Asymptotics, trace, and density results for weighted Dirichlet spaces defined on the halfline

We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, \omega):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \ {\rm and}\ \| u^{'}\|_{L^p(\mathbf{R}_+, \omega)}<\infty \}$, for given continuous weight $\omega$, in terms of the local $B_p$ conditions due to Kufner and Opic, where $1<p<\infty$. Moreover, we propose applications of our results to: analysis of Hardy inequalities, boundary value problems, complex interpolation theory, and to derivation of new Morrey type inequalities.


Introduction
In this paper we are interested in weighted Dirichlet spaces D 1,p (R + , ω) = {u : R + → R : u is locally absolutely continuous on R + and u ′ L p (R+,ω) < ∞}.
In most situations we assume that the weight ω : R + → R + , is continuous and 1 < p < ∞.
In some cases we also assume that ω satisfies the localized at the endpoint variant of the general B p -condition due to Kufner and Opic from [8]: (1,∞) ω(t) −1/(p−1) dt < ∞.
We address and analyze several problems related to such spaces.
Asymptotic behaviour.One of the topics of our interest is asymptotic behaviour at the endpoints for elements of such spaces.Assume for example that ω ∈ B p (0).Among our results in this direction, we show in Theorem 3.1 that when ω ∈ B p (0), u ∈ D p (R + , ω) and c ∈ R, then the conditions (a),(b),(c) are equivalent, where (a) ∃ tn ց 0 : lim As Ω 0 ω (t) → 0 when t → 0, we clearly have (c) ⇒ (b) ⇒ (a).The nontrivial part is to prove that the converse implications hold.Similar analysis is also provided about behaviour near infinity.
Trace operator.There are several ways to define the trace of Sobolev function, see e.g.[6], Section 6.10.5 for the classical approach.We ask about the limit (b) in (1) and we define T r 0 u := lim t→0 u(t). ( Clearly, one has to ask if such limit is well prescribed in our Dirichlet space setting.It is always so, when we assume that ω ∈ B p (0), see Theorem 3.1, part iii) and it is never so, when ω ∈ B p (0), see Theorem 4.4.
The norm on Dirichlet space.Let us note that the quantity u ′ L p (R+,ω) annihilates all constants, therefore u ′ L p (R+,ω) cannot define the norm on Dirichlet space D 1,p (R + , ω).However, for any a ∈ R + , the quantity defines the norm on D 1,p (R + , ω) and makes it a Banach space.Moreover, all such norms • (a) D 1,p (R+,ω) : a ∈ R + , are equivalent.See Fact 2.2.In the case of ω ∈ B p (0), we can extend the definition of the norm (3) also to a = 0, by putting u(0) := T r 0 (u) in place u(a).Such modification gives also the equivalent norm.In such case the trace operator is continuous as functional on our Dirichlet space equipped with any of the proposed norms (3), including a = 0, see Theorem 3.1, part iv).In case of ω ∈ B p (∞), similar property holds with a = ∞, see Theorem 3.2.

Representation of functions.
Let us focus on the case of ω ∈ B p (0).Because in that case the limit lim t→0 u(t) does exist for any u in Dirichlet space and u ′ is integrable near zero, every element u ∈ D 1,p (R + , ω) can be represented as where on the right hand side above we deal with Hardy transform of u ′ , remembering that u ′ belongs to L p (R + , ω).This allows to deduce several further properties, for example applications to Hardy inequality, see Section 5.1.Similar representations hold in case of ω ∈ B p (∞).In this case we use the conjugate Hardy transform as in (33), see Theorem 3.2.

Questions about densities.
Let us denote by D 1,p 0 (R + , ω) the completion of C ∞ 0 (R + ) in D 1,p (R + , ω) in any norm like (3).It is the natural question to ask about characterization of weights ω, for which D 1,p 0 (R + , ω) = D 1,p (R + , ω).If that spaces are not the same, we can ask if it is possible to characterize completely the space D 1,p 0 (R + , ω) by some analytic conditions expressed in terms of the weight ω.Let us focus again on the case of ω ∈ B p (0).In Theorem 4.1 we have proved that for such ω D 1,p  0 (R + , ω) = B 0 p,ω (0) := {u ∈ D 1,p (R + , ω) : lim This gives the analytic characterization of weights for which D 1,p 0 (R + , ω) is the kernel of trace operator as in (2), in the case of ω ∈ B p (0).Let us emphasize that the trace operator u → T r 0 u in such case is well defined and continuous.
As the consequence of Theorem 3.1 and Theorem 4.4, in the case of ω ∈ B p (0), the trace operator T r 0 (•) at zero is not well defined.We can thus ask question if the space D 1,p 0 (R + , ω) could still be characterized by some analytic conditions, without assuming that ω ∈ B p (0).Such characterization is provided in Theorem 4.3, which gives the precise analytic characterization of D 1,p 0 (R + , ω), expressed in terms of the conditions B p (0) and B p (∞).For example, as follows from Theorems: 4.3 and 4.5, among the other statements, we show that D 1,p 0 (R + , ω) = B 0 p,ω (0) ⇐⇒ ω ∈ B p (0) \ B p (∞).

Applications to: Hardy inequality, Boundary Value Problems (B.V.P.) in ODE's, generalized Morrey Theorem and to complex interpolation theory.
Having more precise information about representation of function from our Dirichlet space, or about the asymptotic behaviour of the functions from given Dirichlet space near zero or infinity, one can deduce more precise variants of Hardy and conjugate Hardy inequality (see Section 5.1), or establish if the given boundary value problem, presented in term of vanishing of function near zero or infinity in the analyzed ODE, is well posed or not.The discussion is provided in Section 5.2.Moreover, in Section 5.3 we focus on certain generalization of Morrey Theorem, which deals with B p -conditions.In Section 5.4 we have also presented some new applications of our results to complex interpolation theory, dealing with weighted Dirichlet type spaces, inspired by questions posed recently in [2].
Novelty and link with literature.To our best knowledge, our results concerning the asymptotic behaviour near the endpoints of the interval of functions in the non trivially weighted Dirichlet spaces, as summarized in Theorems 3.1 and 3.2, are new.In the non-weighted setting they are motivated by Morrey theorem, see Section 5.3.However, similar type conclusions can be found also in the case of power weight in [5], on page 9.
Density results in the general Dirichlet space setting, are rather missing in the literature.In the case of ω ≡ 1, they were obtained first by Sobolev in 1963 ([14]) and now they are well understood.See also e.g.[4], Theorem 4 and references therein, where density results are obtained with respect to the semi norm u ′ L p (R+) as in Fact 2.2, instead of the norm u (a) from (5).Our density anaylisis is based on the localized at endpoints B p conditions as in Definition 2.1, which were not considered before.However, some preliminary ideas for such conditions can be found in [8], see Remark 6.1.
Most of the classical density results deal with Sobolev spaces, not Dirichlet spaces.In case of Sobolev spaces the additional restriction on function u is provided, that is its integrability with some power.We would like to emphasise that our density results mostly deal with the norm (3), they are restricted to Dirichlet (not Sobolev) space, and characterize completely the admitted weights.
As about the tools, for the analysis of asymptotic behaviour we use simple computations based on Taylor's formula in 1-d.To study density, we propose the technique, which in our opinion is new in such setting.We call it the energy -caloric approximation, as it is based on the variational technique.More precisely, we first find the function which on segments [a, b] minimize the energy functional Organization of the paper.After the preliminary results presented in Section 2, we analyze questions about the asymptotic behaviour and trace in Section 3, while density results are presented in Section 4. Main applications: to the derivation of Hardy inequality, to the well posedness of B.V.P., to the derivation of Morrey type theorems, as well as to complex interpolation theory in Dirichlet space setting, are discussed in Section 5. Some additional remarks are presented in Section 6, while in Section 7 we enclose some auxiliary computations and complementary results, for reader's convenience.

Basic notation
In most situations we deal with positive continuous functions ω : R + → R + , referred as positive weights, where, by positive expression, we mean that it is strictly larger than zero.However for our purposes we consider continuous weights only, we will sometimes formulate our statements in the more general setting.
We use standard notation: , for smooth compactly supported functions, weighted L p -spaces and their local variants, Lipschitz functions, the classical Sobolev spaces and their local variants.We will also use the more specific notation for the local variants of L p and Sobolev -type spaces.For 1 ≤ p < ∞, by L p loc ([0, ∞)) we denote all functions f ∈ L p loc (R + ) which are p-integrable near zero (shortly 0 |f | p dτ < ∞), while by L p loc ((0, ∞]) we denote all functions f ∈ L p loc (R + ), which are p-integrable near infinity (shortly ∞ |f | p dτ < ∞).Analogous definitions with obvious modifications will be used to denote the corresponding Sobolev spaces: W 1,p loc ([0, ∞)), W 1,p loc ((0, ∞]), and their generalizations.In most situations we will refer to the Lebesgue integral.However, sometimes we will also refer to the Newtonian interpretation of the integral when writing b a f dx(= F (b) − F (a)) where F ′ = f a.e., in place of (a,b) f dx.By measurable sets, we mean sets that are measurable with respect to the Lebesgue measure.
Let X be some subset of Lebesgue measurable functions defined on R + .By X c we will denote its subset consisiting of functions with compact suport in R + .When X ⊆ Z, where (Z, • Z ) is some Banach space, then by X • Z will denote the completion of X in the norm • Z .The symbol Z 0 will be reserved for (C ∞ 0 (R + ) ∩ Z) • Z .In our estimates, we will sometimes write f ∼ 1 if the function f defined on its respective domain can be estimated from both sides by positive constants, while the notation f 1 will mean that the function is bounded from above.

General and local B p -conditions for weights
We will deal with the following variants of the B p -condition introduced by Kufner and Opic in [8].
Definition 2.1 (Bp-conditions) Let ω : R + → [0, ∞) be a measurable function which is positive almost everywhere, 1 < p < ∞.We say that Note that both conditions B p (0) and B p (∞) imply that ω ∈ B p .Moreover, by Hölder inequality, for any measurable set This implies.
Fact 2.1 Let ω, p be as in Definition 2.1.The following statements hold: In our specific situation, we assume that the weight ω is continuous and positive, which guarantees that ω ∈ B p .The conditions B p (0) and B p (∞), which to our best knowledge were not introduced earlier, are motivated by the general B p condition from [8].More precise information about B p -conditions is provided in Remark 6.1.

Weighted Dirichlet spaces
We start with the definition of weighted Dirichlet space.Definition 2.2 (weighted Dirichlet space) Let ω : R + → [0, ∞) be positive weight, that is ω > 0 a.e., 1 < p < ∞.By D 1,p (R + , ω) we will denote the Dirichlet space consisting with all functions u ∈ W 1,1 loc (R + ) such that Clearly, the expression u * D 1,p (R+,ω) annihilates constant functions, so it defines the semi norm on D 1,p (R + , ω) but not the norm.We are interested in Dirichlet spaces in the case when ω is continuous and positive, and so ω ∈ B p .In that case we show that the homogeneous Dirichlet space D1,p (R + , ω) defined below is complete.The proof is enclosed in the Appendix for reader's convenience.is a Banach space.
In the following fact we analyze the norms in Dirichlet spaces.
Fact 2.3 (the norms on D 1,p (R + , ω) and topology of convergence) Let ω, p be as in Definition 2.2.Then for any a ∈ (0, ∞) the expression is the norm on D 1,p (R + , ω), which makes D 1,p (R + , ω) a Banach space.Moreover, for all a ∈ R + the norms •  Proof.We observe that D 1,p (R + , ω) ⊆ W 1,1 loc (R + ) ⊆ C(R + ) and so the value u(a) is well prescribed.In particular Then for u(t) := where . As a consequence of ( 7) we get (6) with any b ∈ I and u n − u in place of u.
More precise analysis, dealing with the conditions B p (0) and B p (∞), will be provided in our next section.
3 Asymptotics and trace We start with the analysis within the case of ω ∈ B p (0).We obtain the following statement, which deals with trace operator defined at zero T r 0 (•), as in (10) below, and precisely describes the elements of weighted Dirichlet space. where The following statements hold.
i) For any c ∈ R, the set R 0 p,ω (c) is a closed subset in D 1,p (R + , ω), equipped with any norm • (a) as in (5), where a ∈ R + .
ii) For any c ∈ R is the norm on D 1,p (R + , ω), which is equivalent to any norm u , where a ∈ R + .
Proof.We observe that the substitution of u − c in place of u, reduces the proofs of i) and ii) to the case of c = 0. Therefore, for that statements, we only show the case of c = 0.
Let us show that both sets in ( 13) are the same as C 0 p,ω (0).Indeed, let us consider u ∈ A 0 p,ω (0).Then, by (15), we deduce that . iii): Consider any sequence t n ց 0.Then, for any fixed t, by using ( 14), we get the boundedness of {u(t n )} n .By Bolzano-Weierstrass Theorem, we can extract a converging subsequence, which we will also denote by {u(t n )} n .Let c be its limit.By taking the limit as n → ∞ in ( 14) we get which implies u(t) → c as t → 0. Hence, any function u ∈ D 1,p (R + , ω) has the limit as t → 0, thus getting the well-posedness of the trace operator T r 0 (•).
We have also proved that any function u ∈ D 1,p (R + , ω) belongs to B 0 p,ω (c) for some c.This, together with (9), gives the decomposition iv): Due to the existence of the limit of u at zero, we can apply the estimate (7) with b := a > 0 and a := 0, thus getting , where Hence On the other hand, by switching the rule of a and 0 in (18), we obtain This together with (18), yields the equivalence of all norms discussed, and completes the proof of the statement.
As a consequence of the above statement, we have is the following remarks.

Analysis in the case
The aim of this section is to establish an analogous results to Theorem 3.1, to representat the Dirichlet space through the trace operator, but in the case ω ∈ B p (∞).
The result stated below can be obatined by using very similar arguments to those used for the proof of Theorem 3.1.Since we will deal with the B p (∞)condition, we have to provide the analysis when t is sufficiently large.The proof is left to the reader with some general suggestions enclosed in order to treat this different setting: • we first modify the appropriate definitions for the sets from Theorem 3.1; • in the proofs, we substitute the previously used limit conditions: t ց 0, t n ց 0 by: t ր ∞, t n ր ∞, respectively; • in place of (12) we deal with the representation • in place of (16) we deal with which forces c = lim t→∞ u(t).
The following statement holds.
For any c ∈ R, let us consider the following subsets in D 1,p (R + , ω): where The following statements hold.
is the norm on D 1,p (R + , ω), which is equivalent to any norm u (a) , where a ∈ R + .
where a ∈ (0, ∞]).Moreover, the mapping In particular in every abstract class in D1,p (R + , ω) there is the representative vanishing at infinity and D1,p (R + , ω) represents as Remark 3.4 (asymptotic behaviour near infinity) The statement ii) in Theorem 3.2 and (19) yield that if ω : R 4 Characterization of D 1,p 0 (R + , ω) and density results 4.1 The space D 1,p 0 (R + , ω) and first density results We start with the following definition.
Definition 4.1 (the space D 1,p 0 (R + , ω), the case of ω ∈ Bp) When ω ∈ Bp, 1 < p < ∞, by D 1,p 0 (R + , ω) we will denote the subset of all functions u ∈ D 1,p (R + , ω), for which there exists the sequence {φn} ⊆ C ∞ 0 (R + ), which satisfies: By Fact 2.3, D 1,p 0 (R + , ω) is the same as the completion of C ∞ 0 (R + ) in the space D 1,p (R + , ω) equipped with any of the norms , where a ∈ R + can be taken arbitrary.In particular, it is the Banach subspace of , with an arbitrary a ∈ R + .
The following fact is rather obvious to the specialists, but for reader's convenience we submit its proof.Proof.Clearly, Hence, it suffices to show that (D 1,p (R + , ω)) c As on compactly supported sets ω ∼ 1, therefore u ∈ D 1,p (R + ) and u is compactly supported.By standard convolution arguments, the convolutions u ǫ (x) := φ ǫ * u, with the classical mollifier functions Moreover, their supports are subsets of J := [a/2, 3/2b] for the sufficiently small ǫ's.Again, as ω ∼ 1 on J, therefore u ǫ 's converge to u also in D 1,p (R + , ω).This shows that u ∈ D 1,p 0 (R + , ω).
In the preceding sections we will analyze independently the cases: ω ∈ B p (0) and ω ∈ B p (∞).
The statement given below answers on this question.
The proof will be based on the following lemma, whose proof is submitted in the Appendix for reader's convenience.

and consider energy functional
Then the minimum of Eω(•) is achieved at We are now to prove Theorem 4.1.
Proof of Theorem 4.1: "⇐=" (E ω -caloric approximation): We have called this part of the proof "ω-caloric approximation", because the construction of the approximation sequence involves the energy minimizers of (23).
Step 1. Reduction argument.We show that it suffices to prove that any u ∈ D 1,p (R + , ω), such that u ≡ 0 near zero and u ≡ 1 on (k, ∞), for some k > 0, belongs to D 1,p 0 (R + , ω).Indeed, let us take u ∈ R 0 p,ω (0), we notice that functions in the form converge to u in (D 1,p (R + , ω), • (0) ) (see (11)), they are zero near zero and constant near infinity.Clearly, if that constant equals zero, according to Lemma 4.1, we have ũn ∈ D 1,p 0 (R + , ω).In the other case, we are left with the proof that ũn ∈ D 1,p 0 (R + , ω).Obviously, it suffices to consider C ũn instead of ũn , with constant C such that C ũn ≡ 1 near infinity.
Step 2. Proof in the special case.We prove that any u ∈ D 1,p (R + , ω) as in (24) belongs to D 1,p 0 (R + , ω).Let u ∈ D 1,p (R + , ω) satisfy (24).For any n ∈ N and k < t n , let where φ (k,tn,1) is as in Lemma 4.2 and t n ր ∞.Clearly, the u n 's are compactly supported.We will show that which, together with Lemma 4.1, will close the assertion for this part of the statement.We have: As ω ∈ B p (∞) and p > 1, This implies (26).
"=⇒": Suppose that D 1,p 0 (R + , ω) = R 1,p (0).We will show that ω ∈ B p (∞).Clearly, the function , and so there is the sequence D 1,p (R+,ω) ), and in all the equivalent norms.In particular for any t n > 2. Let {t n } n∈N be any sequence such that supp u n ⊆ (0, t n ) and t n ր ∞ as n → ∞.According to Lemma 4.2, the energy E ω (φ (2,tn,ξn) ) cannot be larger than E ω (u n ).Therefore we also have it follows that the above converges to zero if and only if equivalently ω ∈ B p (∞).This completes the proof of the statement. .

Analysis in the case of ω ∈ B p (∞)
Let us assume that ω : R Our aim is to analyze the properties of the space D 1,p 0 (R + , ω), the completion of , where the latter space, due to Theorem 3.2, is a closed subspace in D 1,p (R + , ω), we deduce that The goal of this section is the following characterization theorem.
The proof is an easy modification of the proof of Theorem 4.1, so we only sketch it.We start by stating the following result similar to that of Lemma 4.2.Its proof can be easily obtained either by the internal symmetry argument, or by suitable modification of the proof of Lemma 4.2.We leave it to the reader.
and consider energy functional Then the minimum of Ẽω(•) is achieved at We are in position to sketch the proof of Theorem 4.2.
Step 1. Reduction argument.We show that it suffices prove that any u ∈ D 1,p (R + , ω), such that u ≡ 1 on some (0, c) where c > 0, u ≡ 0 near ∞, (29) belongs to D 1,p 0 (R + , ω).To this aim, we note that functions are proportional to functions as in (29), and they converge to u in Step 2. Proof in the special case.
4.4 Analytic description of D 1,p 0 (R + , ω) in general case Our main statement in this section reads as follows.
Proof.i) and ii): Statements i) and ii) have been already obtained in Theorems 4.1 and 4.2.We are left with the proofs of parts iii) and iv).
iii): Assume that ω ∈ B p (0) ∪ B p (∞).Let u ∈ D 1,p (R + , ω) and let us consider the Lipschitz resolution of the unity on R + : φ 0 , φ 1 , defined by We have to prove that u ∈ D 1,p 0 (R + , ω).As u = φ 0 u + φ 1 u, it suffices to consider the following cases: a) u ≡ 0 near 0 and b) u ≡ 0 near ∞.In case a), suppose that u ≡ 0 on (0, a] for some a > 0. Then functions as in (25) converge to u in (D 1,p (R + , ω), • (a) D 1,p (R+,ω) ), they are zero on (0, a] and constant when t > n.Therefore the proof reduces to the case of u ≡ 0 near zero and u ≡ Const near ∞.Then we repeat all the arguments from the proof of Theorem 4.1, Step 2, in the proof of the implication "⇐=".In case b), the argument at the beginning of the proof of Theorem 4.2 reduces that case to the situation when u ≡ Const near 0 and u ≡ 0 near to infinity.In that case we use precisely the same arguments as in the proof of Theorem 4.2, Step 2 in part "⇐=".
ii): Let us assume that the implication "=⇒" does not hold, that is Let us consider the function It belongs to ∈ D 1,p 0 (R + , ω).By arguments as in the proof of part i), we get a contradiction in both cases: a) and b), which proves ii).The proof of the statement is complete.
Let us proceed by proving the converse implications in Theorem 4.3.We state the following Proof.For each of the statements we only have to prove the implication "=⇒"."=⇒:" i): We will show that condition b) cannot hold.We argue by contradiction.If the condition b) was true then, by Theorem 4.3, it would imply . We arrive at contradiction, therefore only the condition a) can be true.This proves the statement i).
ii): We argue similarly as before.As we have D 0) is closed.Hence, by statement ii) in Theorem 4.4, we get ω ∈ B p (∞).
iii): By contradiction, let us assume that D 1,p 0 (R + , ω) = D 1,p (R + , ω) and ω ∈ B p (0)∪B p (∞).Then we have either: a) or b) or c), where a) , respectively.Consequently, we would have either , respectively.However, those identities cannot be true.For example, the function u ≡ 1 belongs to D 1,p (R + , ω), while it does not belong to any of the sets: R 0 p,ω (0), R ∞ p,ω (0), R 0 p,ω (0)∩R ∞ p,ω (0).The contradiction proves iii).iv): Let us suppose that the implication does not hold, that is = ∅, which are false, thanks to either (31) or (32), respectively.The third situation cannot be true also, because the function u ≡ 1 belongs to . In any case we get the contradiction, which proves the validity of iv) and ends the proof of the statement.

Applications
Let us present several example applications of our results.

Application to Hardy inequality
Let us consider classical Hardy and the conjugate Hardy operators, respectively: (33) In Theorems 3.1 and 3.2 we have shown that Hardy type operators are isometric embeddings between L p (R + , ω) and R 0 p,ω (0) or R ∞ p,ω (0), respectively.Precisely, it follows from Theorem 3.1 and 3.2 that D 1,p (R+,ω) ), when ω ∈ B p (0), In the first case the inverse is u → u ′ , while in second case it is −u ′ .Such identification can be further used to obtain the extended variants of Hardy type inequality, where the class of admissible functions is defined in terms of limits of u at 0 or at ∞.
The necessary and sufficient conditions for boundedness of Hardy operator H and conjugate Hardy operator H * as acting from L p (R + , ω) to L q (R + , h), where 1 < p, q < ∞ are known, see e.g.[7], [9], [11], [13].For readers convenience we enclose them in the Appendix in Theorems 41 and 42.Let us call them (C) -in case of conditions for H, and (C * ) -in case of conditions for H * , respectively.We have the following example statement, which deepens our understanding of the Hardy inequality.As the B p (0) condition seems not known before, in our opinion the result is new.
Then the following statemets hold.
ii) When h ∈ L 1 (R + ), then inequality (34) with right hand side finite holds precisely on the set R 0 p,ω (0).In particular the embedding X ⊆ L q (R + , h) cannot be extended to any larger subspace X of D 1,p (R + , ω).
ii): We already know that (34) holds on R 0 p,ω (0).On the other hand, when u ∈ R 0 p,ω (0) then T r 0 (u) = c = 0, and in such case left hand side in (34) cannot be finite as h is not integrable near zero.

Application to formulation of Dirichlet boundary conditions for solutions of ODE's
The following remark contributes to the interpretation and well-posedness of boundary conditions of Dirichlet type, in various problems dealing with ODE's.
Remark 5.2 Our analysis allows to interpret precisely Dirichlet type boundary conditions for u ∈ D 1,p (R + , ω) with 1 < p < ∞: We already know (see Theorems: 3.1 and 3.2) that, for u ∈ D 1,p (R + , ω), in both cases the above conditions can be equivalently stated as As in later conditions the denominators converge to zero, (37) is stronger than (36) if we do not assume that u ∈ D 1,p (R + , ω).We can now confirm that the boundary conditions defined by (37) are well posed and equivalent for functions in the respective Dirichlet space D 1,p (R + , ω).
Remark 5.3 Using simple modification of inequalities (13), we deduce that when where dw,p(x, y) := Observe that the function dω,p(x, y) obeys the properties of distance function on R + and replaces |x − y| in (38).

Application to complex interpolation theory for weighted Dirichlet spaces
In the paper [2], on page 2434, in third question, the authors have asked about complex interpolation results for the weighted homogeneous Sobolev spaces, which in our setting we call Dirichlet spaces: give the isomorphic identification between the two Banach couples Let (X, Y ) θ denote the complex interpolation pair between Banach spaces X, Y .It is deduced from Calderón type generalization of Stein-Weiss Theorem, as in [2], in Remark 3.2 on page 2397, that one has: where ω The precise arguments are submitted in Section A.4 on pages 2439 and 2440 in [2].
In our case Ω = R + and ω ∈ Bp(R + ), so Mp(Ω) = ∅, but the extension of the above definition also to x ∈ Ω in place of x ∈ Ω, would lead in our situation to the validation of conditions Bp(0) and Bp(∞).
Our results can be extended further in several directions.Let us propose some of them.Remark 6.2 (possible extensions) (a) The choice of another domain.All the results that we will stated deal with functions defined on the half line.However, without major changes in the proofs, one can consider instead any interval (a, b) in place of R + .We have focused on functions defined on R + to make our presentation simpler.(b) Possible discontinuities inside the interval.We have assumed in all our statements, that weight function ω is positive and continuous inside the interval R + .It would be interesting to know how much this assumption can be weakened.(c) Higher order Dirichlet spaces.Instead of D 1,p (R + , ω), one could consider for example higher order Dirichlet spaces, for example: D k,p (R + , ω) = {u : R + → R : u is locally absolutely continuous on R + and u (k)  L p (R+,ω) < ∞}.where k ∈ N and u (k) is the distributional derivative of u, and ask similar questions.(d) Fractional order Dirichlet spaces.Instead of D 1,p (R + , ω), one could consider fractional order Dirichlet spaces, where the derivative u ′ is replaced by the fractional one, u (α) , where 0 < α < 1.For example one can use the Caputo, Riemann-Liouville, or Grünwald-Leitnikov derivatives, as discussed for example in the book [10].where I are intervals in R. However our weights are continuous inside R + , they do not need to satisfy the Ap condition.For example, such a one is ω(t) = t p−1 , because ω −1/(p−1) is not integrable near zero.
According to the discussion made in Section 5.4, we address the example open question, which naturally arises from our discussion.

The complementary proofs
Proof of Fact 2.2.The proof is based on modification of arguments from [8], where Sobolev spaces instead of Dirichlet spaces were considered.
Let U n := {u n + c} c∈R be the Cauchy sequence in D1,p (R + , ω).Then for any fixed a ∈ R + the function We have shown that the space D1,p (R + , ω) is complete.
Proof of Lemma 4.2.An easy verification shows that φ (k,K,a) belongs to the admissible class for the functional, that is the non-weighted Sobolev space W 1,p ((k, K)).We will show that φ (k,K,a) is the unique minimizer of (23).We give the proof for a = 1, because φ (k,K,a) (t) = aφ k,K,1 (t).As E(•) is convex functional and the admissible subset of W where v ∈ W 1,p ((k, K), ω), v(k) = v(K) = 0.
By Direct Methods in Calculus of Variations, because the functional is nontrivial, coercive and convex on W 1,p ((k, K)) and ω ∼ 1 on [k, K], we deduce that there exists a unique minimizer of Ẽ.To find the minimizer, we compute Euler-Lagrange equation corresponding to the minimizer.For any v ∈ C ∞ 0 ((k, K)) we have As ω is continuous, the function inside brackets {•} is integrable over (k, K) and its weak derivative is zero.Thus this function is constant and hence u

D 1
,p (R+,ω) := |u(a)| + u ′ L p (R+,ω) , p ω(τ )dτ with the given boundary data at the endpoints {a, b}.In further step we extend such local minimizers to compactly supported functions in the same Dirichlet-Sobolev class.See the considerations in Sections 4.2 and 4.3.

′
in L p (R + , ω) and un → u uniformly on compact sets in R + .
dτ + lim n→∞ u n (a) we have u n − u (a) D 1,p (R+,ω) → 0 as n → ∞.The equivalence of norms is a consequence of the following estimate holding for any closed interval I such that a, b ∈

Theorem 5 . 2 (
Banach's Inverse Mapping Theorem) .Let X, Y be Banach spaces and let T : X → Y be a linear bounded operator.If T is bijective, then T −1 : Y → X is bounded.

Remark 5 . 1 (
possible analysis of conjugate Hardy inequality) Similar considerations based on the analysis of Hardy conjugate transform H * lead to the validity of (34), equipped with the condition lim t→∞ u(t) = 0, under Conditions (C * ) for the admitted weights (see Theorem 7.2).
ω), for i = 1, . . ., n , where U ⊆ R n is an open set and ω : U → R + is a given weight.The authors have focused on the case of p ∈ [1, ∞), n = 1, and U = R for the special class of weights, which satisfy the compact boundedness condition as in Definition 1.3 on page 2383.That condition is satisfied by every positive continuous function defined on R. In that case the mapping: ψ → x 0 ψ(t)dt and its inverse φ → φ ′
As local B p conditions B p (0) and B p (∞) have not been analyzed eariler (see Remark 6.1), in our opinion the result is new.
Remark 6.1 (local Bp conditions in literature) In general the localized Bp conditions are missing in the literature.However, in [8], having positive almost everywhere weight ω defined on open set Ω ⊆ R n , the authors consider the so-called "exceptional set" Mp(Ω) := x ∈ Ω : Ω∩V (x) 1,p ((k, K)) is convex closed set, Direct Methods in the Calculus of Variations (see e.g.[3]), give existence of unique minimizer of E(•).Let us call such a minimizer φ 0 .Let T (t) :=