotropic geometries, where the size of a unit ball B(x, r) around a point x depends not only on the radious r but also on x. In the aforementioned framework, they were introduced by Kerkyacharian and Petrushev in [23] while studying problems related to nonlinear approximation. Above the q -norm is replaced by the sup-norm if q = ∞.
For being able to prove embedding theorems between non-classical Triebel-Lizorkin spaces, we shall further assume the following geometric behavior of the space. The non-exhausting condition holds trivially on R d and on every compact space; i.e. µ(M) < ∞. On the other hand, the reverse doubling condition (4.2) holds always true when the space M is connected and enjoy the doubling property (2.1) [7]. For example, every double volume Riemannian manifold, satis es (4.2).
From (4.2) it follows that there exist constants c ≥ and d > such that |B(x, λr)| ≥ c λ d |B(x, r)| for x ∈ M, r > and λ > . (4.4) The maximal value d * > of the above quantity d will be referred as the lower homogeneous dimension of M.
thanks to (4.9) and the fact that s > s . Having established inequalities (4.11) and (4.12) the remainder of the proof is identical to the one of Theorem 3.1; we omit the details.
We putting this last inequality in the quasi-norm (5.2) and apply Hölder's inequality for the index q /q > to conclude that which completes the proof of claim (ii).
Let us nally point out that claim (i) demands the non-collapsing condition (2.2), while claim (ii) demands the reverse doubling property together with the non-exhausting condition, but not the non-collapsing condition.

. Some relevant problems
Let us nally point out that more general classes of function spaces, the so-called Besov-type and Triebel-Lizorkin-type spaces [33] have been recently introduced in the generality we work [25]. The embeddings between these spaces consist of an interesting open problem.
The homogeneous counterparts of Besov and Triebel-Lizorkin spaces associated with operators have been developed in [15,16]. The embeddings between homogeneous spaces, could be another research direction.

The unit ball
We close this paper with an example where the assumptions of our study as well as the homogeneous dimensions can be illustrated.