Mathematik in den Naturwissenschaften Leipzig On the Trace Space of a Sobolev Space with a Radial Weight

Our concern in this paper lies with trace spaces for weighted Sobolev spaces, when the weight is a power of the distance to a point at the boundary. For a large range of powers we give a full description of the trace space.


Introduction and main result
We consider integer order weighted Sobolev spaces with weights equal to a power of the distance to a point of the boundary and more general weights modelled upon such weights.Our concern in this paper lies with a characterization of trace spaces of these weighted Sobolev spaces.Rather surprisingly there are not too many trace theorems for weighted Sobolev spaces even though traces belong to the fundamental concepts both in the theory and applications, and they have been studied for a very long time.One of the major reasons is that there are no straightforward analogs of methods known from the non-weighted theory, which allow a description of values on manifolds of lower dimensions.Note in passing that the study of traces has

INTRODUCTION AND MAIN RESULT
been closely connected with extension of integer order spaces to spaces with non-integer derivatives, and it was one of the motivation for establishing the general theory of Besov spaces.
The non-weighted theory for the W k p was studied in many papers and it can be found in a number of well known monographs.We shall make no attempt to make an account of that; let us collect just some of the important references.The pioneering works by Aronszajn [4] and Slobodetskii [25] for the Hilbert case and the papers by Gagliardo [11] and Stein [26] should be mentioned.The theory for p = 2 based on abstract methods can be found in Lions and Magenes' monograph [18].The case of general p is treated for instance in monographs by Nečas [20], Adams [2], Kufner, John and Fučík [16], Bergh and Löfström [5], Triebel [28].An immense work has been done by the Soviet school (Lizorkin, Besov, Nikol'skii, Il'in, Uspenskii, and many others).We refer to [28] for a large list of references.
Spaces with weights which equal to a power of the distance to the boundary appeared in many papers; let us refer at least to [14] and [15].A standard approach consists in taking the trace space as a factor space (modulo equality on the boundary).Nikol'skii in his monograph [21] (especially its second edition) established a trace theorem for these Sobolev weighted spaces: For a suitable range of parameters and under assumption on the regularity on ∂Ω, the boundary of Ω, he identified the trace space with an unweighted Besov space with a modified smoothness parameter-the effect of the weight on the domain (Hardy's inequality behind the scenes).
Let us recall the very basic setting of the trace problem.For simplicity we shall consider spaces on R n and traces on R n−1 , that is, on ∂R n + , the boundary of R n + .By virtue of extension theorems the Sobolev space on R n + equals (up to equivalence of norms) to the restriction of the corresponding Sobolev space on R n , equipped with the factornorm (modulo equality on R n + ).This can be transferred to spaces on a smooth domain Ω and its boundary ∂Ω in a standard way-using resolution of unity and local coordinates.Let s > 0 be a non-integer and denote by [s] the integer part of s.
(Note that this is a special case of a general Besov space B s p,q (R n ) for p = q.)Here and in the following we shall use the notation f |X instead of f X whenever it might improve legibility of the text.Recall that . One can prove that there is a bounded linear operator tr : such that tr f (y ′ ) = f (y ′ ) for every y ′ ∈ ∂R n + and every f ∈ C(R n + ).This gives a natural meaning to values of a general f ∈ W 1 p (R n + ) on ∂R n + .Moreover, it is well known that there exists a bounded linear operator ext : Theorems of this kind are now wellknown in a general setting of Besov and Lizorkin-Triebel spaces; we refer to [28].Now let w be a weight function (shortly a weight) in R n , that is, w ∈ L 1,loc and w > 0 a.e. in R n .Let W k p (w) = W k p (R n , w) be the weighted Sobolev space, i.e. the space of all functions f , which together with their generalized derivatives D α f up to the order k belong to Only special weights (of type (1 + |x| 2 ) r/2 and their generalizations) and rather sophisticated methods permit to conclude that a function f belongs to W k p (w) if and only if f w 1/p ∈ W k p (χ Ω ) = W k p , see [24] and [7] for the so called W n classes (one has to assume that the weighted space in question can be extended to the whole of R n , too).In particular, the class W n excludes singularities so that another approach must be used for weights vanishing or blowing-up at the boundary.The situation is now well understood for weights, which equal to a power of the distance to the boundary.(Note also that such weights can be used to characterize zero traces, even in case of a quite general boundary; see e.g.[13].)The trace theorem for such weights was proved by Nikol'skii in [21] with help of real analysis methods.Let us recall Nikol'skii's result.Assume that Ω is a domain with a sufficiently smooth boundary Γ (as to the required smoothness we refer to [21] for details) and let ̺(x) = dist(x, Γ), x ∈ Ω.

INTRODUCTION AND MAIN RESULT
For k ∈ N, 1 ≤ p ≤ ∞, and γ ∈ R, denote by W k p,γ the weighted Sobolev space with the norm and, moreover, there exists a bounded extension operator A by far more general setting-spaces on fractals with this type of weightswas recently considered by Piotrowska in [22].
In the following we shall make use of a Fourier analytic approach to Sobolev spaces and their weighted generalizations, therefore we recall the most important definitions and fix the notation.
Let {ϕ j } ∞ j=0 is the smooth (dyadic) decomposition of unity (see [28], [5]): if q < ∞ and with the finite norm Replacing the L p space in the above definitions by L p (w) we get a formal definition of the weighted Besov space B s p q (R n ; w).Here S(R n ) denotes the space of smooth rapidly decreasing functions f : R n → C and We shall also use the Bessel potential spaces H s p = H s p (R n ) and their weighted clones: For s real and 1 < p < ∞, normed in the obvious way.For Lipschitz domains there exists a universal extension operator working on Sobolev, Besov and Bessel potential spaces (and also on the Lizorkin-Triebel spaces, even for all real s, see Rychkov [23]); this means that many relevant properties of spaces on Lipschitz domains follow from the claims on the whole of R n .That is, one can work either with a formal definition of spaces on domains as factorspaces of spaces on R n modulo equality on the domain in question or with a space on the domain with a usual intrinsic norm (if it is available).This can be partly extended to weighted spaces with the Muckenhoupt weights.Recall that a weight w belongs to the Muckenhoupt class where the supremum is taken over all cubes Q ⊂ R n with edges parallel to the coordinate axes.We shall write simply A p if no misunderstanding can occur.Note in passing that (see e.g.[8]).
We also refer to Chua [6] for an extension theorem for Sobolev spaces on domains and to Rychkov [23] as to the formulae for the norm in Sobolev spaces with A p weights in terms of a weighted Littlewood-Paley decomposition.Specifically, for a positive integer k, 1 < p < ∞, and w ∈ A p , .
This holds even for a bigger class of the so called local A p weights (see [23]) (one requires the condition (1.3)only for small cubes).
In Section 4 we also make use of weighted Sobolev spaces of negative order.It well-known that for 1 < p < ∞ and w ∈ A p , the dual space of L p (w) is given by L p ′ (w ′ ) where Accordingly, for a positive integer k we define For more details about weighted spaces of negative order we refer to [23].
To avoid technicalities we shall not deal with the case of Lipschitz domains and we will concentrate on the basic case of a Sobolev space on R n and a trace on the boundary of a half-space R n + .Our main result is: For the precise definition of the function spaces we refer to Section 2 below.The structure of the paper is as follows: In Section 2 we prove some preliminary results concerning weighted spaces.Then in Section 3 the proof of the Theorem 1.1 for α > 0 is given, based on a suitable estimate of the solution operator to a Dirichlet boundary value problem.Finally, in Section 4 the case α < 0 is proved by a duality argument.

Preliminary results on weighted function spaces
By Garcia-Cuerva and Rubio de Francia [12], Theorem 3.9.the following weighted version of the Hörmander-Mikhlin multiplier theorem holds.
for some constant K > 0. Then T defined by extends to a continuous operator on L q w (R n ) for every q ∈ (1, ∞) and w ∈ A q .In [12] this theorem is stated for an even larger class of multipliers m.The assertion on the operator norm is not mentioned explicitly, but it follows from the same proof.
Moreover, recall that a smooth function p : ) is a Fréchet space e.g. with respect to the semi-norms cf. e.g.[17,27].It is well-known that ξ m ∈ S m 1,0 (R n × R n ), i.e., for every uniformly in ξ ∈ R n .This can e.g.be proved by using the fact that f (a, ξ) := |(a, ξ)| m , (a, ξ) ∈ R n+1 \{0}, is a smooth and homogeneous function of degree m.
For p ∈ S m 1,0 (R n × R n ) the associated pseuododifferential operator is defined by where f = F[f ](ξ) and D x = 1 i ∂ x .Then p(x, D x ) can be extended to a bounded operator on weighted Bessel potential spaces by the following result due to Marschall [19, Theorem 1]: Then p(x, D x ) defined as above extends to a bounded linear operator p(x, D x ) : H s+m q (R n ; w) → H s q (R n ; w).Moreover, there exists N = N (s, m, n, q) ∈ N 0 and C = C(s, m, n, q) > 0 such that Proof: The first part follows directly from [19,Theorem 1].The second part follows easily from the linearity of the mapping ) and the fact that the mapping is bounded, which can be easily checked by observing that all constants in the proof of [19,Theorem 1] only depend on some semi-norm |p| Let ω ∈ A q (R n ), let ϕ j , j ∈ N 0 , be a dyadic decomposition of unity as in the introduction and let s ∈ R, 1 ≤ p, q ≤ ∞.Note that ϕ j , j ∈ N 0 , can be chosen such that ϕ j (ξ) = (ϕ 1 (2 −j+1 ξ) for all j ≥ 1.In particular, this implies uniformly in j ∈ N 0 and for all α ∈ N 0 .
for any weight function ω ∈ A p .
Proof: Use that B s j pq j (R n ; ω) is a retract of ℓ s j q j (N 0 ; L p (R n ; ω)) and apply [5, Theorem 5.6.1].
Proof: The corollary follows directly from Lemma 2.3 and Lemma 2.4.
Proof: The first equality follows from and Lions' trace method for real interpolation, cf.[5, Corollary 3.12.3]or apply [3,Chapter III,Corollary 4.10.2].The second equality follows from the previous corollary and the fact that [10,Lemma 3.1] or [9].
2.2 An Embedding for L q (R n ; |x| α ) Let (M, B, µ) be a measure space and let L p,∞ , 1 ≤ p < ∞, be the corresponding weak L p -space (the Marcinkiewicz space) as e.g.defined in [5,Section 1.3].
Proof: Since the mapping (f, g) → f g is bilinear, it is sufficient to consider the case f L p 1 ,∞ (M,µ) , g L p 2 ,∞ (M,µ) ≤ 1.Let α = p 1 p 2 .Then we either have for every λ > 0, which finishes the proof.
The following theorem is a key result for the proof of Theorem 1.1.
x s n (∇K D , K D ) extends to a bounded operator