The Initial Dirichlet Boundary Value Problem forGeneral Second Order Parabolic Systems in Nonsmooth

Manifolds ∗

Marius Mitrea

0 Introduction

In a series of recent papers [1], [2], [3], [4], [5] we have initiated the study of boundaryvalue problems for (variable coefficients, second order, strongly) elliptic PDE’s in nonsmoothsubdomains of Riemannian manifolds via integral equation methods. Here we take the firststeps in the direction of extending this theory to initial boundary value problems (IBV P ’s)for variable coefficient (strongly) parabolic systems in non-smooth cylinders.

Problems as such have a long history and the field remains a very active area of research.For work in the context of smooth manifolds the reader is referred to [6], [7], [8]. See also[9], [10], [11], [12], [13], [14], [15], [16], for IBV P ’s associated with PDE’s of parabolic typein the smooth Euclidean setting.

With the work of E.B. Fabes, N. Riviere and their collaborators starting in the 1960’s,a new direction of research has emerged, emphasizing Lp-boundary data and less regulardomains and/or coefficients. In this vein see [17], [18], [19], [20], [21], [22], [23], [24], [25], [26],[27]. The papers cited above deal with domains exibiting a limited amount of smoothnessand new techniques needed to be developed in order to treat the nonsmooth case.

After the breakthrough in the elliptic case, cf. [28], [29], [30], [31], [32] and the referencestherein, there has been a substantial amount of work in the direction of solving IBV P ’s forparabolic PDE’s in minimally smooth domains. In flat-space, Euclidean Lipschitz cylinders,∂t −∆ was treated via caloric measure estimates in [33] (inspired by the work in [28]) andvia integral methods in [34], [35], [36] (after the pioneering work in [22] and by adaptingthe approach from [30]). The latter work has been further extended to include second-orderconstant coefficient PDE’s such as the parabolic versions of the Lame system, the linearizedNavier-Stokes system and the Maxwell system in [37], [38], [39]. Higher order, homogeneous,constant coefficient, parabolic PDE’s have been treated in [40], [41], [42], following the workin elliptic case from [43]. The Dirichlet problem for more general, scalar, divergence-formparabolic PDE in Lipschitz cylinders has been considered in [44]. Extensions to time-varyingdomains have been developed in [45], [46], [47], [48], [49], [50].

∗1991 Mathematics Subject Classification: Primary 35K50, 42B20; Secondary 58G20.Key words: Boundary value problems, parabolic systems, Riemannian manifolds, Lipschitz cylinders, layerpotentials, Rellich estimates.

1

In order to explain the main results of this paper we need some notation. Assume thatM is a smooth, compact Riemannian manifold and E → M is a smooth, Hermitian vectorbundle. Let pr : M × R → M stand for the canonical projection and set E := pr∗E, thepull-back vector bundle. Next, consider a second order, formally self-adjoint, strongly ellipticdifferential operator with smooth, real coefficients L : C∞(M,E) → C∞(M,E). Extend Lnaturally as a mapping of C∞(M × R, E) into itself (by making it independent of time),then set P := ∂t − L(x, ∂x). Finally, assume that Ω ⊆ M is a Lipschitz domain, i.e. adomain whose boundary is given in local coordinates by graphs of Lipschitz functions, andfix 0 < T <∞. For 1 < p <∞ we consider the Dirichlet initial boundary value problem

(IBV P )

u ∈ C∞(Ω× (0, T ), E),

Pu = 0 in Ω× (0, T ),

u(·, 0) = 0 on Ω,

u∗ ∈ Lp(∂Ω× (0, T )),

u|∂Ω×(0,T ) = f ∈ Lp(∂Ω× (0, T ), E).

(0.1)

Here u∗ stands for the (parabolic) nontangential maximal function; more precise definitionsare given in the body of the paper.

Our main result states that, with the above assumptions, there exists ε = ε(Ω, L) > 0 sothat for each 2 − ε < p < 2 + ε, the initial boundary problem (0.1) has a unique solution.This solution satisfies

‖u∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E) (0.2)

for some C = C(∂Ω, L, T, p) > 0. Also, when p = 2,

u ∈ H1/4((0, T ), L2(Ω, E)) ∩ L2((0, T ), H1/2(Ω, E)), (0.3)

where Hs is the usual L2-based scale of Sobolev spaces. Furthermore, the solution is moreregular if the boundary datum is so. For a complete statement see Theorem 9.1.

Let us point out that the result sketched above encompasses many particular cases ofindependent interest. Such a list includes the scalar heat operator ∂t −∆, where

∆u =1√g

∑j,k

∂

∂xj

[gjk√g∂u

∂xk

](0.4)

is the Laplace-Beltrami operator associated with the metric tensor g =∑gjkdxj ⊗ dxk on

M or, more generally, when

∆ = dδ + δd (0.5)

is the Hodge-Laplacian on differential forms. Here d and δ are the exterior derivative opera-tor and its adjoint, respectively. In fact, if D : C∞(M,E)→ C∞(M,F ) is an arbitrary firstorder elliptic differential operator then L = DD∗ will do. Many familiar second order, vari-able coefficient, operators arise in this fashion: Lame type operators, symmetric differentialoperators, Dirac Laplacians, etc.

2

The strategy for proving the aforementioned result pertaining to the well-posedness of(IBV P ) consists of reducing the original problem to an integral equation on the boundaryof the Lipschitz cylinder Ω× (0, T ). In this scenario, as a prerequisite, one has to develop aboundary behavior theory for integral operators of the form

Jf(x, t) :=

∫ t

0

∫∂Ω

〈k(x, y, t, s), f(y, s)〉 dσ(y)ds, (0.6)

where k is the Schwartz kernel of a classical (casual) parabolic pseudodifferential operatorin OPS−1,+

cl,2 whose principal symbol p−1(x, ξ, τ) is odd in ξ. Also, dσ is the surface measureon ∂Ω and 〈·, ·〉 refers to the Hermitian structure on the fibers of E .

Among the issues of interest for us here are the nontangential maximal function estimate

‖(Jf)∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E) (0.7)

and the jump-formula

limx→ w

x ∈ γ±(w)

Jf(x, t) =∓ 12i p−1(w, ν(w), 0)f(w, t)

+ limε→0

∫ t−ε

0

∫∂Ω

〈k(x, y, t, s), f(y, s)〉 dσ(y)ds,

(0.8)

valid at a.e. (w, t) ∈ ∂Ω× (0, T ). Here ν is the outward unit conormal to ∂Ω and γ±(w) areappropriate nontangential regions (in Ω+ := Ω and Ω− := M \ Ω, respectively).

It should be mentioned that, as far as the boundedness of the operators involved in thispaper is concerned, several techniques can be adapted to the present context. For example,(0.7) can be proved via the powerful methods developed in [49], [50], [46], [47] which canactually handle even more general domains. However, given that we work in a cylinder, weprefer to present an alternative proof, more akin to the original approach in [22]. This makesthe presentation somewhat more uniform and self-contained. The approach just alluded toessentially consists of working on the Fourier transform side in time and reducing mattersto the elliptic theory. This also sets the stage for the proof of the jump-formula (0.8) which,once again, we treat via elliptic theory.

In the Euclidean setting, the Dirichlet problem for P = ∂t − L is typically solved bylooking for a candidate u in the form of a double layer potential operator. Going further,the kernel of this latter integral operator is obtained by taking the conormal derivative of k,a suitable fundamental solution for P . Now, in the invariant setting and in the absence ofa global ‘product’ structure of the second-order operator L, one lacks a canonical choice ofa conormal derivative. We overcome this problem by working with the dual of an operatorwhich resembles the one used to solve the oblique derivative problem for P . This is anadaptation of an idea which goes back to A. P. Calderon ([51]) in the case of the (flat-space)scalar Laplacian. The rest of the argument in [51] involves a decomposition of the boundaryintegral operator into an accretive part and a symmetric, compact part. This works wellin the elliptic case and yields that the operator in question is bounded from below modulocompacts; cf. [1]. In the parabolic setting, we adapt a somewhat more circuitous approachbased on Rellich type estimates to reach essentially the same conclusion.

3

The organization of the paper is as follows. Sections §1–§2 contain a discussion ofanisotropic symbols and pseudodifferential operators, respectively. Layer potential opera-tors of parabolic type on Lipschitz cylinders are introduced and studied in §3. Among otherthings, here we give a proof of (0.7). The jump-formula (0.8) is proved in §4. In Section 5we discuss similar issues for fractional time derivatives of parabolic ‘single’ layer potentials.Section 6 deals with square-function estimates, while Section 7 treats Rellich type identi-ties and applications. In §8 we then proceed to prove invertibility results for the relevantboundary integral operators. Finally, in §9, we state and prove the Dirichlet and the Reg-ularity initial boundary problems for second order, strongly parabolic operators of the typeP = ∂t − L(x, ∂x) in cylinders of the form Ω× (0, T ) where Ω is a Lipschitz subdomain of asmooth Riemannian manifold.

Acknowledgments. It is a pleasure to thank Steve Hofmann for several inspiring discussionsand suggestions. This work was supported in part by the NSF grant DMS-9870018.

1 Anisotropic Symbols

We consider a class of symbols depending on a parameter. The basic feature is that theparameter is not treated as a lower order perturbation, but rather built into the leadingsymbol.

Let U ⊆ Rm be open, k ∈ R the anisotropy factor, ` ∈ R the order, N ∈ N the class, and0 ≤ 1−ρ ≤ δ < ρ ≤ 1. Also, fix d1, d2 ∈ N. We say that p : U×R×Rm×R→ Hom (Cd1 ,Cd2)belongs to

CNS`ρ,δ,k(U × R× Rm × R; Hom (Cd1 ,Cd2)) (1.1)

if p(x, t, ξ, τ) is of class CN in (x, t), C∞ in (ξ, τ) and, for each K ⊂ U×R compact, α ∈ Nm,ν ∈ N, β ∈ Nm, γ ∈ N with |α|+ |ν| ≤ N ,

|∂αx∂νt ∂βξ ∂

γτ p(x, t, ξ, τ)| ≤ Cα,β,γ,ν,K(1 + |ξ|+ |τ |1/k)`+δ(|α|+|ν|)−ρ|β|−k|γ|,

uniformly for (x, t) ∈ K, ξ ∈ Rm, τ ∈ R.(1.2)

To the point that no ambiguities arise, we shall try to simplify the notation as much aspossible. For example, we may write CNS`ρ,δ,k (if U, d1, d2 are clear from the context) and,

further, S`ρ,δ,k if N =∞.Let us point out that

CNS`ρ,0,k(U × R× Rm × R) ⊆ CNS`/k

min ρk ,1,0(U × R× Rm+1), (1.3)

where CNS`ρ,δ is the ordinary Hormander’s class of symbols (with a limited amount of smooth-ness in (x, t)). Also,

p ∈ CNS`ρ,δ,k ⇒ p(·, ·, ·, 0) ∈ CNS`ρ,δ. (1.4)

Call p ∈ CNS`ρ,δ,k mixed homogeneous of degree d in (ξ, τ) if

4

p(x, t, λξ, λkτ) = λdp(x, t, ξ, τ), ∀λ ≥ 1. (1.5)

Asymptotic sums are defined in the usual way:

p ∼∑j

pj ⇔ ∀M > 0,∃ `M such that

p−∑j≤`

pj ∈ CNS−Mρ,δ,k, ∀ ` ≥ `M .(1.6)

A symbol p is called classical, and we write p ∈ CNS`cl,k, if p ∼ p` + p`−1 + p`−2 + · · · , where

pj ∈ CNSj1,0,k, j = `, ` − 1, . . ., and pj is mixed homogeneous of degree j in (ξ, τ). Two

other subclasses of CNS`ρ,δ,k which are important for us are as follows. First, p ∈ CNS`ρ,δ,k is

called a Volterra symbol, and we write p ∈ V±CNS`ρ,δ,k if p extends holomorphically in τ toC∓(:= z ∈ C; Re z ≶ 0) and such that

|p(x, t, ξ, z)| ≤ CK(1 + |z|)M (1.7)

for some M > 0, uniformly for (x, t) ∈ K, compact subset of U × R, and |ξ| = 1.Second, call p ∈ V±CNS`ρ,δ,k a casual symbol, and we write p ∈ CNS`,±ρ,δ,k, if p = p(x, ξ, τ),

i.e. p is independent of t. It is clear that all these classes are stable undertaking asymptoticsums.

Later on, we shall need the following lemma, part of the classical Paley-Wiener Theorem.

Lemma 1.1. If p : R→ C extends to C∓ holomorphically and with polynomial growth, i.e.|p(z)| ≤ c(1 + |z|)N for some N ≥ 0, ∀z ∈ C∓, then F(p), the Fourier transform of p,vanishes on R±.

2 Anisotropic Pseudodifferential Operators

Here we shall introduce some classes of pseudodifferential operators associated with anisotropicsymbols.

Let U ⊆ Rm be open, d1, d2 ∈ N, k, ` ∈ R, N ∈ N, 0 ≤ 1 − ρ ≤ δ, ρ ≤ 1. We say thatp(x, t, ∂x, ∂t) ∈ OPCNS`ρ,δ,k(U × R; Hom (Cd1 ,Cd2)) if, for any u ∈ C∞comp(U ×R),

p(x, t, ∂x, ∂t)u(x, t)

=

∫R×Rm

eitτ+i〈x,ξ〉p(x, t, ξ, τ)(FtFxu)(ξ, τ) dξdτ(2.1)

for some symbol p(x, t, ξ, τ) ∈ CNS`ρ,δ,k(U × R × Rm × R; Hom (Cd1 ,Cd2)). Note that theintegral in (2.1) is absolutely convergent.

Remark 1. It is clear that the Schwartz kernel of p(x, t, ∂x, ∂t) in (2.1) is, modulo a normal-ization constant, (FξFτp)(x, t, y − x, s− t).

5

Remark 2. For ` ≤ 0, k > 0, we have, thanks to (1.3),

OPCNS`ρ,0,k(U × R) ⊆ OPCNS`/kminρ/k,1,0(U × R) (2.2)

where the latter class is that of ordinary pseudodifferential operators (with symbols in

S`/kminρ/k,1,0 and CN regularity in (x, t)).

As in the case of anisotropic symbols, we single out several important subclasses ofOPCNS`ρ,δ,k. First, p ∈ OPCNS`ρ,δ,k is called of Volterra type, and we write p ∈ V±OPCNS`ρ,δ,k,if for each T ∈ R it satisfies

supp [p(x, t, ∂x, ∂t)u(x, t)] ⊆ U × [T,∞) (or U × (−∞, T ], respectively),

whenever supp u(x, t) ⊆ U × [T,∞), (or U × (−∞, T ], respectively).

Second, p ∈ V±OPCNS`ρ,δ,k is called a casual pseudodifferential operator, and we write p ∈OPCNS`,±ρ,δ,k, if p(x, t, ξ, τ) is independent of t and p(x, t, ∂x, ∂t) commutes with translationsin time.

For the remainder of the paper we shall assume that

all anisotropic symbols are smooth and independent of t. (2.3)

Thus, N = ∞ and we shall drop the dependence on the class in the notation for symbolsand pseudodifferential operators. It is not hard to relax the smoothness assumption a bit,although the case when only a very limited amount of smoothness is assumed requires furtherarguments. We hope to return to this problem in a separate paper.

The classes OPS`ρ,δ,k, V±OPS`ρ,δ,k, turn out to be stable under composition. Also, taking

adjoints intertwines V+OPS`ρ,δ,k with V−OPS

`ρ,δ,k and OPS`,+ρ,δ,k with OPS

`,−ρ,δ,k. Moreover, if

OPS`ρ,δ,k 3 p(x, ∂x, ∂t)σprinc7−→ p(x, ξ, τ) ∈

S`ρ,δ,k

S`−(2ρ−1)ρ,δ,k

(2.4)

is the principal symbol map, then

σprinc(p1 p2) = σprinc(p1)σprinc(p2), ∀ pi ∈ OPS`iρ,δ,k, i = 1, 2. (2.5)

In particular,

σprinc :OPS`ρ,δ,k

OPS`−(2ρ−1)ρ,δ,k

∼−→S`ρ,δ,k

S`−(2ρ−1)ρ,δ,k

(2.6)

is an isomorphism.Everything extends to manifolds via partitions of unity and pull-backing. More specif-

ically, if Φ : U∼−→ V is a smooth diffeomorphism then Φ∗, acting as composition by Φ,

maps S`ρ,δ,k(V ×Rm×R) isomorphically onto S`ρ,δ,k(U ×Rm×R). Also, defining (Φ∗p)(u) :=[p(u Φ−1)] Φ, it follows that

Φ∗ : OPS`ρ,δ,k(U)∼−→ OPS`ρ,δ,k(V ) (2.7)

6

is an isomorphism and

σprinc(Φ∗p)(x, ξ, τ) = σprinc(p)(Φ−1(x), [dΦ−1(x)]tξ, τ). (2.8)

In other words, the principal symbol transforms covariantly under a change of coordinates.It should be noted here that in the class of classical pseudodifferential operators OPS`cl,k,the principal symbol map is unequivocally defined by

σprinc(p) := p` Φ−1 if the symbol of Φ∗p expands as p` + p`−1 + .... (2.9)

Throughout the paper M is going to be a smooth, compact, boundaryless, Riemannianmanifold of real dimension m and E,F → M are two complex vector bundles. We letE := pr∗E, F := pr∗F be the pull-backs of E and F , respectively, under the canonicalprojection pr : M×R→M . We can then define the class of symbols S`ρ,δ,k(M×R; E ,F) and

operators OPS`ρ,δ,k(M ×R; E ,F) in a natural fashion. For example, the latter is the class oflinear operators

p(x, ∂x, ∂t) : C∞(M × R, E)→ C∞(M × R,F) (2.10)

such that, when suitably localized and pulled back to an Euclidean domain U , they belongto OPS`ρ,δ,k(U ; Hom (Cd1 ,Cd2)), where d1 := rank E , d2 := rankF . In particular,

σprinc(p)(x, ξ, τ) ∈ Hom (Ex, Fx), x ∈M, ξ ∈ T ∗xM, τ ∈ R. (2.11)

Volterra and casual operators are defined similarly. A simple but useful observation iscontained in the proposition below.

Proposition 2.1. Let p ∈ OPS`ρ,δ,k(M × R; E ,F). Then p is of Volterra type if and only if

σprinc(p) ∈V±S`ρ,δ,k

V±S`−(2ρ−1)ρ,δ,k

. Moreover, p is casual if and only if σprinc(p) ∈S`,±ρ,δ,k

S`−(2ρ−1),±ρ,δ,k

.

We now make the standing assumption that M is Riemannian and E ,F are Hermitian.A symbol p ∈ OPS`ρ,δ,k(M × R; E ,F) is called strongly parabolic if E = F and

Re 〈σprinc(p)(x, ξ, τ)η, η〉 ≥ C(|ξ|+ |τ |1/k)`|η|2, (2.12)

uniformly for x ∈M , ξ ∈ T ∗xM , τ ∈ R with (ξ, τ) 6= 0 and η ∈ Ex, provided that |ξ|+ |τ |1/kis sufficiently large.

Proposition 2.2. Let Ω ⊆M be a smooth domain, −m < ` < −1 (recall that m := dimM),p ∈ OPS`,±ρ,0,k(M × R; E ,F). Denote by dσ the surface measure on ∂Ω.

For each u ∈ C∞comp(∂Ω×R, E), regard u(x, t) dσdt as a distribution on M ×R (supported

on ∂Ω×R). Then the section [p(x, ∂x, ∂t)u dσdt] |Ω×R extends to an element in (Ω×R,F).Moreover,

(pu)(x, t) := limx→ xx ∈ Ω

[p(x, ∂x, ∂t)u dσdt](x, t), (2.13)

7

x ∈ ∂Ω, t ∈ R, has the property that p ∈ OPS`+1,±ρ,0,k (∂Ω × R; E |∂Ω,F |∂Ω) and, in the sense

of equivalent classes,

σprinc(p)(x, ξ, τ) =1

2π

∫ +∞

−∞σprinc(p)(x, ξ + tν, τ) dt, (2.14)

for any x ∈M , ξ ∈ T ∗xM , τ ∈ R. Here ν is the outward unit conormal to ∂Ω.Furthermore, if p is homogeneous of degree d, then p is homogeneous of degree d+ 1, and

if p is strongly parabolic, then so is p.

Proof. In the class of ordinary pseudodifferential operators, a similar statement is essentiallywell-known and the proof can be adopted without difficulty to the present anisotropic setting(cf., e.g., [8, Vol. II, p. 34], [52]) to obtain that p ∈ OPS`+1

ρ,0,k(∂Ω × R; E |∂Ω,F |∂Ω) and that(2.14) holds.

Now, the fact that p is casual follows because p is so with the aid of Proposition 2.1.Also,

σprinc(p)(x, λξ, λkτ) =

1

2π

∫ +∞

−∞σprinc(p)(x, λξ + tν, λkτ) dt

= λd+1σprinc(p)(x, ξ, τ),

(2.15)

by a simple change of variables.Finally, if p is strongly parabolic then, for |ξ|+ |τ |1/k large,

Re 〈σprinc(p)(x, ξ, τ)η, η〉 =1

2π

∫ +∞

−∞Re 〈σprinc(p)(x, ξ + tν, τ)η, η〉dt

≥ C|η|2∫ +∞

−∞(|ξ + tν|+ |τ |1/k) dt

≈ C|η|2(|ξtan|+ |τ |1/k)`+1

≥ C|η|2(|ξ|+ |τ |1/k)`+1.

(2.16)

This finishes the proof of the proposition.

Next, call p ∈ OPS`,±ρ,δ,k(M ×R; E ,F) left-parabolic if σprinc(p) is left-invertible, i.e. there

exists q ∈ S−`,±ρ,δ,k such that

q σprinc(p)− I ∈ S−(2ρ−1),±ρ,δ,k . (2.17)

Similarly, one defines right-parabolic (casual) pseudodifferential operators and two-sided(casual) pseudodifferential operators. It follows (from definitions and (2.6)) that a stronglyparabolic casual pseudodifferential operator is two-sided parabolic.

A basic result for us in the sequel is as follows.

Theorem 2.3. If p ∈ OPS`,±ρ,δ,k(M×,R; E ,F) is left parabolic (right-parabolic, or two sided

parabolic, respectively) then there exists q ∈ OPS−`,±ρ,δ,k (M × R;F , E) such that p q = I (orq p = I, or both, respectively).

8

This is a version of Theorem 30 on p. 90 in [53]. In what follows, we denote q by p−1, if p istwo-sided parabolic.

Let L : C∞(M, E) → C∞(M, E) be a strongly elliptic, second order differential operatorwith smooth coefficients on M , and set P (x, ∂x, ∂t) := ∂t−L(x, ∂x) ∈ OPS2,+

1,0,2(M×R; E , E).

Then P is strongly parabolic and, hence, the inverse P−1 ∈ OPS−2,+1,0,2 (M ×R; E , E) exists, is

strongly parabolic and has a mixed homogeneity of degree −2.The single layer potential operator associated with P on ∂Ω is defined by

Sf := P−1(fdσdt) |∂Ω×R, f ∈ C∞comp(∂Ω× R, E). (2.18)

Then, invoking Proposition 2.2 we see that S ∈ OPS−1,+cl,2 (∂Ω × R; E) is strongly parabolic

and has a mixed homogeneity of degree −1. In particular, thanks to Theorem 2.3, for eachT > 0,

S : C∞(∂Ω× (0, T ), E)→ C∞(∂Ω× (0, T ), E) is onto (2.19)

(actually invertible). This is going to be of importance for us later on.

Next we shall study mapping properties of S, ∇S, D1/2t S, etc., in the case when S is

associated with a nonsmooth domain. We take up this task in the next four sections.

3 Boundedness Properties of Layer Potentials on Lip-

schitz Domains

Let E ,F →M be as in §2 and let p ∈ OPS−1,+cl,2 (M × R; E ,F) be such that σprinc(p)(x, ξ, τ)

is odd in ξ ∈ T ∗xM . Denote by k(p)(x, y, t, s) the Schwartz kernel of p and, for each ε > 0,set

Jεf(x, t) : =

∫ t−ε

0

∫∂Ω

〈k(p)(x, y, t, s), f(y, s)〉y dσ(y)ds,

(x, t) ∈ ∂Ω× R+,

(3.1)

where Ω ⊆M is a Lipschitz domain and dσ is the surface area measure on ∂Ω. Also, set

J∗f(x, t) := supε>0|Jεf(x, t)|, (x, t) ∈ ∂Ω× R+, (3.2)

and, for x 6∈ ∂Ω, t > 0,

Jf(x, t) :=

∫ t

0

∫∂Ω

〈k(p)(x, y, t, s), f(y, s)〉y dσ(y)ds. (3.3)

To state the main result of this section we need some more notation. Set Ω+ := Ω,Ω− := M \ Ω and, at each x ∈ ∂Ω, consider appropriate nontangential approach regionsγ±(x) ⊆ Ω±; cf. [30], [2] for more on this. Then, ·|∂Ω±×R will stand for the nontangentialboundary trace operators on ∂Ω± × R. That is, if u : Ω± × R→ E , then

9

u|∂Ω±×R(x, t) := limy → x

y ∈ γ±(x)

u(y, t), x ∈ ∂Ω, t ∈ R, (3.4)

whenever this exists. Also, the (parabolic) nontangential maximal function (·)∗ is definedfor sections u : Ω± × R→ E by

u∗(x, t) := sup |u(y, t)|; y ∈ γ±(x) x ∈ ∂Ω. (3.5)

Proposition 3.1. Let Ω ⊂M be a Lipschitz domain, and fix 0 < T <∞, 1 < p <∞. Thenthe following hold.

(i) There exists C = C(∂Ω, T, p) > 0 such that

‖J∗f‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E). (3.6)

(ii) For any f ∈ Lp(∂Ω× (0, T ), E) the limit

Jf(x, t) := limε→0

Jεf(x, t) (3.7)

exists at almost every (x, t) ∈ ∂Ω× (0, T ) and defines a bounded operator

J : Lp(∂Ω× (0, T ), E)→ Lp(∂Ω× (0, T ),F). (3.8)

(iii) There exists C = C(∂Ω, T, p) > 0 so that

‖(Jf)∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E). (3.9)

The strategy for proving (i) and (ii) is as follows:

Step 1. In local coordinates, the total symbol p(x, ξ, τ) can be decomposed as p = p−1 +p−2

where p−1 is the principal symbol and p−2 ∈ S−21,0,2 is regarded as residual.

Accordingly, we have the splitting k(p) = k−1 + k−2 for the kernel. Now, direct sizeestimates for k−2 give the right bounds for the contribution coming from this part of thekernel, as long as T > 0 is finite and 1 < p <∞.

Step 2. To handle the contribution from k−1 we first consider the L2 context. The pointis that we can work on the Fourier transform side (assuming T =∞); Plancherel’s theoremallows are to eventually return to the original setting.

In the process, a substantial piece of the operator (containing the delicate cancellations)can be handled via the elliptic theory from [1], where as the remainder can be controlled interms of maximal functions.

Step 3. Passing to the general case 1 < p < ∞ is then done more or less automaticallybased as the L2 analysis in Step 2 and Calderon-Zygmund theory in the context of spaces ofhomogeneous type. For this latter segment we need to check a cancelation condition for thekernel of Hormander type.

10

Finally, a Cotlar type inequality (relating J∗ and J) allows one to transfer the Lp-boundedness of J to J∗. The case of J is handled similarly.

Details of Step 1. First, it is standard that, locally,

k(p)(x, y, t, s) = cm

∫∫p(x, ξ, τ)ei〈x−y,ξ〉ei(t−s)τ dξdτ, (3.10)

where the integral is interpreted in the oscillatory sense. Estimates for the right side of(3.10) can be derived using the following lemma.

Lemma 3.2. Let p ∈ S`1,0,k with ` ∈ R and k ≥ 1. Then

∣∣∣∣∫∫ ξατβ∂γxp(x, ξ, τ)ei〈z,ξ〉eitτdξdτ

∣∣∣∣≤ Cα,β,γ[|z|+ |t|1/k]−(m+`+k+|α|+k|β|)

(3.11)

uniformly for x in compacts.

In particular, in the context of Step 1, Lemma 3.2 gives

|k−2(x, y, t, s)| ≤ C min

1

|t− s|m/2,

1

|x− y|m

. (3.12)

Since the right-side of (3.12) is absolutely integrable on ∂Ω × [0, T ] uniformly in (x, t) ∈∂Ω × [0, T ], Schur’s lemma shows that the integral operator with kernel k−2 is bounded onLp(∂Ω× (0, T )), as desired.

We now turn to the

Proof of Lemma 3.2. Let the double hat · stand for FξFτ . With this piece of notation,(3.11) reads

|∂γx∂αz ∂βtp(x, z, t)| ≤ Cα,β,γ[|z|+ |t|1/k]−(m+`+k+|α|+k|β|). (3.13)

Next, observe that x can be treated as a parameter and, hence, it suffices to treat the casewhen γ = 0 and p is independent of x, which we shall assume in the sequel.

In order to estimate ∂αz ∂βtp we make a parabolic Littlewood-Paley decomposition. Fix

η ∈ C∞comp(Rm × R) such that η(ξ, τ) = 1 for |ξ| + |τ |1/k ≤ 1 and η ≡ 0 for |ξ| + |τ |1/k ≥ 2.Also, set

δ(ξ, τ) := η(ξ, τ)− η(2ξ, 2kτ), (3.14)

i.e. δ := η−ηD where D is an “anisotropic” dilation operator. It follows that δ is supportedwhere 1

2≤ |ξ|+ |τ | 1k ≤ 2 and

1 = η(ξ, τ) +∞∑j=1

δ(2−jξ, 2−kjτ). (3.15)

Accordingly, write

11

p(ξ, τ) = p0(ξ, τ) +∞∑j=1

pj(ξ, τ), (3.16)

where

p0(ξ, τ) := p(ξ, τ)η(ξ, τ) and pj(ξ, τ) := p(ξ, τ)δ(2−jξ, 2−kjτ). (3.17)

Next, define

qj(ξ, τ) := p(2jξ, 2kjτ)δ(ξ, τ), j ≥ 1, (3.18)

so that pj(ξ, τ) = qj(2−jξ, 2−kjτ) and, consequently,

p(ξ, τ) = p0(ξ, τ) +∞∑j=1

qj(2−jξ, 2−kjτ). (3.19)

Now we make a general claim to the effect that

if p ∈ S`1,0,k and pr(ξ, τ) := r−`p(rξ, rkτ) then

prr≥1 is bounded in C∞(2−1 ≤ |ξ|+ |τ |1/k ≤ 2).(3.20)

To see this, simply note that

|∂αξ ∂βτ pr(ξ, τ)| = r`+|α|+k|β||(∂αξ ∂βτ p)(rξ, rkτ)|≤ Cα,β r

−`+|α|+k|β|(1 + r|ξ|+ r|τ |1/k)`−|α|−k|β|

≈ Cα,β,

(3.21)

uniformly for r ≥ 1, as long as 2−1 ≤ |ξ|+ |τ |1/k ≤ 2. In particular, (3.20) gives that

2−j`qj(ξ, τ)j≥1 is bounded in S(Rm × R), (3.22)

since δ(ξ, τ) has the effect of a cut off function in (3.18). In turn, (3.22) implies that for allα, β and N > 0,

|∂αz ∂βtqj(z, t)| ≤ CN,α,β 2j`(1 + |z|+ |t|1/k)−N , (3.23)

since 1 + |z|+ |t|1/k ≤ C(1 + |z|+ |t|). Now, from (3.19),

p(z, t) = p0(z, t) +∑j≥1

2jm2jkqj(2jz, 2jkt), (3.24)

so that

∂αz ∂βtp(z, t) = ∂αz ∂

βtp0(z, t)

+∑j≥1

2jm+jk+j|α|+jk|β|(∂αz ∂βtqj)(2jz, 2jkt). (3.25)

12

Introduce θ := log2(|z| + |t|1/k) and L := m + k + ` + |α| + k|β|. Then, by (3.23), thesum in (3.25) is bounded by

CN,α,β∑j≥1

2jL(1 + 2θ+j)−N = CN,α,β∑j′≥θ+1

2(j′−θ)L(1 + 2j′)−N

≤ CN,α,β2−θL∑j′≥1

2j′L(1 + 2j

′)−N

≤ CL,α,β2−θL

= CL,α,β[|z|+ |t|1/k]−(m+k+`+|α|+k|β|),

(3.26)

where the last inequality follows by choosing N sufficiently large. Clearly, this bound is of

the right order. Finally, since p0 ∈ C∞comp(Rm × R) it follows that p(z, t) ∈ S(Rm × R) and(3.11) is proved.

This completes Step 1 and now we provide the

Details of Step 2. Consider p ∈ S−11,0,2 such that p is odd in ξ and homogeneous of degree −1

in (ξ, τ). From (3.10) we know that locally

k(p)(x, y, t, s) = cm p(x, y − x, s− t), (3.27)

where, recall that the double hat · stands for the iterated Fourier transform FξFτ . Thus,

Jεf(x, t) = cm

∫R

∫∂Ω

⟨p(x, y − x, s− t)χ(−∞,−ε)(x− t), f(y, s)⟩dσ(y)ds. (3.28)

Since Jε is of convolution type in time, denoting · := Ft, we have

Jεf(x, τ) = cm

∫∂Ω

⟨∫ ∞ε

p(x, y − x, s)e−isτ ds , f(y, τ)⟩dσ(y)

=: I + II + III + IV + V,

(3.29)

where

13

I : = cm

∫1√|τ |≥ |x− y| ≥

√ε

y ∈ ∂Ω

⟨∫ ∞0

p(x, y − x, s)ds , f(y, τ)⟩dσ(y)

II : = cm

∫|x− y| ≤

√ε

y ∈ ∂Ω

⟨∫ ∞ε

p(x, y − x, s)e−isτds , f(y, τ)⟩dσ(y)

III : = −cm∫|x− y| ≥

√ε

y ∈ ∂Ω

⟨∫ ε

0

p(x, y − x, s)e−isτds , f(y, τ)⟩dσ(y)

IV : = cm

∫|x− y| ≥ max

√ε, 1√

|τ |

y ∈ ∂Ω

⟨∫ ∞0

p(x, y − x, s)e−isτds , f(y, τ)⟩dσ(y)

V : = cm

∫1√|τ |≥ |x− y| ≥

√ε

y ∈ ∂Ω

⟨∫ ∞0

p(x, y − x, s)(e−isτ − 1)ds , f(y, τ)⟩dσ(y).

(3.30)

The claim is that

∥∥∥ sup1|τ |>ε>0

|I|∥∥∥L2(∂Ω)

≤ C‖f(·, τ)‖L2(∂Ω),

uniformly for τ ∈ R,(3.31)

and

|II|+ |III|+ |IV | ≤ C(M∂Ωf(·, τ)|)(x),

uniformly for x ∈ ∂Ω and τ ∈ R,(3.32)

where M∂Ω stands for the Hardy-Littlewood maximal function on ∂Ω. Of course, granted(3.31)–(3.32), the conclusion in Step 2 follows from the boundedness of M∂Ω on L2(∂Ω) andPlancherel’s Theorem.

We now check (3.31)–(3.32) starting with the first. To this end, let

b(x, z) :=

∫ ∞0

p(x, z, s) ds. (3.33)

The fact that p(x, ξ, τ) is odd in ξ entails that p(x, z, s) is odd in z and, further, that b(x, z)is odd in z. Also, since p is mixed homogeneous of degree −1, a simple calculation showsthat

b(x, λz) = λm−1b(x, z), (3.34)

i.e. b is homogeneous of degree (m − 1) in z. Hence, (3.31) follows from the elliptic theoryin [2], [1].

In order to check (3.32), we shall use the estimates

14

|p(x, y − x, s)| ≤ C(|x− y|+ s1/2)−(m+1),

|∂sp(x, y − x, s)| ≤ C(|x− y|+ s1/2)−(m+3).(3.35)

They both follow from Lemma 3.2 since p ∈ S−11,0,2. Based on these, we may write

|II| ≤ C

∫|x− y| ≤

√ε

y ∈ ∂Ω

(∫ ∞ε

1

s12

(m+1)ds

)|f(y, τ)| dσ(y)

≤ C1

(√ε)m−1

∫|x− y| ≤

√ε

y ∈ ∂Ω

|f(y, τ)| dσ(y) ≤ C(M∂Ωf(·, τ)|)(x).

(3.36)

Also,

|III| ≤ C∞∑k=0

(∫2k√ε≤|x−y|≤2k+1

√ε

∫ ε

0

ds

|x− y|m+1|f(y, τ) |dσ(y)

)≤ C

∞∑k=0

2−2k+m−1 1

(2k+1√ε)m−1

∫|x−y|≤2k+1

√ε

|f(y, τ)| dσ(y)

≤ C( ∞∑k=0

2−2k+m−1)

(M∂Ω|f(·, τ)|)(x)

≤ C(M∂Ω|f(·, τ)|)(x).

(3.37)

To estimate |IV |, introduce

Fx,y(τ) :=

∫ ∞0

p(x, y − x, s)e−isτ ds. (3.38)

The decay of Fx,y(τ) as |τ | → ∞ can be justified via the classical Lebesgue-Riemann lemma.Specifically,

τFx,y(τ) = −ip(x, y − x, 0) + i

∫ ∞0

∂sp(x, y − x, s)e−isτ ds, (3.39)

via an integration by parts and the fact that lims→∞ p(x, y − x, s) = 0; cf. (3.35). Then, itfollows from (3.39), (3.35) and some algebra that

|Fx,y(τ)| ≤ C|τ |−1|x− y|−(m+1). (3.40)

Consequently,

15

|IV | ≤ C

∞∑k=0

∫2k√|τ |≤|x−y|≤ 2k+1√

|τ |

|Fx,y(τ)||f(y, τ)| dσ(y)

≤ C∞∑k=0

∫2k√|τ |≤|x−y|≤ 2k+1√

|τ |

( 2k√|τ |

)−(m+1)

|f(y, τ)| dσ(y)

≤ C

[∞∑k=0

1

|τ |

( 2k√|τ |

)−(m+1)( 2k√|τ |

)m−1]

(M∂Ω|f(·, τ)|)(x)

≤ C(M∂Ω|f(·, τ)|)(x).

(3.41)

Finally, to estimate |V |, we use |eisτ − 1| ≤ Cs|τ | and (3.35) to write

|V | ≤ C

∫|x−y|≤ 1√

|τ |

|τ |(∫ ∞

0

s

(|x− y|+ s1/2)m+1ds

)|f(y, τ)| dσ(y)

≈ C

∫|x−y|≤ 1√

|τ |

(|τ |

|x− y|m−3

)|f(y, τ)| dσ(y)

≤ C∞∑k=0

∫2−k−1√|τ |≤|x−y|≤ 2−k√

|τ |

|τ ||x− y|m−3

|f(y, τ)| dσ(y)

≤ C

[∞∑k=0

|τ |( 2−k√|τ |

)−(m−3)( 2−k√|τ |

)m−1]

(M∂Ω|f(·, τ)|)(x)

≤ C(M∂Ω|f(·, τ)|)(x).

(3.42)

This concludes the detailed presentation of Step 2.

Next we give the

Details of Step 3. First we concentrate on the Hormander type estimate

∫∫∂Ω×R\[S2r(y)×(s−4r2,s+4r2)]

|p(x, y − x, s− t)− p(x, y′ − x, s′ − t)| dσ(x)dt ≤ C, (3.43)

which, so we claim, is valid whenever

|y′ − y|+ |s′ − s|1/2 ≤ r, (3.44)

uniformly for r > 0, x, y, y′ ∈ ∂Ω, s, s′, t ∈ R. Hereafter, Sr(y) ⊆ ∂Ω, y ∈ ∂Ω, r > 0, willdenote the surface ball of radius r centered at y.

The domain of integration in (3.43) is covered by D1 ∪D2 where

16

D1 : = S2r(y)× [R\(s− 4r2, s+ 4r2)],

D2 : = [∂Ω\S2r(y)]× R.(3.45)

In D1 we shall simply use size estimates for p, whereas in D2 we shall apply the Mean-ValueTheorem. More concretely, granted (3.44) it is easy to check that

|s′ − t| ≥ c|s− t|, uniformly for (x, t) ∈ D1. (3.46)

This and (3.35) then give

|p(x, y − x, s− t)|, |p(x, y′ − x, s′ − t)| ≤ C

|s− t|(m+1)

2

(3.47)

on the domains of integration, provided (3.44) holds. Thus,

∫∫D1

|p(x, y − x, s− t)− p(x, y′ − x, s′ − t)| dσ(x)dt

≤ C

∫∫D1

dσ(x)dt

|s− t|(m+1)

2

≤ Crm−1

∫ ∞4r2

dt

t(m+1)

2

= Cm,(3.48)

as desired. There remains the contribution from integrating over D2. First, by the Mean-Value Theorem and (3.13),

|p(x, y − x, s− t)− p(x, y′ − x, s′ − t)|≤ |p(x, y − x, s− t)− p(x, y − x, s′ − t)|

+ |p(x, y − x, s′ − t)− p(x, y′ − x, s′ − t)|≤ |s− s′| sup

θ∈[s,s′]

|∂3p(x, y − x, θ − t)|

+ |y − y′| supz∈[y,y′]

|∂2p(x, z − y, s′ − t)|

≤ C supθ∈[s,s′]

r2

(|x− y|+ |θ − t| 12 )m+3

+ C supz∈[y,y′]

r

(|x− z|+ |s′ − t| 12 )m+2,

(3.49)

where (3.44) has also been used in the last step.Going further, note that,

17

∫∂Ω\S2r(y)

(∫ +∞

−∞

r2

[|x− y|+ |θ − t| 12 ]m+3dt)dσ(x)

≤ Cr2

∫|x| ≥ 2rx ∈ Rm−1

(∫ +∞

−∞

dt

[|x|+ |t| 12 ]m+3

)dx

= Cr2(∫

|x| ≥ 2rx ∈ Rm−1

dx

|x|m+1

)(∫ +∞

−∞

dt

[1 + |t| 12 ]m+3

)= Cm,

(3.50)

which has the right form. Also, granted (3.44), |x − z| ≥ c|x − y| uniformly for z ∈ [y, y′]and x ∈ ∂Ω\S2r(y) so that

∫∂Ω\S2r(y)

(∫ +∞

−∞

r

[|x− z|+ |s′ − t| 12 ]m+2dt)dσ(x)

≤ Cr

∫∂Ω\S2r(y)

(∫ +∞

−∞

dt

[|x− y|+ |t| 12 ]m+2

)dσ(x)

≤ Cr

∫|x| ≥ 2rx ∈ Rm−1

(∫ +∞

−∞

dt

[|x|+ |t| 12 ]m+2

)dx

= Cr(∫

|x| ≥ 2rx ∈ Rm−1

dx

|x|m)(∫ +∞

−∞

dt

[1 + |t| 12 ]m+2

)= Cm.

(3.51)

This finishes the proof of (3.43).Given what we have proved so far, all the conclusions in Proposition 3.1 follow by routine

arguments. Here we only outline the main steps. First, regarding ∂Ω × R as a space ofhomogeneous type (equipped with the quasi-distance |x− y|+ |t− s| 12 ) one concludes, basedon [54, Theorem 3, p. 19] that J−1, the integral operator corresponding to the kernel k−1, isbounded from Lp(∂Ω × R, E) into Lp(∂Ω × R,F) for each 1 < p ≤ 2. The class of formaladjoints of operators like J−1 can also be handled along similar lines since the correspondingsymbols are amenable to virtually an identical treatment (we refer the interested reader to[2], [1] for a more detailed discussion in the elliptic case). The bottom line is that our resultcan be dualized, so that J−1 turns out to be bounded on Lp for 2 ≤ p <∞ also.

Now, Cotlar’s inequality (cf. [34, Theorem 4.5, p. 70], for a relevant discussion) yieldsthat J∗ : Lp(∂Ω× R, E)→ Lp(∂Ω× R) is bounded for each 1 < p <∞.

Finally, the boundedness of this maximal operator together with the existence of the a.e.pointwise limit for a dense subclass of Lp(∂Ω×R, E) (e.g., C∞comp(∂Ω×R, E) will do) entailsthe pointwise existence of limε→0 Jεf a.e. on ∂Ω×R, for each f ∈ Lp(∂Ω×R, E), 1 < p <∞.

This proves (i)− (ii) in Proposition 3.1.

We now tackle the

Proof of (iii) in Proposition 3.1. Again, decompose the total symbol p = p−1 + p−2 withp−1 ∈ S−1

1,0,2, p−2 ∈ S−21,0,2, so that k(p) = k−1 + k−2 and J = J−1 + J−2.

18

Next, for each 0 < T <∞, 1 < p <∞, the estimate

‖(J−2f)∗‖Lp(∂Ω×(0,T )) ≤ C(∂Ω, T, p)‖f‖Lp(∂Ω×(0,T ),E) (3.52)

can be established directly, by reducing matters to the analysis of a convolution type operatorwith an absolutely integrable kernel. There remains the contribution from J−1. Below weassume that p ∈ S−1

1,0,2 drop the index −1. Fix 1 < p < ∞, f ∈ Lp(∂Ω × R, E), x0 ∈ ∂Ω,x ∈ γ(x0) (the nontangential approach region with “vertex” at x0), ε := |x − x0| anddecompose

Jf(x, t) = (−J2εf(x0, t) + Jf(x, t)) + J2εf(x0, t). (3.53)

Now |J2εf(x0, t)| ≤ J∗f(x0, t) and, thus, by (i) in Proposition 3.1, the contribution fromthis term has the appropriate control. As for the first term in the right side of (3.53), wefirst estimate the contribution of the piece of Jf(x, t) corresponding to integrating near x0.More concretely, consider∫ t

−∞

∫|y − x0| ≤ 2εy ∈ ∂Ω

|k(p)(x, y, t, s)||f(y, s)| dσ(y)ds. (3.54)

Note that |x− y| ≥ C dist(x, ∂Ω) ≈ |x− x0| = ε. Thus, since |k(p)(x, y, t, s)| ≤ C(|x− y|+|t− s| 12 )−(m+1), it suffices to bound

∫y∈Rm−1

∫ +∞

−∞χB2(0)

(x0 − yε

)×

× χ(0,∞)

(t− sε2

)ε−(m+1)

(1 +

∣∣∣t− sε2

∣∣∣ 12)−(m+1)

|f(y, s)| dyds.(3.55)

To this end, we shall employ the following lemma

Lemma 3.3. Let Φ ∈ L1(Rm−1 × R) be so that

Φ(x, t) := sup |Φ(x, t)|; x ∈ Rm−1, t ∈ R, |x| ≥ x|, |t| ≥ |t|

is integrable on Rm−1 × R. Also, introduce the maximal function

Mf(x, t) := supr>0,µ>0

1

µ

1

rm−1

∫|t− s| ≤ µs ∈ R

∫|x− y| ≤ ry ∈ Rm−1

|f(y, s)| dyds. (3.56)

Then there exists C > 0 so that, for each x, t,

supr>0,µ>0

∣∣∣∣ 1µ 1

rm−1

∫R

∫Rm−1

Φ(x− y

r,t− sµ

)f(y, s) dyds

∣∣∣∣ ≤ C‖Φ‖L1Mf(x, t). (3.57)

19

The proof of lemma is an exercise; cf. [55, Theorem 2, p. 62]. For a related version see also[34, Lemma 4.4, p. 70].

Returning to the analysis of (3.55) and taking

Φ(x, s) := χB2(0)(x)χ(0,∞)(s)(1 + |s|12 )−(m+1) and r := ε, µ := ε2,

Lemma 3.3 gives that the integral in (3.55) is ≤ CMf(x0, t), uniformly in ε. Again, sinceM is bounded on Lp(Rm−1 ×R), this leads to a bound of the right order. At this point, weare left with estimating the contribution of∫ t

0

∫|y − x0| ≥ 2εy ∈ ∂Ω

|k(p)(x, y, t, s)− k(p)(x0, y, t, s)||f(y, s)| dσ(y)ds. (3.58)

Now

|k(p)(x, y, t, s)− k(p)(x0, y, t, s)| ≤ |p(x, y − x, s− t)− p(x0, y − s, s− t)|+ |p(x0, y − x, s− t)− p(x0, y − x0, s− t)|=: A+B.

(3.59)

We regard part A as residual (here we use 0 < t < T ). Note that

|A| ≤ C|x− x0| supw∈[x,x0]

|∂1p(w, y − x, s− t)|

≤ C|x− x0|1

(|x− y|+ |s+ t| 12 )m+1,

(3.60)

by (3.13). Observe that on the domain of integration in (3.58), we have |x − y| ≈ |y − x0|,due to the definition of ε. Also, |x− x0| = ε ≤ 1

2|y − x0|. Thus,

∫ t

0

∫|y−x0|≥2ε

|A||f(y, s)| dσ(y)ds

≤∫ T

0

∫∂Ω

|y − x0|(|y − x0| − |s− t|

12 )m+1

|f(y, s)| dσ(y)ds

(3.61)

since the last double integral above is a convolution type operator with an absolutely inte-grable kernel (evaluated at (x0, t)), it follows that the contribution fromA in ‖(Jf)∗‖Lp(∂Ω×(0,T ))

is ≤ C‖f‖Lp(∂Ω×(0,T ),E).As for the contribution from B, first note that, if |y − x0| ≥ 2ε, then

|B| ≤ C|x− x0| supw∈[x,x0]

|∂2p(x0, y − w, s− t)|

≤ Cε

[|y − x0|+ |t− s|12 ]m+2

,(3.62)

20

where we have used the fact that |x−x0| = ε and |w− y| ≈ |x0− y| uniformly for w ∈ [x, x0]and |y − x0| ≥ 2ε. Thus,

∫ t

−∞

∫|y−x0|≥2ε

|B||f(y, s)| dσ(y)ds

≤ C

∫y∈Rm−1

∫s∈R

1

εm−1

1

ε2χRm−1\B2(0)

(y − x0

ε

)χ(0,∞)

(t− sε2

)×

× 1(∣∣y−x0

ε

∣∣+∣∣ t−sε2

∣∣ 12

)m+2 |f(y, s)| dyds

≤ C

(∫Rm−1

∫R

χRm−1\B2(0)(y)χ(0,∞)(s)ds dy

[|y|+ |s| 12 ]m+2

)Mf(x0, t)

≤ CMf(x0, t),

(3.63)

where the second inequality utilizes Lemma 3.3. SinceM is bounded on Lp(Rm−1×R), thelast bound has the right size.

This completes the proof of (iii) in Proposition 3.1.

4 Jump Relations

Consider P ∈ S−1,+cl,2 (M ×R; E ,F) whose principal symbol p is odd in ξ, and let k = k(p) be

the Schwartz kernel of p. Recall that this implies

k(x, y, s, t) = cm p(x, y − x, s− t). (4.1)

Also, denote by d(x, y) the geodesic distance between x, y ∈M .

Lemma 4.1. Let Ω be a Lipschitz domain, Ω ⊆ M , 0 < T < ∞, 1 < p < ∞, f ∈Lp(∂Ω× (0, T ), E). Then for a.e. x ∈ ∂Ω, t ∈ (0, T ),

limε→0

∣∣∣∣∫ t−ε

0

∫∂Ω

〈k(x, y, s, t), f(y, s)〉 dσ(y)ds

−∫ t

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈k(x, y, s, t), f(y, s)〉 dσ(y)ds

∣∣∣∣∣∣ = 0.

(4.2)

Proof. Let

Aεf(x, t) : =

∫ t−ε

0

∫d(x, y) ≤

√ε

y ∈ ∂Ω

〈k(x, y, s, t), f(y, s)〉 dσ(y)ds, (4.3)

Bεf(x, t) : =

∫ t

t−ε

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈k(x, y, s, t), f(y, s)〉 dσ(y)ds. (4.4)

21

We shall prove that:

limε→0

Aεf(x, t) = 0 ∀ f in a dense subclass of Lp(∂Ω× (0, T ), E), (4.5)

limε→0

Bεf(x, t) = 0 ∀ f in a dense subclass of Lp(∂Ω× (0, T ), E), (4.6)

‖ supε>0|Aεf |‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E), (4.7)

‖ supε>0|Bεf |‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E). (4.8)

Then the lemma follows from (4.5)-(4.8) in the usual fashion.

Proof of (4.5). Step I. Work in local coordinates and decompose p = p−1 + p−2 wherep−1 ∈ S−1,+

1,0,2 , p−2 ∈ S−2,+1,0,2 . Accordingly, k = k−1 + k−2 and, further, Aε = Aε,−1 + Aε,−2,

with some self-explanatory notation. Since k−2 has an absolutely integrable singularity,Aε,−2f(x, t) → 0 by Lebesgue’s Dominated Convergence Theorem. Hence, it is enough toconsider Aε,−1; in what follows, we drop the subindex −1.

Step II. As in [2, Appendix B],

limε→0

Aεf(x, t) = limε→0

∫ t−ε

0

∫|x− y| ≤

√ε

y ∈ ∂Ω

〈k(x, y, t, s), f(y, s)〉 dσ(y)ds, (4.9)

i.e. d(x, y) can be replaced by |x− y|, the Euclidean distance.

Step III. Assume that x = (0, 0) ∈ Rm−1 × R, ∂Ω is the graph of a Lipschitz functionϕ : Rm−1 → R such that ϕ(0) = 0, ∇ϕ(0) = 0. In particlar, there exists ω ∈ L∞, ω ≥ 0 suchthat

|ϕ(y)| ≤ |y|ω(|y|), ‖ω‖L∞ ≤ ‖∇ϕ‖L∞ , limλ→0+

ω(λ) = 0. (4.10)

Change coordinates so that we work on Rm−1 in place of ∂Ω, absorb the Jacobian of thetransformation (∈ L∞) into f and, at that stage, restrict attention to functions f of the typeu(x)v(t) with u, v ∈ C∞comp (note that the linear span of this class is a dense subspace of Lp).Adding and subtracting u(0) in the integral (and utilizing the smoothness of u), matters canfinally be reduced to showing that

limε→0

∫ t−ε

0

∫|y| ≤

√ε

y ∈ Rm−1

k((0, 0), (y, ϕ(y)), s, t) dyds = 0. (4.11)

What is crucial is the fact that k((0, 0), (y, 0), s, t) = p((0, 0), (y, 0), s − t) is odd in y. Inparticular, ∫ t−ε

0

∫|y| ≤

√ε

y ∈ Rm−1

k((0, 0), (y, 0), s, t) dyds = 0. (4.12)

22

Thus, it suffices to analyze∫ t−ε

−∞

∫|y| ≤

√ε

y ∈ Rm−1

|k((0, 0), (y, ϕ(y)), s, t)− k((0, 0), (y, 0), s, t)|dyds. (4.13)

The integrant is, by the Mean-Value Theorem,

≤ C|ϕ(y)| supθ∈[0,ϕ(y)]

|∂2k((0, 0), (y, θ), s, t)|

≤ C|y|ω(|y|) 1

[|y|+ |s− t| 12 ]m+2,

(4.14)

where the last inequality above utilizes (4.10), (4.1) and (3.13). Thus, the integral in (4.13)is dominated by

∫ t−ε

−∞

∫|y| ≤

√ε

y ∈ Rm−1

|y|ω(|y|)(|y|+ |s− t| 12 )m+2

dyds ≤∫ ∞ε

∫|y| ≤

√ε

y ∈ Rm−1

|y|ω(|y|)(|y|+ s

12 )m+2

dyds. (4.15)

Making the change of variables y =√ε y′, s = εs′, the last integral in (4.15) is

≤ C

(∫ ∞1

1

|s′|m+22

ds′)(∫

|y′|≤1

ω(√ε|y′|) dy′

), (4.16)

i.e. o(1) as ε→ 0 by Lebesgue’s Dominated Convergence Theorem. This proves (4.5).

Proof of (4.6). This is pretty similar to that of (4.5). Once again, following the same re-duction procedure as before, the crux of the matter is showing that

limε→0

∫ t

t−ε

∫|y| ≥

√ε

y ∈ Rm−1

k((0, 0), (y, ϕ(y)), t, s) dyds = 0. (4.17)

Again, it suffices to show that actually

limε→0

∫ t

t−ε

∫|y| ≥

√ε

y ∈ Rm−1

|k((0, 0), (y, ϕ(y)), t, s)− k((0, 0), (y, 0), t, s)| dyds = 0. (4.18)

As in (4.14), the above integrand is ≤ C|y|ω(|y|)[|y| + |s − t| 12 ]−(m+2) so that we are led toconsider ∫ ε

0

∫|y| ≥

√ε

y ∈ Rm−1

|y|ω(|y|)(|y|+ s

12 )m+2

dyds. (4.19)

Making the change of variables y =√ε y′, s = εs′, this becomes

23

∫ 1

0

∫|y′| ≥ 1

y′ ∈ Rm−1

|y′|ω(√ε|y′|)

(|y′|+ |s′| 12 )m+2dy′ds′

≤ C

∫|y′| ≥ 1

y′ ∈ Rm−1

ω(√ε|y′|)

|y′|m+1dy′ = o(1) as ε→ 0,

(4.20)

by Lebesgue’s Dominated Converge Theorem. This takes care of (4.6).

Proof of (4.7). Working in local coordinates and pull-backing ∂Ω to Rm−1 a direct estimateon Aε gives

|Aεf(x, t)|

≤ C1

ε

1

(√ε)m−1

∫R

∫Rm−1

χB1(0)

(x− y√ε

)χ(1,∞)

(t−sε

)|f(y, s)|[∣∣∣x−y√ε ∣∣∣+

∣∣ t−sε

∣∣ 12

]m+1 dyds.(4.21)

Hence Lemma 3.3 with

Φ(y, s) := χB1(0)(y)χ(1,∞)(s)[|y|+ |s|12 ]−(m+1) ∈ L1(Rm−1 × R)

applies and gives that

supε>0|Aεf(x, t)| ≤ CMf(x, t). (4.22)

From this, (4.7) follows.

Proof of (4.8). This is virtually identical to that of (4.7) except that, this time, we need totake

Φ(y, s) := χRm−1\B1(0)(y)χ(0,1)(s)[|y|+ |s|12 ]−(m+1) ∈ L1(Rm−1 × R). (4.23)

Thus, by the same token,

supε>0|Bεf(x, t)| ≤ CMf(x, t). (4.24)

Now (4.8) follows from this, thanks to the boundedness ofM on Lp. This finishes the proofof (4.8) and, with it, the proof of Lemma 4.1.

After these preparations, we are ready to discuss the main result of this section.

Theorem 4.2. Let Ω ⊆M be a Lipschitz domain with outward unit conormal ν, 0 < T <∞,1 < p <∞. Also, let p ∈ S−1,+

cl,2 (M ×R; E ,F) have principal symbol p−1 which we assume isodd in ξ. Let J be associated with p and Ω as in (3.3). Then for each f ∈ Lp(∂Ω× (0, T ), E)and at a.e. w ∈ ∂Ω, t ∈ (0, T ),

limx→ w

x ∈ γ±(w)

Jf(x, t) = ∓12i p−1(w, ν(w), 0)f(w, t) + lim

ε→0Jεf(w, t). (4.25)

24

Proof. Given the results in §3, it suffices to assume that f ∈ C∞comp(∂Ω × (0, T ); E). In thiscase matters can be reduced to proving (4.25) on the Fourier transform side in time, i.e. toshow that for a.e. w ∈ ∂Ω, τ ∈ R,

limx→ w

x ∈ γ±(w)

Jf(x, τ) = ∓12i p−1(w, ν(w), 0)f(w, τ) + lim

ε→0Jεf(w, τ), (4.26)

where hat denotes Ft. Indeed, integrating (4.26) against ϕ(τ)dτ , for τ ∈ R, (where ϕ ∈C∞comp(R) is arbitrary) and using Plancherel’s formula allows us to return to (4.25). Moreover,it is clear that we can assume that J is associated with p−1; in the sequel, we drop thesubscript −1. Now,

Jf(x, τ) =

∫∂Ω

⟨∫ ∞0

p(x, y − x, s)e−isτ ds , f(y, τ)⟩dσ(y)

=

∫∂Ω

⟨∫ ∞0

p(x, y − x, s)[e−isτ − 1] ds , f(y, τ)⟩dσ(y)

+

∫∂Ω

⟨∫ ∞0

p(x, y − x, s)ds , f(y, τ)⟩dσ(y)

=: A(x, τ) +B(x, τ).

(4.27)

Next, as x→ w,

A(x, τ) −→∫∂Ω

⟨∫ ∞0

p(w, y − w, s)[e−isτ − 1] ds , f(y, τ)⟩dσ(y), (4.28)

by Lebesgue’s Dominated Convergence Theorem. To justify this, note that by the elementaryestimate

|eisτ − 1| ≤ C min|τ ||s|, 1 (4.29)

and the fact that |x− y| ≥ C|y − w| uniformly for w, y ∈ ∂Ω, x ∈ γ±(w), we have

|p(x, y − x, s)[e−isτ − 1]| ≤ Cτ|s| 12

[|w − y|+ |s| 12 ]m+1(4.30)

and

∫∂Ω

∫ ∞0

|s| 12(|w − y|+ |s| 12 )m+1

dsdσ(y)

≤(∫

∂Ω

1

|w − y|m−2dσ(y)

)(∫ ∞0

|s′| 12(1 + |s′| 12 )m+1

ds′)< +∞.

(4.31)

Going further, if b(x, z) :=∫∞

0p(x, z, s) ds then b(x, z) is odd and homogeneous of degree

(−m+ 1) in z. The elliptic theory from [2],[1] applies in this case to give

25

limx→ w

x ∈ γ±(w)

B(x, τ) =∓ 12i(Fzb)(w, ν(w))f(w, τ)

+ limε→0

∫d(w, y) ≥

√ε

y ∈ ∂Ω

⟨∫ ∞0

p(w, y − w, s) ds , f(y, τ)⟩dσ(y).

(4.32)

Now, by assumptions and Lemma 1.1, p vanishes for s < 0. In particular, b(x, z) =∫ +∞−∞

p(x, z, s) ds = (Fξp)(x, z, 0) so that (Fzb)(x, ξ) = p(x, ξ, 0). Thus, the jump term in(4.32) agrees with that of (4.26). Combining the right side of (4.28) with the limit in theright side of (4.32) we obtain

limε→0

∫d(w, y) ≥

√ε

y ∈ ∂Ω

⟨∫ +∞

−∞

p(w, y − w, s)e−isτds , f(y, τ)⟩dσ(y)

= Ft

limε→0

∫d(w, y) ≥

√ε

y ∈ ∂Ω

∫ +∞

−∞〈p(w, y − w, t− s), f(y, s)〉 dsdσ(y)

(τ)

= Ft

limε→0

∫d(w, y) ≥

√ε

y ∈ ∂Ω

∫ t

0

〈p(w, y − w, t− s), f(y, s)〉 dsdσ(y)

(τ)

= Ft[limε→0

∫ t−ε

0

∫∂Ω

〈k(w, y, t, s), f(y, s)〉 dσ(y)ds

](τ)

= Ft[limε→0

Jεf(w, t)](τ) = limε→0

Jεf(w, τ),

(4.33)

as desired. Note that, in the second equality, the support conditions on p and f have beenused. Also, Lemma 4.1 is invoked in the third equality. Finally commuting Ft with limε→0

can be done due to, e.g., the continuity of Ft in L2.This finishes the proof of Theorem 4.2.

5 Fractional Time-Derivative Layer Potentials

Let f ∈ L1(−∞, T ) which decays fast enough at −∞. For 0 < σ ≤ 1, introduce the fractionalintegral operator of order σ (i.e. the one-dimensional Riemann-Liouville integral; cf. [56,p. 217]), by

Iσf(t) :=1

Γ(σ)

∫ t

−∞

f(s)

(t− s)1−σ ds, (5.1)

where Γ is the usual Euler’s gamma function. Hence,

Iσf =1

Γ(σ)(tσ−1χ(0,∞)) ∗ f. (5.2)

26

Finally, for σ = 0, take I0 to be the identity operator.Next, define the family of fractional time-derivative operators by

Dσt :=

∂

∂tI1−σ, 0 ≤ σ ≤ 1. (5.3)

In particular, D1t = ∂

∂t, the ordinary time-derivative. Some of the immediate properties of

the operators Iσ, Dσt are summarized below.

Lemma 5.1. For each 0 < σ < 1, the following hold:

(i) the operators Iσ, Dσt are casual, in the sense that they commute with translations and

if f ≡ 0 for t ≤ t0 then Dσt f, Iσf ≡ 0 for t ≤ t0;

(ii) Dσt [f(λt)] = λσ(Dσ

t f)(λt) and Iσ[f(λt)] = λσ−1(Iσf)(λt);

(iii) Dσ1t ·Dσ2

t = Dσ1+σ2t and Iσ1 · Iσ2 = Iσ2+σ2 for 0 ≤ σ1, σ2 ≤ 1, σ1 + σ2 ≤ 1;

(iv) Dσt (f ∗ g) = (Dσ

t f) ∗ g = f ∗ (Dσt g) and Iσ(f ∗ g) = (Iσf) ∗ g = f ∗ (Iσg);

(v) Dσt f(τ) = (2π)σ|τ |σ 1√

2(1 + i sign(τ))f(τ);

(vi) Iσf(τ) = −i(2π)1−σ |τ |1−στ

1√2(1 + i sign(τ))f(τ).

For proofs see, e.g., [56, p. 217], [34, p. 35], and [41, p. 40-41].

To state our next result recall that, as before, · denotes the iterated Fourier transformFξFτ .

Proposition 5.2. Assume that m + ` + k > 1. If p ∈ S`,−1,0,k then, for each 0 < σ < 1 andα, β, γ, there holds

|Iσ∂γx∂αz ∂βtp(x, z, t)| ≤ Cα,β,γ,σ

tσ

|z|m+`+k+|α|+k|β|

[min

1,|z|k

t

]|β|+1

, (5.4)

∀ z, ∀ t > 0, uniformly for x in compacts. Also,

|DσtD

γx∂

αz ∂

βtp(x, z, t)| ≤ Cα,β,γ,σ

t1−σ

|z|m+`+2k+|α|+k|β|

[min

1,|z|k

t

]|β|+2

, (5.5)

∀ z, ∀ t > 0, uniformly for x in compacts.

Proof. Invoke Lemma 1.1 and decompose

Iσ∂γx∂

αz ∂

βtp(x, z, t) =

1

Γ(σ)

∫ t

0

∂γx∂αz ∂

βsp(x, z, s) 1

(t− s)1−σ ds

=1

Γ(σ)

∫ t2

0

(· · · ) ds+1

Γ(σ)

∫ t

t2

(· · · ) ds =: I + II.

(5.6)

To estimate I, we integrate by parts ν times, 0 ≤ ν ≤ β, and write

27

I =ν∑j=1

cj,σ(−1)j+1(t2

)j−σ ∂γx∂αz ∂β−jtp(x, z, t

2

)+ cν,σ

∫ 12

0

∂γx∂αz ∂

β−νsp(x, z, s) 1

(t− s)ν−σ+1ds.

(5.7)

Estimating each ∂γx∂αz ∂

β−νsp according to (3.13) then gives

|I| ≤ C

ν∑j=1

1

tj−σ1

[|z|+ |t| 1k ]m+`+k+|α|+k(|β|−j)

+ C1

tν−σ+1

∫ t

0

ds

[|z|+ |s| 1k ]m+`+k+|α|+k(|β|−ν)

≤ C1

tν−σ+1

∫ t

0

ds

[|z|+ |s| 1k ]m+`+k+|α|+k(|β|−ν),

(5.8)

where the last inequality follows by observing that a multiple of the last expression abovedominates each term of the sum in (5.8). Going further, we have

|II| ≤[

supt2≤s≤t|∂γx∂αz ∂βs p(x, z, s)|](∫ t

0

ds

s1−r

). (5.9)

Once again, (a multiple of) the last expression in (5.8) dominates the last expression above.Thus, at this point, we have shown that the left-hand side of (5.4) is dominated by the lastexpression in (5.8).

In order to continue, we need an elementary result to the effect that, for a, b > 0, θ ∈ (0, 1)and N > 1/θ, there holds ∫ a

0

ds

(b+ sθ)N≈ min

a

bN,

1

bN−1/θ

. (5.10)

Returning to the task of further estimating the last expression in (5.8), we utilize (5.10) towrite

1

tν−σ+1

∫ t

0

ds

[|z|+ |s| 1k ]m+`+k|α|+k(|β|−ν)

≈ C1

tν−σ+1min

t

|z|m+`+k+|α|+k(|β|−ν),

1

|z|m+`+|α|+k(|β|−ν)

= C

1

tν−σ+1

1

|z|m+`+|α|+k(|β|−ν)min

t

|z|k, 1

= C

1

t1−σ1

|z|m+`+|alpha|+k|β|

(|z|k

t

)νmin

t

|z|k, 1

.

(5.11)

28

When |z|k

t≤ 1, we make the choice ν = β in which case the last expression in (5.11) becomes

C 1t1−σ

1|z|m+`+|α|+k|β|

(|z|kt

)β, in agreement with (5.4). If, on the other hand, |z|

k

t≥ 1, then we

select ν = 0 and, once again, the bound provided by (5.11) agrees with (5.4). This concludesthe proof of (5.4). The proof of (5.5) is similar and we omit it.

Proposition 5.3. Assume that P ∈ OPS−2,+cl,2 (M × R; E ,F) and let k(x, y, t, s) denote the

Schwartz kernel of P . Also, for Ω ⊆ M Lipschitz, introduce the associated single layerpotential operator, i.e.

Jf(x, t) :=

∫ t

−∞

∫∂Ω

〈k(x, y, t, s), f(y, s)〉 dσ(y)ds, x 6∈ ∂Ω, t ∈ R, (5.12)

and fix some 1 < p <∞, 0 < T <∞.

(i) For each ε > 0 there holds

‖(Jf)∗‖Lp(∂Ω×(0,T )) ≤ Cε maxT12−ε, 1‖f‖Lp(∂Ω×(0,T ),E) (5.13)

uniformly in f .

(ii) For each f ∈ Lp(∂Ω× (0, T ), E),

limx→ w

x ∈ γ±(w)

Jf(x, t) =

∫ t

0

∫∂Ω

〈k(w, y, t, s), f(y, s)〉 dσ(y)ds (5.14)

at a.e. w ∈ ∂Ω, t ∈ (0, T ).

(iii) There holds

‖(D12t Jf)∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T ),E), (5.15)

uniformly in f .

(iv) For each f ∈ Lp(∂Ω× (0, T ), E),

limx→ w

x ∈ γ±(w)

D12t Jf(x, t) = lim

ε→0

∫ t−ε

0

∫∂Ω

〈(D12t k)(w, y, t, s), f(y, s)〉 dσ(y)ds

= limε→0

∫ t

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈(D12t k)(w, y, t, s), f(y, s)〉 dσ(y)ds

(5.16)

at a.e. w ∈ ∂Ω, t ∈ (0, T ).

29

Proof. First we consider the operator

Tε : f 7→∫ t

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈(D12t k)(x, y, t, s), f(y, s)〉 dσ(y)ds, (5.17)

x ∈ ∂Ω, 0 < t < T . Our aim is to show that this is well-defined and bounded in L2(∂Ω ×(0, T ), E) uniformly in ε.

Work in local coordinates and let p ∈ S−2,+cl,2 be the total symbol of P . Further, decompose

p = p−2 + p−3 with p−2 ∈ S−2,+1,0,2 , p−3 ∈ S−3,+

1,0,2 so that, accordingly, k = k−2 + k−3. The

contribution coming from k−3 can be handled directly, based on size estimates for p−3 (cf.Lemma 3.2). Thus, we may restrict attention to p−2 and we drop the subindex −2 in thesubsequent analysis. The advantage in this scenario is that p(x, ξ, τ) is mixed homogeneousof degree −2, i.e. p(x, λξ, λ2τ) = λ−2p(x, ξ, τ). In particular,

(Fξp)(x, z, τ) = λ2−m(Fξp)(x,z

λ, λ2τ

). (5.18)

We are also interested in the decay properties of (Fξp)(x, z, τ) as |τ | → ∞. To this effect,note that, when x and z belong to compacts,

|(iτ)γ(Fξp)(x, z, τ)| =∣∣∣∣∫ +∞

−∞eiτs∂γs

p(x, z, s) ds∣∣∣∣≤ C

∫ +∞

−∞

ds

(1 + |s| 12 )m+2|γ|≤ Cγ < +∞,

(5.19)

by (3.13). Thus, (Fξp)(x, z, τ) decays fast as |τ | → ∞, uniformly for x and z in compacts.Going further, since Tε is a convolution operator in the time variable, using (5.18)-(5.19) wemay write (with hat denoting the Fourier transform in time):

|Tεf(x, τ)| ≤ C

∫d(x, y) ≥

√ε

y ∈ ∂Ω

|τ |12 |(Fξp)(x, y − x, τ)||f(y, τ)| dσ(y)

= C

∫d(x, y) ≥

√ε

y ∈ ∂Ω

|τ |12 |y − x|2−m

∣∣∣(Fξp)(x, y − x|y − x|, τ |y − x|2

)∣∣∣|f(y, τ)| dσ(y)

≤ CN

∫y∈Rm−1

|τ |12

1

|y − x|m−2max|τ |y − x|2|N , 1|f(y, τ)| dy,

(5.20)

for any N > 0. Hence, by Plancherel, it is enough to ultimately show that, for a fixed, largeN , the convolution operator with kernel

Kτ (x) := |τ |12

1

|x|m−2max|τ |N |x|2N , 1 (5.21)

is bounded in L2(Rm−1) uniformly in τ . However,

30

∫Rm−1

|Kτ (x)|dx =

∫Rm−1

|τ | 12|x|m−2

max[|τ ||x|2]N , 1 dx

=

∫Rm−1

1

|x|m−2max|x|2N , 1 dx ≤ Cm < +∞

(5.22)

if N = 1. Thus, supτ ‖Kτ‖L1 ≤ Cm < +∞, so that supε ‖Tε‖L2→L2 < +∞. Similarconsiderations also show that

limη,ε→0

‖Tεf − Tηf‖L2 = 0; (5.23)

cf. also [34, p. 67]. Hence

Tf := limε→0

Tεf exists in L2 and

T : L2 → L2 is a bounded operator.(5.24)

As before (cf. §3) proving that T extends to a bounded operator in Lp, 1 < p < ∞,depends on duality and a Hormander type cancelation property for the kernel. Since T is ofconvolution type in the time variable, one can show that its formal adjoint satisfies similarproperties as T itself. In particular, it is amenable to the same analysis as above.

There remains to show that

∫∫∂Ω×R\[S2r(y)×(s−4r2,s+4r2)]

|D12t [p(x, y − x, s− t)]−D

12t [p(x, y′ − x, s′ − t)]| dσ(x)dt ≤ C < +∞,

whenever |y′ − y|+ |s′ − s|12 ≤ r.

(5.25)

However, given the estimates (5.4)-(5.5) this closely parallels the calculation done in §3, StepIII. For example,

∫∫S2r(y)×[R\(s−4r2,s+4r2)]

|D12t [p(x, y − x, s− t)]| dxdt

≤ C

∫|x− y| ≤ 2rx ∈ Rm−1

(∫|t|≥4r2

t12

|x− y|m+2

(min

1,|x− y|2

t

)2

dt)dx

≤ C

(∫|x−y|≤2r

|x− y|4

|x− y|m+2dx

)(∫t≥4r2

dt

t32

)≈ C · r · 1

r= C < +∞.

(5.26)

We omit the remaining details.At this stage we have proved the Lp-boundedness of the operator (5.17), uniformly in ε.

Next, semi-standard arguments entail the boundedness of the maximal operator

31

T∗f(x, t) := supε>0|Tεf(x, t)|. (5.27)

This step is based on a Cotlar type inequality which, in turn, can be deduced from theestimates (5.4), (5.5). We refer the interested reader to [34, Theorem A.5, p. 70] for detailsin similar circumstances. Next, much as in [34, Proposition 1.7, p. 74], we can prove that

(D12t Jf)∗ ≤ C(T∗f +Mf) pointwise on ∂Ω× (0, T ), (5.28)

where, recall that M stands for the (parabolic) Hardy-Littlewood maximal function on∂Ω× (0, T ) from Lemma 3.3. This allows us to finally conclude that (5.15) holds.

Now we turn our attention to (5.16). First, for each f ∈ C∞comp(∂Ω × (0, T )) we have

D12t Jf = J(D

12t f); cf. Lemma 5.1. Granted (5.14), the nontangential trace of the latter

function is

limε→0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

(∫R

〈k(x, y, t, s), (D12s f)(y, s)〉 ds

)dσ(y)

= limε→0

∫ t

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈(D12t k)(x, y, t, s), f(y, s)〉 dσ(y)ds.

(5.29)

Hence, we are left with proving that for each f ∈ Lp and a.e. (x, t) ∈ ∂Ω× (0, T ),

limε→0

∫ t−ε

0

∫∂Ω

〈(D12t k)(x, y, t, s), f(y, s)〉 dσ(y)ds

= limε→0

∫ t

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈(D12t k)(x, y, t, s), f(y, s)〉 dσ(y)ds.

(5.30)

The strategy is to show that: (I) the two limits coincide for a dense subclass of Lp, and(II) the corresponding maximal operators are bounded in Lp. We now tackle these two pointsstarting with the first. To this effect, fix f ∈ C∞comp(∂Ω×(0, T ), E) so that, via an integrationby parts,

∫ t−ε

0

∫∂Ω

⟨− ∂

∂s

[I 1

2(p(x, y − x,−·))(t− s)], f(y, s)

⟩dσ(y)ds

=

∫ t−ε

0

∫∂Ω

⟨I 1

2(p(x, y − x,−·))(t− s), ∂f

∂s(y, s)

⟩dσ(y)ds

−∫∂Ω

⟨I 1

2(p(x, y − x,−·))(ε), f(y, t− ε)

⟩dσ(y).

(5.31)

Now, the last integral converges to zero as ε → 0+ by Lebesgue’s theorem since p vanishesfor τ > 0. As for the first integral in the right side of (5.31), the kernel has an integrable

32

singularity so passing to the limit ε→ 0+ is straightforward. All in all, the left side of (5.31)can be written in the form

limε→0

∫R

∫d(x, y) ≥

√ε

y ∈ ∂Ω

⟨I 1

2(p(x, y − x,−·))(t− s), ∂sf(y, s)

⟩dσ(y)ds

= limε→0

∫R

∫d(x, y) ≥

√ε

y ∈ ∂Ω

⟨p(x, y − x, s− t),(I 12

∂

∂s

)f(y, s)

⟩dσ(y)ds

= limε→0

∫R

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈p(x, y − x, s− t), (D 12s f)(y, s)〉 dσ(y)ds

= limε→0

∫R

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈D12t [p(x, y − x, s− t)], f(y, s)〉 dσ(y)ds,

(5.32)

as desired.Consider next (II) in the strategy outlined above. As usual, it suffices to prove that for

x ∈ ∂Ω, t ∈ (0, T ),

supε>0

∣∣∣∣∣∣∫ t−ε

0

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈D12t k(x, y, t, s), f(y, s)〉 dσ(y)ds

∣∣∣∣∣∣ ≤ CMf(x, t) (5.33)

and

supε>0

∣∣∣∣∣∣∫ t

t−ε

∫d(x, y) ≥

√ε

y ∈ ∂Ω

〈D12t k(x, y, t, s), f(y, s)〉 dσ(y)ds

∣∣∣∣∣∣ ≤ CMf(x, t). (5.34)

Passing to local coordinates and pull-backing ∂Ω to Rm−1, we see that the left-side of (5.33)is bounded by

C

∫ t−ε

−∞

∫|x− y| ≤

√ε

y ∈ Rm−1

|s− t| 12|x− y|m+2

(min

1,|x− y|2

|s− t|

)2

|f(y, s)| dyds

≤ C1

ε

1

(√ε)m−1

∫R

∫Rm−1

χ(1,∞)

(s− tε

)χB1(0)

(x− y√ε

)∣∣∣s− tε

∣∣∣1/2∣∣∣x− y√ε

∣∣∣m+2

×(

min

1,∣∣∣x− y√

ε

∣∣∣2/∣∣∣s− t√ε

∣∣∣)2

|f(y, s)| dyds.

(5.35)

Now (5.33) follows from (5.35) with the aid of Lemma 3.3 in which we choose

Φ(y, s) := χ(1,∞)(s)χB1(0)(y)s1/2

|y|m+2

(min

1,|y|2

s

)2

∈ L1(Rm−1 × R). (5.36)

The estimate (5.34) is proved in a similar fashion; we omit the details. This completes theproof of (iii) and (iv) in Proposition 5.3.

Granted (5.4)–(5.5), the estimates (i) and (ii) are elementary and we leave them to theinterested reader. The proof of Proposition 5.3 is finished.

33

6 Square-Function Estimates

The main aim of this section is to prove the following.

Proposition 6.1. Let P ∈ OPS−2,+cl,2 (M × R; E ,F) have an even principal symbol (in ξ),

Ω ⊆M Lipschitz, and denote by J the single layer potential operator associated with P (andΩ) as in (5.12).

Then, for each 0 < T < ∞ there exists C = C(Ω, P, T ) > 0 so that if u := Jf withf ∈ L2(∂Ω× (0, T ), E), we have∫ T

0

∫∫Ω

|∇2u|2dist (·, ∂Ω) dVol dt ≤ C‖f‖2L2(∂Ω×(0,T ),E), (6.1)

∫ T

0

∫∫Ω

|D1/4t ∇u|2 dVol dt ≤ C‖f‖2

L2(∂Ω×(0,T ),E). (6.2)

Proof. This follows from the work of S. Hofmann and J. Lewis [46], where very general resultsof this type are proved; we shall only sketch the main steps.

Dealing first with (6.1), work in local coordinates and decompose the symbol σprinc(P )as p−2 + p−3 where p−2 ∈ S−2,+

1,0,2 , p−2(x, ξ, τ) is even in ξ, and p−3 ∈ S−3,+1,0,2 . Thus, J splits

accordingly, as J−2 + J−3. Now, the contribution from J−3 is easily controlled, due to theweak singularity in the kernel. In fact this leads to an estimate like (6.1) with C(Ω, P, T )→ 0as T → 0.

Next, extend f to ∂Ω × R by setting f ≡ 0 outside ∂Ω × (0, T ) and concentrate on thecontribution from

J−2f(x, t) :=

∫ +∞

−∞

∫∂Ω

〈p−2(x, y − x, t− s), f(y, s)〉 dσ(y)ds. (6.3)

Note that by localizing the problem there is no loss of generality in assuming that f is scalar-valued, has small support and Ω is Euclidean so that ∂Ω is the graph of a Lipschitz functionϕ : Rm−1 → R. To proceed, for z ∈ Rm, r ∈ R, let ρ = ρ(z, r) > 0 solve the equation|z|2ρ2 + r2

ρ4 = 1. In particular, ρ(λz, λ2r) = λρ(z, r) and (z/ρ, r/ρ2) belongs to Sm, the unitsphere in Rm × R.

Let now ψj : j ≥ 1 be an orthonormal basis of L2(Sm) consisting of eigenfunctions of

the Laplace-Beltrami operator on Sm and, for each fixed x, expand p−2

(x, z

ρ, rρ2

)in spherical

harmonics:

p−2

(x,z

ρ,r

ρ2

)=∑j≥1

aj(x)ψj

(zρ,r

ρ2

),

with ‖aj‖C1 · ‖ψj‖CN ≤ CNj−2, ∀ j.

(6.4)

Given that p−2 is even in the second variable, we may replace each ψj above by ψ+j , where

ψ+j (z, r) := 1

2(ψj(z, r) + ψj(−z, r)). It follows that

34

p−2(x, z, r) =∑j≥1

aj(x)bj(z, r),

where bj(z, r) := ρ−mψ+j

(zρ,r

ρ2

).

(6.5)

In particular,

each bj is even in z for fixed r

and bj(λz, λ2r) = λ−mbj(z, r).

(6.6)

At this stage, the arguments in [46] (cf. especially Lemma 5.11 and Theorem 4.1) apply toeach

uj(x, t) :=

∫ +∞

−∞

∫∂Ω

bj(x− y, t− s)f(y, s) dσ(y)ds (6.7)

to yield ∫ +∞

−∞

∫∫Ω

|∇2uj|2dist(·, ∂Ω) dxdt ≤ C‖ψj‖CM · ‖f‖2L2(∂Ω×R,E) (6.8)

for some fixed M > 0, uniformly in j. Clearly, since u =∑

j ajuj, (6.8) and (6.4) yield (6.1).The reader should be aware that the estimate (6.8) was obtained in [46] in the particular

case when P = (∂t − ∆)−1. Three proofs of this were given, one of which has a purelyreal-variable nature. Inspection of the arguments reveals that the techniques developed inthe course of this latter proof are in fact powerful enough to handle (6.7). There is only oneaspect which we would like to emphasize here. Specifically, if θ is a nice bump function andx = (x′, φ(x′) + λ) ∈ Ω, y = (y′, φ(y′)) ∈ ∂Ω then, for b(x, t) as in (6.6),∫

Rm

(∂xj∂xkb)(x′ − y′, 〈x′ − y′,∇(θλ ∗ φ)(x′)〉+ λ, t− s) dy′ds = 0, (6.9)

for all λ, x′, t, as long as 1 ≤ j, k ≤ m.Indeed, if we set F (x) :=

∫R(∂xj∂xkb)(x, t) dt, then F becomes an even function which is

(positive) homogeneous of degree −m in Rm. In this scenario, (6.9) is going to be a simpleconsequence of the elementary identity

∫Rm−1

F (y′, 〈a, y′〉+ λ) dy′ =1

2λ

∫Sm−2

∫R

F (ω, s) dsdω

=

∫Rm−1

F (y′, λ) dy′(6.10)

valid for each λ > 0 and a ∈ Rm−1; see Lemma 1.3 in [1] for a proof.Finally, (6.2) is proved much as in [46] based on (6.1), this finishes the proof of the

proposition.

35

In the sequel, we shall not utilize the full force of (6.2). The case of interest to us hereis when P = (∂t − L)−1 with L : C∞(M, E) → C∞(M, E) formally self-adjoint, real, secondorder, strongly elliptic differential operator. At least for some common, special cases, otheralternative approaches to (6.2) can be developed. Below we briefly elaborate on this idea.

First, anticipating notation, terminology and results which are actually developed in §7,let us assume that L is locally given by (7.1) and that the coefficients of the principal partcan be chosen so that for all ζ = (ζαj )

aαβjk (x)ζαj ζβk ≥ C|ζ|2, uniformly in x. (6.11)

This is always possible when rank E = 2, i.e. when 1 ≤ α, β ≤ 2. To see this, introduceA := (a11

jk)j,k, B := (a12jk)j,k, C := (a22

jk)j,k so that

(aαβjk )α,β,j,k =

[A BBt C

]. (6.12)

Also, note that A,C are symmetric and λ2A + λ(B + Bt) + C is positive semi-definite foreach λ ∈ R. Then there exists an anti-symmetric n× n matrix D = −Dt so that[

A B +DBt −D C

](6.13)

is positive semi-definite. This result may be deduced from [57, Theorem 5.51]. A directproof is given in [58]. The interested reader is also referred to [59] which contains a survey ofwork on necessary and sufficient conditions for a pair of quadratic forms to admit a positivedefinite linear combination. Clearly, applying this result to (aαβjk ) − εI for ε > 0 small itfollows that in the case when 1 ≤ α, β ≤ 2 matters can always be arranged so that (6.11) istrue.

Another important situation in which (6.11) holds is when L = L0−DD∗+lower order terms,where D is a first-order differential operator, D∗ is its formal adjoint, and L0 is a second-order differential operator for which (6.11) is valid. This is easily seen by a direct calculation:if the principal part of D is locally given by Aαβj ∂j, then L is locally given by (7.1) with

aαβjk = Aαγj Aβγk . Thus, ∑

aαβjk ζαj ζ

βk =

∑β

(∑α,j

aαβjk ζαj

)2

≥ 0. (6.14)

Returning to the main-stream discussion and assuming that (6.11) holds, we may use

(7.13) with v := w := D14t u to get, by virtue of (6.11) and (7.12), that

36

∫ T

0

∫∫Ω

|D14t ∇u|2 ≤ C

∫ T

0

∫∫Ω

∑∂j(D

14t u

α)aαβjk ∂k(D14t u

β)

≤ −C2

∫ T

0

∫∫Ω

∂

∂t|D

14t u|2

+O(‖D12t u‖L2(∂Ω×(0,T ))‖∇u‖L2(∂Ω×(0,T )))

+ C

∫ T

0

∫∫Ω

|D14t u|2.

(6.15)

The first term after the second inequality sign is −12

∫∫Ω|D

14t u(·, T )|2 and can be dropped

based on, e.g., sign considerations. Now all the remaining terms are ≤ C‖f‖2L2(∂Ω×(0,T ),E) by

the results in §§4-5. This yields (6.2).Another approach to (6.2) in the case when P = (∂t−L)−1, is based on Rellich identities

and duality. An approach in the Fourier transform side, i.e. for a single layer potentialassociated with (λ − L)−1 where λ ∈ C is regarded as a parameter has been developed in[60]. This also can be adapted to the situation under discussion to give an alternative proofof (6.2). We leave the details to the interested reader.

7 Parabolic Rellich Estimates

Let L : C∞(M,E) → C∞(M,F ) be a second order differential operator with smooth, real-valued coefficients. We assume that, in local coordinates over which E,F can be trivialized,L takes the form

(Lu)α =∑β,j,k

∂j(aα,βjk (x)∂ku

β) +∑β,j

bαβj (x)∂juβ +

∑β

cαβ(x)uβ. (7.1)

Recall the canonical projection pr : M × R → M and the pull-backs E := pr∗E, F :=pr∗F →M × R. Then L extends in a natural fashion as a mapping

L : C∞(M × R, E)→ C∞(M × R,F). (7.2)

The same applies to P := ∂t − L ∈ OPS2,+cl,2 (M × R; E ,F). Then, in local coordinates,

p(x, ξ, τ) := σprinc(P )(x, ξ, τ) = iτ +∑j,k

aαβjk (x)ξjξk. (7.3)

This clearly satisfies the hypothesis in Lemma 1.1 (relative for C−) and it accounts for thefact that p is casual.

Assume next that L is strongly elliptic, i.e. E = F and

C|η|2x ≤ Re 〈−σprinc(L)(x, ξ)η, η〉x = Re

[ ∑j,k,α,β

aαβjk (x)ξjξkηαηβ

],

∀x ∈M, ξ ∈ T ∗xM, |ξ|x = 1, η ∈ Ex.

(7.4)

37

Then, it follows that P is strongly parabolic, i.e.

Re 〈p(x, ξ, τ)η, η〉 ≥ C(|ξ|+ |τ |12 )2|η|2, (7.5)

uniformly for x ∈ M , ξ ∈ T ∗xM\0, τ ∈ R+, η ∈ Ex. In particular, by Theorem 2.3,P−1 ∈ OPS2,+

cl,2 (M × R; E) exists and, locally,

σprinc(P−1)(x, ξ, τ) =

[iτ +

∑j,k

aαβjk (x)ξjξk

]−1

. (7.6)

Next consider k(x, y, t, s), the Schwartz kernel of P−1, and for a Lipschitz domain Ω ⊆Mintroduce the single layer potential operator (associated with P := ∂t − L), i.e.

Sf(x, t) :=

∫ t

−∞

∫∂Ω

〈k(x, y, t, s), f(y, s)〉 dσ(y)dt; (7.7)

recall that dσ is the surface measure on ∂Ω.Finally, fix 0 < T <∞, and for arbitrary f ∈ L2(∂Ω× (0, T ), E) set

u := Sf in Ω× (0, T ). (7.8)

Our aim is to show the following.

Proposition 7.1. For each ε > 0 there holds

∫ T

0

∫∂Ω

|∇tanu|2 dσdt+

∫ T

0

∫∂Ω

|D12t u|2 dσdt ≈

∫ T

0

∫∂Ω

|∇u|2 dσdt

modulo ε‖f‖2L2(∂Ω×(0,T ),E) and terms which are small with T.

(7.9)

Generally speaking, writing ‘A(s) ≈ B(s) modulo C’ indicates that A ≤ κB + C andB ≤ κA + C for some constant κ > 0 independent of the relevant parameters in A, B,C. In (7.9), the equivalence constant may depend on ε but not on f . Also, a quantity iscalled “small with T” if its absolute value can be bounded by C(T )‖f‖L2(∂Ω×(0,T ),E) withC(T ) = o(1) as T → 0+.

With an eye on (7.9), observe that it suffices to show a local version of this estimate,i.e. when ∂Ω is contained in a coordinate patch over which E trivializes. Indeed, there is noloss of generality assuming that f has small support and we may also truncate u, i.e. workwith ψu in place of u, where ψ ∈ C∞comp(M) has a suitable support. This latter reduction isrelatively harmless: the only thing we loose is the quality of u of being a null solution for∂t−L in Ω. However, as an a posteriori inspection of the proof shows, (∂t−L)u = O(|∇u|)is just as good.

Assuming the above reductions and working in local coordinates where L has the form(7.1) we now proceed to present the

Proof of Proposition 7.1. Let h = (hj)j be a smooth, arbitrary vector field. Then the fol-lowing Rellich type identity is known to hold in the case under discussion

38

∑α,β,j,k,`

∂

∂x`

[(h`a

αβjk − hja

αβ`k − hka

αβj` )

∂uα

∂xj

∂uβ

∂xk

]

= −2∑`,α

h`∂uα

∂x`

(∑β,j,k

∂

∂xj

(aαβjk

∂uβ

∂xk

))+O(|∇u|2 + |u|2).

(7.10)

See, e.g., [61]. Under the current assumption that (∂t−L)u = O(|∇u|), the entire right sideof (7.10) becomes O(|∇u|2 + |u|2). Consequently, much as in the elliptic case (cf. [1])

∫ T

0

∫∂Ω

|∇u|2 dσdt ≤ C

∫ T

0

∫∂Ω

|∇tanu|2 dσdt

+ C

∫ T

0

∫∫Ω

∑α

∂uα

∂h· ∂u

α

∂tdVol dt

+ C

∫ T

0

∫∫Ω

[|∇u|2 + |u|2] dVol dt

=: A+B + C.

(7.11)

In the sequel, we shall keep A. Also, in the applications that we have in mind, C is going tobe small with T , which suits our purposes. There remains to estimate B.

In order to continue, we need two auxiliary facts. First, if f, g are sufficiently smoothand decay fast enough at −∞, then∣∣∣∣∫ T

−∞(D

14t f)g

∣∣∣∣ ≤ C

(∫ T

−∞|f |2) 1

2(∫ T

−∞|D

14t g|2

) 12

(7.12)

with C > 0 independent of T . See [34] for a proof. Second, we need an identity to the effectthat for each v, w, there holds

∫ T

0

∫∫Ω

∑α,β,j,k

∂wα

∂xjaαβjk

∂vβ

∂xkdVol dt+

∫ T

0

∫∫Ω

∑α

wα∂vα

∂tdVol dt

=

∫ T

0

∫∂Ω

∑α,β,j,k

wαnjaαβjk

∂vβ

∂xkdσdt+

∫ T

0

∫∫Ω

∑α

wα[(∂t − L)v]α dVol dt

+

∫ T

0

∫∫Ω

O(|w||∇v|) dVol dt,

(7.13)

where n = (nj)j and dσ are, respectively, the Euclidean unit normal and surface measure on∂Ω (the latter differs from the surface measure induced by the Riemannian structure by afactor ρ with ρ, ρ−1 ∈ L∞). Indeed,

39

∫∫Ω

∑α,β,j,k

∂wα

∂xjaαβjk

∂vβ

∂xk=

∫∫Ω

∑ ∂

∂xj

(wαaαβjk

∂vβ

∂xk

)−∫∫

Ω

∑wα

∂

∂xj

(aαβjk

∂vβ

∂xk

)=: I + II.

(7.14)

An integration by parts gives

I =

∫∂Ω

wαnjaαβjk

∂vβ

∂xkdσ, (7.15)

whereas for each α,∑ ∂

∂xj

(aαβjk

∂vβ

∂xk

)=∂vα

∂t− [(∂t − L)v]α +O(|∇u|). (7.16)

Using these back in (7.14) and integrating in t on (0, T ) yields (7.13).

Returning now to the task of estimating B in (7.11) we write ∂t = D14t D

34t so that, by

(7.12),

|B| ≤ C

(∫ T

0

∫∫Ω

|D14t ∇u|2 dVol dt

) 12

·(∫ T

0

∫∫Ω

|D34t u|2 dVol dt

) 12

=: B1 ·B2.

(7.17)

To estimate B2 we set v := I 14u, w := ∂t(I 1

4u) = D

34t u so that, availing ourselves of (7.13),

we get∑

αwα ∂vα

∂t= |D

34t u|2 and

∫ T

0

∫∫Ω

∑ ∂wα

∂xjaαβjk

∂vβ

∂xkdVol dt

=1

2

∫ T

0

∫∫Ω

∑ ∂

∂t

[∂(I 1

4uα)

∂xjaαβjk

∂(I 14uβ)

∂xk

]dVol dt

=1

2

∫∫Ω

∑ ∂(I 14uα)

∂xj(·, T )aαβjk

∂(I 14uβ)

∂xk(·, T ) dVol

≤ C

∫∫Ω

|I 14∇u(·, T )|2 dVol = small with T.

(7.18)

The last equality is easy to check based on the estimates established in §5. Thus,

|B2|2 =

∫ T

0

∫∫Ω

|D34t u|2 dVol dt

= O(‖∇u‖L2(∂Ω×(0,T )) · ‖D

12t u‖L2(∂Ω×(0,T ))

)+ terms which are small with T.

(7.19)

40

As for B1 in (7.17), we may invoke Proposition 6.1 to write

|B1| ≤ C‖f‖L2(∂Ω×(0,T ),E). (7.20)

In particular, since for each ε > 0 there holds |B| ≤ B1 · B2 ≤ εB21 + CεB

22 , (7.11), (7.19)

and (7.20) give the inequality “≥” in (7.9).To see the opposite inequality, fix a smooth field h = (hj)j which is transversal to ∂Ω

and write (recall that n is the outward unit normal to ∂Ω)

∫ T

0

∫∂Ω

|D12t u|2 dσdt ≤ C

∫ T

0

∫∂Ω

〈h, n〉|D12t u|2dσdt

= C

∫ T

0

∫∫Ω

∑hj(∂jD

12t u

α)(D12t u

α) dVol dt

+ C

∫ T

0

∫∫Ω

O(|D12t u|2)

≤ C

(∫ T

0

∫∫Ω

|D14t ∇u|2 dVol dt

) 12

·(∫ T

0

∫∫Ω

|D34t u|2 dVol dt

) 12

+ small terms with T,

(7.21)

via the divergence theorem and (7.12). Note that the last product above is the familiarB1 ·B2 (cf. (7.17)). Thus, granted (7.19) and (7.20), for each given ε > 0, (7.21) leads to

∫ T

0

∫∂Ω

|D12t u|2 dσdt ≤ Cε

∫ T

0

∫∂Ω

|∇u|2 dσdt+ ε‖f‖2L2(∂Ω×(0,T ),E)

+ small terms with T.

(7.22)

With this at hand, the inequality “≤” in (7.9) readily follows. This finishes the proof ofProposition 6.1.

8 Inverting Parabolic Layer Potentials

Retain the hypotheses made on the differential operator L from §7. Also, fix an arbitraryLipschitz domain Ω ⊆M and let wj1≤j≤m be a family of C1-vector fields on M which arelinearly independent in a neighborhood of ∂Ω. Hence, in local coordinates wj =

∑k cjk

∂∂xk

,

where C := (cjk)jk has C1 entries and is invertible near ∂Ω.Next, equip E with a smooth connection ∇, so that locally

(∇wju)α =∑k

cjk∂kuα +O(|u|), ∀ j, (8.1)

41

for any smooth section u. It follows that, locally,

|∇u| ≈∑j

|∇wju| modulo O(|u|), pointwise near ∂Ω, (8.2)

where ∇u in the left side stands for the Euclidean gradient acting componentwise on u.Denote now by Γ(x, y, t, s) the Schwartz kernel of the operator P := (∂t − L)−1 ∈

OPS−2,+cl,2 (M × R; E), so that (∂t − Lx)Γ(x, y, t, s) = δy(x)δs(t), and introduce the single

layer potential J associated with P and Ω. Also, denote by S the trace of S on ∂Ω× (0, T ).Going further we also introduce

T : ⊕1≤j≤m

Lp(∂Ω× (0, T ), E)→ Lp(∂Ω× (0, T ), E) (8.3)

by setting

T(g1, g2, . . . , gm) := limε→0

∫ t−ε

0

∫∂Ω

m∑j=1

〈∇wj,yΓ(x, y, t, s), gj(y, s)〉 dσ(y)ds. (8.4)

Here ∇wj ,y indicates that ∇wj acts in the y-variable. Denote by ν ∈ T ∗M the outward unitconormal to ∂Ω.

Theorem 8.1. For each 0 < T < ∞, 1 < p < ∞, the operator T in (8.3) is well definedand bounded. Moreover, there exists ε = ε(∂Ω, L) > 0 such that for each 2 − ε < p < 2 + εand T > 0, the assignment

(g1, . . . , gm) 7→ 1

2[σprinc(L)(·, ν)]−1

(∑j

ν(wj)gj

)+ T(g1, . . . , gm) (8.5)

maps ⊕1≤j≤m

Lp(∂Ω× (0, T ), E) onto Lp(∂Ω× (0, T ), E).

A remark is in order here. As the proof will show the operator in (8.5) satisfies a slightlystronger property. More specifically, there exists 0 < C = C(∂Ω, L, T ) such that

∀ f ∈ Lp(∂Ω× (0, T ), E), ∃ (gj)j ∈ ⊕jLp(∂Ω× (0, T ), E) with

1

2

∑j

ν(wj)[σprinc(L)(·, ν)]−1gj + T(g1, . . . gm) = f

and∑j

‖gj‖Lp(∂Ω×(0,T ),E) ≤ C‖f‖Lp(∂Ω×(0,T ),E).

(8.6)

Proof. Let us consider the operator

T′ : Lp(∂Ω× (0, T ), E)→ ⊕

1≤j≤mLp(∂Ω× (0, T ), E) (8.7)

defined by

42

(T′f)j := limε→0

∫ t−ε

0

∫∂Ω

〈∇wj,xΓ(x, y, t, s), f(y, s)〉 dσ(y)ds (8.8)

from the results in §5, T′ is well-defined and bounded for each 1 < p <∞ and 0 < T <∞.We now make two basic observations regarding the operator (8.7)-(8.8). First, for f ∈

Lp(∂Ω× (0, T ), E) let u := Sf in Ω± × (0, T ). Then, for each j, so we claim,

(∇wju)|∂Ω±×(0,T ) = ±1

2ν(wj)[σprinc(L)(·, ν)]−1f + (T′f)j. (8.9)

Our second claim is that

R (T′)∗ R = T, modulo operators

whose norms are small with T.(8.10)

Here (Rf )(u, t) := f(x, T − t) is a time-reflection and the asterisk indicates the adjoint.To prove (8.9), note that ∇wj P ∈ OPS−1

cl,2 has the principal symbol (cf. the discussionin §7)

σprinc(∇wj P )(x, ξ, τ) = σprinc(∇wj)(x, ξ, τ)σprinc(P )(x, ξ, τ)

= iwj(ξ)[iτ +∑

aαβjk (x)ξjξk]−1.

(8.11)

Also, the Schwartz kernel of ∇wj P is ∇wj,xΓ(x, y, t, s). Thus, the jump formula (8.9) is aconsequence of (4.25).

As for (8.10) we first observe that, like ∂t − L, the operator P = (∂t − L)−1 is invariantunder changing t into −t and taking the adjoint. At the level of Schwartz kernels thisproperty reads

[Γ(x, y, t, s)]t = Γ(x, y, t, s), (8.12)

where the superscript t indicates adjunction (in Hom (E , E)). Also, as is well known,

(∇wj)∗ = −∇wj + zero order terms. (8.13)

Now (8.10) follows from (8.12), (8.13) and (5.13). Armed with (8.9)-(8.10) we are now readyto prove the ontoness of the operators (8.5).

First, we consider the case p = 2. Let f ∈ L2(∂Ω×(0, T ), E) be arbitrary and set u := Sfin Ω± × (0, T ). Then Proposition 7.1 and (8.2) give that for each ε > 0 there holds

∫ T

0

∫∂Ω

|∇tanu|2 dσdt+

∫ T

0

∫∂Ω

|D12t u|2 dσdt

≈∑j

∫ T

0

∫∂Ω

|∇wju|∂Ω±×(0,T )|2 dσdt

modulo ε‖f‖2L2(∂Ω×(0,T ),E) and terms which are small with T.

(8.14)

43

Also, given that wjj are linearly independent near ∂Ω and that L is strongly elliptic, weget from (8.9) and the triangle inequality that

‖f‖L2(∂Ω×(0,T ),E) ≤Cm∑j=1

‖(∇wju)|∂Ω+×(0,T )‖L2(∂Ω×(0,T ),E)

+ C

m∑j=1

‖(∇wju)|∂Ω−×(0,T )‖L2(∂Ω×(0,T ),E)

(8.15)

where C is independent of T . In order to continue, we note that, for each tangent field h,∇hu does not jump, i.e. (

∇hu)∣∣∣∂Ω+×(0,T )

=(∇hu

)∣∣∣∂Ω−×(0,T )

. (8.16)

This is a consequence of (8.9) given that ν(h) = 0. A similar comment applies to D1/2t u; cf.

(5.16). These observations in concert with (8.14) give that

‖f‖L2(∂Ω×(0,T ),E) ≤ Cm∑j=1

‖(∇wju)|∂Ω±×(0,T )‖L2(∂Ω×(0,T ),E)

+ ε‖f‖L2(∂Ω×(0,T ),E) + C(T )‖f‖L2(∂Ω×(0,T ),E),

(8.17)

where C(T ) → 0 as T → 0. Choosing first ε sufficiently small and then making T smallenough, it follows that

‖f‖L2(∂Ω×(0,T ),E)

≤ C∥∥∥(1

2ν(wj)[σprinc(L)(·, ν)]−1

1≤j≤m

+ T′)f∥∥∥L2(∂Ω×(0,T ),E)

,(8.18)

for some C = C(∂Ω, T ) > 0 independent of f , granted that T is small enough.At this stage we invoke a simple functional analysis lemma to the effect that if a linear

operator A maps a Hilbert space H into itself so that, for some δ > 0,

δ‖x‖H ≤ ‖Ax‖H , ∀x ∈ H, (8.19)

then

∀ y ∈ H, ∃x ∈ H with ‖x‖ ≤ δ−1‖y‖ and A∗x = y. (8.20)

This follows easily from [62, Theorem 4.13, p. 100]. When applied to (8.18) this gives, in thelight of (8.10) and the fact that R is an isomorphic isometry, that

12

∑j

ν(wj)[σprinc(L)(·, ν)]−1πj + T maps

⊕1≤j≤m

Lp(∂Ω× (0, T ), E) onto Lp(∂Ω× (0, T ), E)

when p = 2 and T > 0 is sufficiently small.

(8.21)

44

Here πj : ⊕1≤j≤m

Lp(∂Ω × (0, T ), E) → Lp(∂Ω × (0, T ), E) is the canonical projection on the

j-th component.Now, since the operator in (8.21) is well-defined and bounded for each 1 < p < ∞

and since the property of being onto is stable on complex interpolation scales (cf. [63]),we get that the condition p = 2 in (8.21) can be relaxed to 2 − ε < p < 2 + ε for someε = ε(∂Ω, L) > 0.

Finally, the smallness assumption on T in (8.21) can be eliminated thanks to an ab-stract argument which, for the convenience of the reader, we formulate below. Modulo this,Theorem 8.1 is proved.

Here is the lemma which finishes the proof of Theorem 8.1. This is taken from [39].Similar results have been used in [22], [34], [37].

Lemma 8.2. Let X be an arbitrary, fixed set, and let V be a certain vector space of functionsdefined on X × R. We assume that f(·, · + h) ∈ V for any f ∈ V and any h ∈ R. SetV0 := f ∈ V ; f |X×(−∞,0] ≡ 0 and, for each T > 0, VT := f |X×(0,T ); f ∈ V0. Suppose thatB : V → V is a linear operator so that

i) B(f(·, ·+ h)) = B(f)(·, ·+ h) for any f ∈ V and for any h ∈ R;

ii) V0 is an invariant subspace of B;

iii) there exists T0 > 0 such that B : VT → VT is a bijection for any T0 > T > 0.

Then B : VT → VT is a bijection for each T > 0.

Remark. If F ∈ VT for (x, t) ∈ X × (0, T ), we define B(f)(x, t) := B(F )(x, t), where F ∈ V0

is such that F |X×(0,T ) = f . The properties of B ensure that this definition does not dependon the particular extension F of f .

To state our next result we introduce the space Lp1, 1

2

(∂Ω × (0, T ), E) as the collection of

all sections f : ∂Ω× (0, T )→ E such that |∇tanf | and |D12t f | belong to Lp(∂Ω× (0, T )).

Theorem 8.3. Let L be a second order, strongly elliptic, formally self-adjoint differentialoperator with smooth, real coefficients and fix Ω ⊆ M , arbitrary Lipschitz domain. SetP := (∂t−L)−1 and denote by S the (nontangential trace on ∂Ω× (0, T ) of the) single layerpotential associated with P and ∂Ω.

Then there exists ε = ε(Ω, L) > 0 such that for each 2− ε < p < 2 + ε and 0 < T < ∞,the operator

S : Lp(∂Ω× (0, T ), E)→ Lp1, 1

2

(∂Ω× (0, T ), E) (8.22)

is an isomorphism.

Let us point out that, as the proof will show, one also has

‖S−1‖Lp1, 12

→Lp ≤ C(∂Ω, L, T, p). (8.23)

45

Proof. For an arbitrary f ∈ L2(∂Ω× (0, T ), E), set u := Sf in Ω± × (0, T ). Select now thevector fields wjj as in the proof of Theorem 8.1 so that, much as before,

‖f‖L2(∂Ω×(0,T )) ≤ C∑j

‖∇wju|∂Ω×(0,T )‖L2(∂Ω×(0,T ),E)

+ C∑j

‖∇wju|∂Ω+×(0,T )‖L2(∂Ω×(0,T ),E)

+ terms which are small with T

≤ C‖∇tanu‖L2(∂Ω×(0,T ),E) + C‖D12t u‖L2(∂Ω×(0,T ),E)

+ terms which are small with T

≤ C‖∇tanSf‖L2(∂Ω×(0,T ),E) + C‖D12t Sf‖L2(∂Ω×(0,T ),E)

+ terms which are small with T.

(8.24)

Therefore,

‖f‖L2(∂Ω×(0,T ),E) ≤C‖∇tanSf‖L2(∂Ω×(0,T ),E)

+ C‖D12t Sf‖L2(∂Ω×(0,T ),E),

(8.25)

if T > 0 is sufficiently small. Now, ∇tanS, D12t S : Lp(∂Ω × (0, T ), E) → Lp(∂Ω × (0, T ), E)

are bounded for each 1 < p < ∞ and 0 < T < ∞. Thus, utilizing the fact that estimatesfor below like (8.25) on complex interpolation scales are stable under small perturbations ofthe scale parameter ([63]), we finally obtain that there exists ε > 0 so that

‖f‖Lp(∂Ω×(0,T ),E) ≤ C‖∇tanSf‖Lp(∂Ω×(0,T ),E) + C‖D12t Sf‖Lp(∂Ω×(0,T ),E)

= C‖Sf‖Lp1, 12

(∂Ω×(0,T ),E),(8.26)

uniformly in f for each 2− ε < p < 2 + ε, provided T > 0 is sufficiently small.It is important to point out that the constant C in (8.26) depends exclusively on p, T, L

and the Lipschitz character of ∂Ω. Thus, S in (8.22) is injective and with closed range if|2− p| and T are small, which we shall assume for now.

If we now take Ωj Ω, ∂Ωj ∈ C∞, a suitable approximating sequence and denote bySj the corresponding single layer potential on ∂Ωj, it follows from (2.19) that Sj : Lp(∂Ωj ×(0, T ), E) → Lp

1, 12

(∂Ωj × (0, T ), E) is invertible. This, the fact that supj ‖S−1j ‖ < +∞, and

a standard limiting argument (cf. [30], [2] for details in similar circumstances) then implythat S in (8.22) also has dense range.

Summarizing, at this point we have proved that S in (8.22) is invertible provided 2− ε <p < 2 + ε and T > 0 is sufficiently small. Finally, the latter restriction can be lifted byinvoking Lemma 8.2. This completes the proof of the theorem.

46

9 Initial Boundary Value Problems

Let L : C∞(M, E)→ C∞(M, E) be as in §8, i.e. a second order, formally self-adjoint, stronglyelliptic differential operator with smooth, real coefficients.

For Ω ⊆ M Lipschitz domain, 1 < p < ∞ and 0 < T < ∞ we consider the Dirichletinitial boundary value problem

(IBV P )

u ∈ C∞(Ω× (0, T ), E),

(∂t − L)u = 0 in Ω× (0, T ),

u|t≤0 = 0 on Ω,

u∗ ∈ Lp(∂Ω× (0, T )),

u|∂Ω×(0,T ) = f ∈ Lp(∂Ω× (0, T ), E).

(9.1)

Theorem 9.1. With the above assumptions, there exists ε = ε(Ω, L) > 0 so that for each2 − ε < p < 2 + ε, 0 < T < ∞, the initial boundary problem (9.1) has a unique solutionwhich also satisfies

‖u∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp(∂Ω×(0,T )) (9.2)

for some C = C(∂Ω, L, T, p) > 0. Also, when p = 2 there holds

u ∈ H1/4((0, T ), L2(Ω, E)) ∩ L2((0, T ), H1/2(Ω, E)), (9.3)

where Hs is the usual L2-based scale of Sobolev spaces.Moreover, the following regularity statement is true:

(∇u)∗, (D12t u)∗ ∈ Lp(∂Ω× (0, T ))⇔ f ∈ Lp

1, 12

(∂Ω× (0, T ), E). (9.4)

In this case, one also has

‖(∇u)∗‖Lp(∂Ω×(0,T )) + ‖(D12t u)∗‖Lp(∂Ω×(0,T )) ≤ C‖f‖Lp

1, 12

(∂Ω×(0,T ),E), (9.5)

for some C = C(∂Ω, L, T, p) > 0. If, in addition, p = 2 then

u ∈ H3/4((0, T ), L2(Ω, E)) ∩ L2((0, T ), H3/2(Ω, E)). (9.6)

Finally, in each case, there is an integral representation formula for the solution.

Proof. Let ε > 0 be such that the invertibility results of §8 hold. Also, select wjj a familyof smooth vector fields as in §8. Assume that 2− ε < p < 2 + ε.

For a system of functions (gj)j=1,...,m ∈ ⊕1≤j≤m

Lp(∂Ω× (0, T ), E) yet to be determined, we

look for a solution u of (9.1) in the form

u(x, t) :=

∫ t

0

∫∂Ω

∑j

〈∇wj,yΓ(x, y, t, s), gj(y, s)〉 dσ(y)ds, (x, t) ∈ Ω× (0, T ). (9.7)

47

Here, as before, Γ is the Schwartz kernel of P := (∂t −L)−1. Clearly, the function u in (9.7)satisfies the first four conditions in (9.1), while the fifth condition amounts to

1

2

∑j

ν(wj)[σprinc(L)(·, ν)]−1gj + T(g1, . . . , gm) = f. (9.8)

By Theorem 8.1 this system has a solution (g1 . . . gm) satisfying∑‖gj‖Lp ≤ C‖f‖Lp . Thus,

(9.1) admits a solution for which (9.2) is verified uniformly in f .Next, the global regularity statement (9.3) follows from the layer potential representation

of the solution, used in concert with the square-function estimates from §6.There remains uniqueness for the Lp-Dirichlet initial boundary problem, 2−ε < p < 2+ε.

Since this proceeds much as in the elliptic case (cf. [1]; compare also with [22, Theorem 2.3,p. 188] and [37, p. 332]), we only sketch the main steps. Let Ωj Ω, ∂Ωj ∈ C∞, andfix x0 ∈ Ω, t0 ∈ (0, T ). Using the invertibility of the single layer potential Sj on ∂Ωj (cf.Theorem 8.3) we can construct, for each j, a Green function Gj with pole at (x0, t0) ∈ Ωj

for the problem (9.1) written for Ωj. The important feature is that

supj‖(∇Gj)

∗‖Lq(∂Ωj×(0,T )) ≤ C < +∞, (9.9)

where 1p

+ 1q

= 1. Indeed, this is more or less a direct consequence of (8.23).

Let now u be a null solution for (9.1). Having constructed Gj we then produce a Poissontype integral representation formula for u|Ωj×(0,T ) for each j. In turn, this leads to theestimate

|u(x0, t0)| ≤ C(x0, t0)‖(∇Gj)∗‖Lq(∂Ωj×(0,T )) · ‖u‖Lp(∂Ωj×(0,T ),E). (9.10)

Passing to the limit in (9.10) and utilizing (9.9) plus the fact that u|∂Ωj×(0,T ) → 0 as j →∞,Lebesgue’s Dominated Convergence Theorem allows us to conclude that u(x0, t0) = 0. Since(x0, t0) ∈ Ω× (0, T ) was arbitrary, the desired conclusion follows.

Turning our attention to the regularity statement, we only need to observe that, in thiscase, the solution of (9.1) is given by

u := S(S−1f) in Ω× (0, T ). (9.11)

The rest is a consequence of this integral representation formula and the results in §4–§6.

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[3] Mitrea, M.; Taylor, M. Potential theory on Lipschitz domains in Riemannian manifolds:Lp, Hardy, and Holder space results. Communications in Analysis and Geometry 2001, inpress.

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Department of MathematicsUniversity of Missouri-Columbia202 Mathematical Sciences BuildingColumbia, MO 65211, USAe-mail: marius@math.missouri.edu

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