Introduction to Lp Sobolev Spaces via Muramatu’s Integral Formula

Muramatu’s integral formula is a very useful tool for the study of Sobolev spaces, although this does not seem to be widely recognized. Most theorems in Sobolev spaces can be proved by this formula combined with basic inequalities in analysis, and it is possible to directly treat not only the whole space but also a special Lipschitz domain. In this paper, we present an introduction to Lp-based Sobolev spaces of integer order by making Muramatu’s integral formula play a central role, as Cauchy’s integral formula does in complex analysis. The topics we take up are approximation by smooth functions, the interpolation inequality, the Sobolev embedding theorems, the trace theorem, construction of an extension operator, complex interpolation of Sobolev spaces and real interpolation of Sobolev spaces.


Introduction
There are many books that cover Sobolev spaces (see [1,7,17,18,30,32,40] for the books whose main themes are Sobolev spaces, and [6,9,10,11,13,14,16,31,33,34] for the books which are related to partial differential equations), and the standard way seems to have been established. Nevertheless, we wish to propose another way of using Muramatu's integral formula [22,Corollary 1,p341] to study Sobolev spaces, especially as an introductory course of L p -based Sobolev spaces. We collect features of Muramatu's integral formula.
• It is simple and easy to use. • Most theorems can be proved by it with the help of basic inequalities such as Hölder's inequality, Minkowski's inequality for integrals, Young's inequality and the lemma on boundedness of integral operators. This means that we can make Muramatu's integral formula play a central role in the theory of Sobolev spaces, as Cauchy's integral formula does in complex analysis.
• If you understand the proof of a theorem, then you can think of the proof of another theorem by yourself. This is in contrast with the standard way that requires different techniques to prove different theorems. • It enables us to prove some theorems in one shot. In other words, we do not have to use induction on the regularity order of Sobolev spaces. • It enables us to treat directly a special Lipschitz domain as if the domain were the whole space R d . It works for L p -based Sobolev spaces on a special Lipschitz domain as well as the Fourier transform does for L 2 -based Sobolev spaces on R d (see [12,15,33]). • It works well also for Sobolev spaces of negative order. In order to demonstrate how useful Muramatu's integral formula is, we give here a short proof of the embedding theorem W m p (R d ) ⊂ L q (R d ) for 1 ≤ p < q < ∞ and m − d/p > −d/q with a positive integer m. Let f ∈ W m p (R d ). Then Muramatu's formula or its primitive form is given by with suitable functions ϕ ∈ C ∞ 0 (R d ) and K α ∈ C ∞ 0 (R d ) for every multi-index α, where ϕ * f stands for the convolution of ϕ and f . Noting and taking into account the definition of weak derivatives (or integrating by parts if f ∈ W m p (R d ) ∩ C ∞ (R d )), we can move ∂ α from K α to f to get for |α| = m. Taking r ∈ (1, ∞) so that p −1 + r −1 = 1 + q −1 , and using Young's inequality, Minkowski's inequality and the relations Vol. 85 (2017) Introduction to Lp Sobolev Spaces 105 Introduction to L p Sobolev Spaces 3 Muramatu [22] devised not only formula (1.1) but also the second integral formula [22,Theorem 1,p344], which is obtained by a repeated use of (1.1), and which is more complicated than (1.1). Combining the second formula with the Littlewood-Paley theory of L p boundedness of operators acting on Hilbert-space-valued functions, he investigated Sobolev spaces W σ p (Ω) as well as Besov spaces B σ pq (Ω) for all σ ∈ R. As for Sobolev spaces, he showed that a function f (x) belongs to W σ p (Ω) if and only if its regularization u(t, x) = R d t −d K((x − y)/t)f (y) dy with a suitable C ∞ function K satisfies t −σ u(t, x) ∈ L p (Ω, L 2 (I, t −1 dt)) with I = (0, 1), and proved the theorems on Sobolev spaces by reducing the matters to the properties of L p (Ω, L 2 (I, t −1 dt)).
In this paper, we give an introduction to Sobolev spaces W m p (Ω) with integer m and 1 ≤ p < ∞ on a domain Ω of R d , making full use of Muramatu's integral formula. Our viewpoint in this paper is different from that of Muramatu [22] who aimed at full generalities. We put emphasis on simplicity and try to make the presentation as accessible as possible. Since formula (1.1) is powerful enough for Sobolev spaces of integer order, we can prove the theorems without using the second integral formula and the characterization of Sobolev spaces by L p (Ω, L 2 (I, t −1 dt)), which are required for Sobolev spaces of fractional order. The domain we treat in this paper is the whole space R d or a special Lipschitz domain including the half space. As for the treatment of bounded Lipschitz domains we will make some remarks in Section 11. This paper is organized as follows. In Section 2 we derive Muramatu's integral formula and apply it to the interpolation inequality. Sections 3 and 4 are devoted to the Sobolev embedding theorems into another Sobolev space and the Hölder-Zygmund space, respectively. In Section 5 we prove the trace theorem for the half space after defining the Besov space which is a suitable function space for describing the boundary values. In Section 6 we present the convergence theorem which enables us to investigate Sobolev spaces further. The proof for p = 2 is given, and the proof for p = 2 is postponed to the Appendix. As an application we show the interpolation inequality concerning pure derivatives. In Section 7 we construct an extension operator from W m p (Ω) to W m p (R d ). Sections 8 and 9 are devoted to complex and real interpolations of Sobolev spaces, respectively. In Section 10 we define Sobolev spaces of negative order, and show that Muramatu's formula also works well for the case of negative order. As an application we give a short proof of a lemma which leads to the regularity theorem for elliptic equations in divergence form. In Section 11 we give a survey of the related literature and comparisons with the methods presented there. In the Appendix we complete the proof of the convergence theorem.
The interdependence of the sections is indicated in Figure 1. This paper is self-contained except for the proof of Theorem 3.1 in a certain critical case. Here an open cone is a subset of R d which is written in the form with a unit vector ν ∈ R d and an angle θ ∈ (0, π/2]. Let us derive Muramatu's integral formula, assuming 1 ≤ p < ∞. Let Γ be the cone satisfying (2.2), and set Inclusion (2.2) implies x + tΓ 0 ⊂ Ω for all x ∈ Ω and t > 0. For a function K on R d and t > 0 we define the rescaling K t by When we consider a function K α , which depends on a multi-index α, we write its rescaling as For f ∈ L p (Ω) we consider the convolution of the rescaling ϕ t and f : ∞) × Ω) as a function of t and x, and If we set Let E 0 f denote the extension of f to R d by zero, namely Then ϕ * f is considered as the restriction to Ω of the function ϕ * E 0 f in R d . By Young's inequality where the integral λ 0 should be interpreted as lim →0 λ in the norm of L p (Ω).
It is essential that K in (2.11) consists of the derivatives of some functions, (2.6), and that the support of K j is also contained in −Γ 0 . The advantage of supp This combined with (2.11) yields Thus by moving the derivatives from K to f we have succeeded in producing a factor t in the integrand. If we could move the derivatives k times from K to f , then we could produce a factor t k , which makes the integrand more integrable in t.
Remark 2.6. Expression (2.15) implies that we may interchange the order of integration and differentiation, when applying ∂ α to (2.14). We will discuss the problem of differentiation under integral sign in the other situations in Proposition 2.9 and Corollary 6.2. At this stage we may not interchange the order of integration and summation in Muramatu's formula, since it has not been guaranteed that the integral of each term in the sum converges in L p (Ω). Proposition 2.9 and Theorem 6.1 will give criteria for the L p convergence of the integral. Proof. Take ω ∈ K 0 Γ satisfying R d ω(x) dx = 1, and define ϕ and K as in Lemma 2.3 with N replaced by N + m. Then we find that (2.11) is valid, since R d ϕ(x) dx = 1 by integration by parts. Applying (2.11) to f (α) ∈ L p (Ω) with |α| ≤ m, we get In view of (2.13) with N replaced by N + m we know that K is written as We note that in general ψ * f (α) = (∂ α ψ) * f (2.16) holds for ψ ∈ K 0 Γ . In fact, by the definition of distributional derivatives we have As an application of Muramatu's integral formula we show the interpolation inequality. We use the symbol Theorem 2.7 (Interpolation inequality). Let 1 ≤ p < ∞, m ∈ N, k ∈ N and 1 ≤ k < m. Then there exists C = C(m, p, d, Ω) > 0 such that the inequalities Proof. We use Muramatu's formula (2.15) for f (α) ∈ L p (Ω) with |α| = k. Minkowski's inequality and Young's inequality give and thereby We get (2.17) by setting λ = 1. Inequality (2.18) follows by setting Remark 2.8. If Ω satisfies Ω = λΩ for all λ > 0, then inequality (2.19) follows from One advantage of Muramatu's formula is that we can directly obtain (2.19) without passing through (2.20), and that there is no need to assume that Ω satisfies Ω = λΩ for all λ > 0.
It is possible to relax the assumption in the interpolation inequality. To this end, we prepare a lemma on convergence in a simple case.
In particular, if p = ∞, then λ 0 u(t, · ) dt ∈ C(Ω). (ii) In addition to the above assumption, suppose that , and they are equal: Proof. The first half of (i) follows from Minkowski's inequality and the completeness of L p (Ω). The assertion for p = ∞ follows from the fact that convergence in the L ∞ norm implies uniform convergence. We get (ii) by (i) and Lemma 2.1, noting that Proof. From the proof of Theorem 2.5, we see that (2.14) and (2.15) with |α| = m are valid. In order to show that (2.15) is also valid for 0 < |α| < m, we use (2.14). Observing for |α| < m, and using Proposition 2.9, we get (2.15) for |α| < m by applying ∂ α to (2.14). Now that (2.15) holds for all α with |α| ≤ m, the proof of Theorem 2.7 works as it is.
We conclude this section with the theorem on approximation by C ∞ functions.
, and consider the convolution ϕ t * f defined by (2.5). Let E 0 be the zero-extension defined by (2.9). Then ϕ t * f is the restriction Thus we obtain the theorem.

Embeddings into other Sobolev spaces
In this section we consider the embeddings such as W m p (Ω) ⊂ W n q (Ω) with the Sobolev inequality We shall show by the scaling method that the conditions are required for inequality (3.1) to hold, assuming for a moment that Ω satisfies Vol.85 (2017) Introduction to Lp Sobolev Spaces 113 Introduction to L p Sobolev Spaces 11 know that (3.2) is necessary for (3.1). Following Runst and Sickel [26], we call the quantity which is the exponent appearing in the leading term of f (λ · ) W m p as λ → ∞, the differential dimension of W m p (Ω). Generally, when a function space X is continuously embedded into another function space Y , the differential dimension of X is equal to or greater than that of Y .
and the inequality Proof. First of all we note that m > n, We use Muramatu's formula (2.15) for W m p (Ω), and apply Proposition 2.9 for L q (Ω). Minkowski's inequality and Young's inequality with (K (α) L p f Lp → 0 as λ → ∞, where 1/p + 1/p = 1, and note that the integral λ 0 is the limit of λ in L p (Ω) as → 0. Hence there exists a sequence of positive numbers { j } ∞ j=1 tending to 0 such that , for a.e. x ∈ Ω, which gives By the Hardy-Littlewood-Sobolev inequality, which is a special case of [12,Theorem 6.36], we obtain f (α) , which gives the desired result. Thus we complete the proof for p = 1 and m ≥ 2.
It is a good exercise to prove Nash's inequality by Muramatu's formula. Indeed, we can prove it in the same spirit as in the proof of Theorem 3.1 for the case

Embeddings into Hölder-Zygmund spaces
When the differential dimension m−d/p is large enough, the functions in W m p (Ω) are differentiable in the classical sense. To describe the statement precisely we introduce the Hölder-Zygmund space.
The difference operators ∆ h and ∆ 2 h are defined by Then the Hölder-Zygmund space C σ (Ω) is the collection of all functions f which are [σ] − times continuously differentiable and satisfy We equip C σ (Ω) with the norm · C σ (Ω) .
The Hölder-Zygmund space C σ (Ω) is a Banach space. We shall find necessary conditions for m ∈ N, 1 ≤ p < ∞ and σ > 0 that guarantee the embedding W m p (Ω) ⊂ C σ (Ω) with the ineqaulity assuming for a moment that Ω satisfies Ω = λΩ for all λ > 0. Applying So this inequality is a necessary condition for (4.3). In view of the above argument it is reasonable to call σ, which is the exponent appearing in the leading term of f (λ · ) C σ (Ω) , the differential dimension of the Hölder-Zygmund space C σ (Ω).
where the integral ∞ 0 should be interpreted as the limit of λ as → 0 and λ → ∞ in the topology of L p (Ω) + L ∞ (Ω). We use the inequality Hölder's inequality and the change of variables t = |h|s give Since the last integral is finite, we obtain f ∈ C σ (Ω) with (4.6). Also, (4.4) follows from (4.6) and the estimates obtained in Step 1.
Step 3. When {σ} + = 1, we can make the same argument as in Step 2 if we replace ∆ h by ∆ 2 h and use, instead of (4.7), follows by Taylor's formula.

Trace theorem
In this section we consider the trace or the restriction to a hyperplane of a function belonging to Sobolev spaces. To this end, we first give an intrinstic definition of the Besov space. It is convenient to use the symbols for 1 ≤ q ≤ ∞ and R + = (0, ∞). Recall that Ω 1,h and Ω 2,h are defined by (4.1).
Definition 5.1 (Besov space). Let 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. We define the seminorm |f | B τ pq (Ω) for 0 < τ ≤ 1 by with the usual modification for q = ∞ if 0 < τ < 1, and The Besov space B σ pq (Ω) is a Banach space. By the same argument as in the case of Sobolev spaces we find that it is reasonable to call σ − d/p the differential dimension of B σ pq (Ω). We will give a statement of the trace theorem only when the domain is the half space

and the inequality
Tr In particular, We note that the differential dimension of B The statement is also valid if we replace the integral kernel K(|h|/t) by K(t/|h|). When d = 1, the assertion remains valid if we replace R d and L q (R d ) by R + and L q (R + ), respectively.
Proof. The assertion for f (t) is valid by the theorem about boundedness of integral operators (see [12,Theorem 6.18] where ω d−1 is the area of the unit sphere in R d . The other cases can be proved similarly. and 0 ≤ |α | < m. We use Muramatu's formula (2.15). Since f (β) ∈ C(R d + ) for |β| = m, and since for x ∈ R d + , the integral in (2.15) can be also regarded as the pointwise limit of with y = (y , y d ), and use the inequality where χ is the characteristic function of [0, 1]. Setting λ = 1, and using Minkowski's inequality with Young's inequality, we have Since χ( · /t) L p (R + ) = t 1−1/p with 1/p + 1/p = 1, Hölder's inequality gives Step 2. Next we shall evaluate the seminorms |∂ α f and |α| = |α | = m − 1. Since the same calculations as in Step 1 give β ) t ( · , y d ) L∞ by Taylor's formula, and taking into account the support of K (α) β , we have , and using Minkowski's inequality with Young's inequality, we have Changing the variables y d → s and t → |h|t, and setting we get  and using Minkowski's inequality, we have where χ is the characteristic function of [0, 1]. Since Step 2. We shall evaluate the L p norm of the higher-order derivatives of u defined by with ψ l (x ) = −x l ϕ(x ). Repeating this procedure, we get, for |α| ≤ m − 1, For |α| = m − 1 and 1 ≤ j ≤ d we have with the functions where δ jl is Kronecker's delta. Since R d−1 ψ jlαβ (x ) dx = 0, we apply the result of Step 1 to get Step 3. Let g ∈ B m−1/p pp (R d−1 ). We take a function η ∈ C ∞ 0 (R) satisfying η(t) = 1 for |t| ≤ 1, and setf (x) = η(x d )u(x), where u is defined by (5.6). It easily follows from (5.7) and (5.8) . It remains to check that Tr f = g. To this end, we consider the translation Hence the boundedness of the trace operator yields Tr . On the other hand, by definition of f we see that Tr τ h f = ϕ h * g for 0 < h < 1, and hence Tr τ h f → g in L p (R d−1 ) as h → 0. Therefore Tr f = g.

Convergence theorem
In this section we consider the convergence of the integral as → 0 in the L p norm for f ∈ L p (Ω), K ∈ K 0 Γ and z ∈ C. If Re z > 0, Proposition 2.9 guarantees its convergence since If Re z = 0, the matter of convergence is subtle. When z = 0, and K is written as (2.6) with a function ϕ ∈ K 0 Γ , we know that T f converges to by (2.7), although this case is beyond the scope of Proposition 2.9. Theorem 6.1 below enables us to assert that T f converges even when T f is not necessarily written in the form of ϕ * f − ϕ λ * f , if the additional condition K ∈ K 1 Γ is assumed. It is also important to evaluate T f with a bound independent of z, which can not be obtained from estimate (6.2) depending on z.
Γ . For f ∈ L p (Ω), 0 < < λ < ∞ and z ∈ C with 0 ≤ Re z ≤ b we consider T f defined by (6.1). Then we have with C = C(p, d, K, λ b ). Furthermore, T f converges uniformly with respect to z in L p (Ω) as → 0. The limit function T f = lim →0 T f satisfies, with the same C as in (6.3), Proof. Extending f ∈ L p (Ω) to R d by zero, we may assume Ω = R d . First we shall show (6.3). For the convergence of T f , it is crucial that the constant C in (6.3) is independent of . We need to take an approach different from the calculations in (6.2).

Vol.85 (2017)
Introduction to Lp Sobolev Spaces 123 Introduction to L p Sobolev Spaces 21 Let p = 2. We denote by Ff the Fourier transform of f : Then By assumption we may write Plancherel's theorem gives Thus we obtain (6.3) for p = 2. In order to show (6.3) for p = 2 we require the Calderón-Zygmund theory of singular integral operators. We postpone the proof of (6.3) for p = 2 to the Appendix.
Assuming that (6.3) is true, we proceed to complete the proof. Let f ∈ C ∞ 0 (R d ). For 0 < < δ < λ, Finally, let f ∈ L p (R d ). Writing with any g ∈ C ∞ 0 (R d ), and using (6.3) and the above inequality for functions in Since {T f } 0< <λ satisfies Cauchy's convergence criterion. Therefore T f converges in L p (R d ). Letting → 0 in (6.3), we get T f Lp ≤ C f Lp . Thus we complete the proof.
With Theorem 6.1 and Corollary 6.2 in hand, we can improve Corollary 2.10, the interpolation inequality, in terms of pure derivatives.
holds for 1 ≤ k ≤ m with C = C(m, p, d, Ω). In particular, Proof. We write Muramatu's formula (2.11) as For each β with |β| = md there exists j such that β ≥ me j , where e j is the multi-index of length 1 whose jth component is 1; otherwise we would have |β| ≤ (m − 1)d < md, a contradiction. For β with |β| = md and β ≥ me j we have with some K j ∈ K 1 Γ . Applying Corollary 6.2, we find that f ∈ W m p (Ω) and for |α| ≤ m. Using Proposition 2.9 for |α| < m and Theorem 6.1 with b = 0 for |α| = m, we get Estimate (6.4) follows by setting λ = 1, and (6.5) follows by letting λ → ∞ for k = m.

Extension to the whole space
In this section we construct an extension operator from W m p (Ω) to W m p (R d ). Let R Ω be the restriction, that is, Theorem 7.1. Let 1 < p < ∞ and m ∈ N 0 . Then there exists a bounded linear operator with C = C(m, p, d, Ω).
Proof. We use Muramatu's formula (2.14) with λ = 1: Recall that E 0 is the zero-extension to R d given by (2.9), and define E Ω by Let f and F be defined by the right-hand sides of (7.1) and (7.2), respectively, with 1 0 replaced by 1 . Since g ∈ L p (Ω) satisfies R Ω (K * E 0 g) = K * g for K = K β or Using Proposition 2.9 for |α| < m and Theorem 6.1 for |α| = m, we have which gives the boundedness of E Ω .

Complex interpolation of Sobolev spaces
In this section we show that the complex interpolation of two Sobolev spaces is a Sobolev space. To this end, we quickly review the method of complex interpolation. See [3,35] for the details. Let X and Y be Banach spaces with Y ⊂ X, and suppose that the embedding Y → X is continuous. Let H(X, Y ) be the space of all X-valued functions F satisfying the following conditions: We equip H(X, Y ) with the norm The space [X, Y ] θ is a Banach space.
Obviously, F (θ) = f . Also, ϕ * f is a constant term belonging to W m p (Ω). In order to show F ∈ H(W n p (Ω), W m p (Ω)), we define F (z) to be the right-hand side of the above equation with it follows by Theorem 6.1 and Corollary 6.2 that F (z) converges to F (z) in W n p (Ω) as → 0 uniformly with respect to z ∈ S. Also, F (z) converges to F (z) in W m p (Ω) uniformly on the line Re z = 1. Therefore we conclude that for all g ∈ C ∞ 0 (Ω) and |α| ≤ k with 1/p + 1/p = 1. To this end, we view g ∈ C ∞ 0 (Ω) as a function belonging to C ∞ 0 (R d ) ⊂ L p (R d ), and use Muramatu's formula (2.11) for ∂ α g, taking −Γ as the cone associated with the whole space R d ; recall that Γ is the cone associated with Ω. Then we have and ψ ∈ K 0 −Γ . Taking this formula into account, we define the function z → G(z) ∈ L p (R d ) on S by Obviously, G(θ) = ∂ α g. Since G(z) is written as with some H β ∈ K 1 −Γ , and since Re (k −|α|−(1−z)n−zm+m+1) ≥ 1 for z ∈ S, the same argument as for F (z) in Step 1 shows that G(z) is an L p (R d )-valued function that is analytic in S, and bounded and continuous on S.
where we interchanged the order of ∂ β and integration by Proposition 2.9 and Theorem 6.1. By integration by parts and Theorem 6.1 we have Since the maximum principle for analytic functions yields (8.1). From the duality between L p (Ω) and L p (Ω) (see [12,Theorem 6.15]), and the fact that C ∞ 0 (Ω) is dense in L p (Ω) it follows that ∂ α f ∈ L p (Ω) for |α| ≤ k, and that f W k p (Ω) ≤ C F H(W n p (Ω),W m p (Ω)) . Taking the infimum of the right-hand side, we conclude that

Real interpolation of Sobolev spaces
In this section we show that the real interpolation of two Sobolev spaces is a Besov space. To this end, we quickly review the method of real interpolation. See [1,3,35] Vol.85 (2017) Introduction to Lp Sobolev Spaces 129 Introduction to L p Sobolev Spaces 27 for the details. There are several ways to define real interpolation spaces, of which we adopt the K-method. Recall that L * q (R d ) and L * q (R + ) are defined by (5.1).
Definition 9.1. Let X and Y be Banach spaces with Y ⊂ X, and suppose that f X ≤ f Y . For f ∈ X and t > 0 we set For 0 < θ < 1 and 1 ≤ q ≤ ∞ we define the real interpolation space (X, Y ) θ,q by (X, Y ) θ,q = {f ∈ X : t −θ K(t, f ) ∈ L * q (R + )}, and equip it with the norm by the triangle inequality and the assumption v X ≤ v Y . Taking the infimum, we get f X ≤ K(t, f ) and hence where (θq) −1/q should be replaced by 1 if q = ∞. Also, it follows from For m ∈ N and h ∈ R d we set for a function f on Ω. We note that ∆ m h f = ∆ h (∆ m−1 h f ) for m ≥ 2. To state the theorem for the real interpolation of Sobolev spaces, we introduce the space B m,σ pq (Ω), which will turn out to coincide with the Besov space B σ pq (Ω) defined in Definition 5.1. For the proof we prepare several lemmas. Lemma 9.5. For m ∈ N define the sequence {b j } m j=0 so that the identity holds as polynomials in t. Let K ∈ C ∞ 0 (R d ) be written in the form Then Proof. We write U (t, x) for the right-hand side of (9.3), which is well defined since the set {(y, z) ∈ R 2d : z ∈ R d , y ∈ Ω m,z } is open, and hence measurable. If we set for k ∈ {0, 1, . . . , m}, we have Indeed, W t (x − y, x − y − mz) = 0 implies y ∈ x + tΓ 0 ⊂ x + Γ and y + mz ∈ x + Γ, and the convexity of Γ gives y + kz ∈ x + Γ ⊂ Ω for all k with 0 ≤ k ≤ m, so we may restrict the domain of y-integration for U k (t, x) to Ω m,z . Thus it suffices to evaluate each U k (t, x) for x ∈ Ω.
Vol.85 (2017) Introduction to Lp Sobolev Spaces 131 Introduction to L p Sobolev Spaces 29 Let k = 0. The substitution w = x − y − mz gives W t (x − y, w)E 0 f (y) dy dw.