Limiting absorption principles for the Navier equation in elasticity

Limiting absorption principles for the Navier equation in elasticity

JUANANTONIOBARCELO´, MAGALIFOLCH-GABAYET, SALVADORP ´EREZ-ESTEVA, ALBERTORUIZ ANDMARICRUZVILELA

Abstract. We prove some a priori estimates for the resolvent of Navier equa- tion in elasticity, when one approaches the spectrum (Limiting Absorption Prin- ciples). They are extensions of analogous estimates for the resolvent of the eu- clidean Laplacian inRⁿ. As a consequence, we get some results for the evolution equation and for the spectral measure.

Mathematics Subject Classification (2010):35J47 (primary); 74B05, 42B37 (secondary).

1. Introduction and statement of results

The Navier equation of the dynamic linearized elasticity in a homogeneous and isotropic medium is ruled by the wave operator

∂²

∂t² −"^∗, (1.1)

where the operator"^∗, acting on thex-variable of vector-valued functionsu(x,t)∈ C^nreads

"^∗u=µ"Iu+(λ+µ)∇divu, (1.2)

and"I denotes the diagonal matrix with the Laplace operator on the diagonal and λ, µare the Lam´e constants.

In the case of time harmonic solution of frequencyωthe operator to consider, known asspectral Navier operator, is

Lu(x)="^∗u(x)+ω²u(x),

whereω>0, x ∈R^{n, n}≥2,anduis a vector-valued function fromR^ntoC^n.

The first and fourth authors were supported by Spanish Grant MTM2008-02568, the second and the third by Proyecto CONACyT-DAIC U48633-F, and the fifth by Spanish Grant MTM2007- 62186.

Received December 20, 2010; accepted April 19, 2011.

Throughout this paper we will assume thatµ >0 and 2µ+λ> 0 so that the operator "^∗ is strongly elliptic and, we will denote bykp andks respectively the speed of propagation of longitudinal and transversal waves, which are given by

k²_p= ω²

(2µ+λ) and k_s²= ω²

µ. (1.3)

It is well known that any solutionu,of thehomogeneous spectral Navier equation Lu(x)="^∗u(x)+ω²u(x)=0, (1.4) with "^∗ given by (1.2), in a domain, can be written as the sum of the so called compressional part, denoted byu_p,and the shear part, denoted byu_s,where

up =− 1

k²_p∇divu and us =u−up. (1.5) Observe thatupandusare solutions of thevectorial homogeneous Helmholtz equa- tions"Iup(x)+k²_pup(x)=0and"Ius(x)+k_s²us(x)=0,respectively.

Besides, if u is an entire solution (i.e. a solution in the wholeR^{n) of (1.4)} satisfying the Kupradze outgoing radiation conditions:

(∂_r −ikp)up = o(r^−(n−1)/2), r = |x|→ ∞, (1.6) (∂_r−iks)us = o(r^−(n−1)/2), r = |x|→ ∞, (1.7) then,u=0(see [19] for the three-dimensional case).

As a consequence, for a vector-valued functionf∈C₀^∞,if there exists a solu- tion of theNavier equation

"^∗u(x)+ω²u(x)=f(x), ω>0, x ∈R^{n, n}≥2, (1.8) satisfying the Kupradze outgoing radiation conditions (1.6) and (1.7), whereupand usare given by (1.5) out of the support off,then the solution is unique.

Similar statements hold true for the Kupradze incoming radiation conditions (∂_r +ikp)up = o(r^−(n−1)/2), r = |x|→ ∞, (1.9)

(∂_r+iks)us = o(r^−(n−1)/2), r = |x|→ ∞. (1.10) Equation (1.8) and conditions (1.6) and (1.7) are genuine vectorial versions of the scalar Helmholtz equation, given by

"v(x)+k²v(x)=g(x), k>0, x∈R^{n, n}≥2, (1.11) and the Sommerfeld outgoing radiation condition

(∂_r−ik)v=o(r^−(n−1)/2), r = |x|→ ∞. (1.12)

Throughout this paper, given a Banach space of scalar valued functions B(Rⁿ⁾= {g:Rⁿ−→C:(g(B <∞},

its vector-valued version will be denoted by

B(Rⁿ⁾= {f:Rⁿ −→Cⁿ :(f(B<∞},

where(f(B=(|f|(B, and|f|²= |(f1, . . . , fn)|² = |f1|²+. . .+ |fn|²and its dual space byB^∗(R^n).

Limiting absortion principles. Ifz = γ +iεbelongs to the resolvent set of"^∗, i.e. ε)=0 (see (2.7) below), there exists a constantc(z) >0 independent offsuch that the following estimate holds:

(("^∗+zI)⁻¹f(L²≤c(z)(f(L²,

where I denotes the identity matrix of order n. The constant c(z)blows up as z approaches the spectrum of"^∗,that isR^+.An interesting question is the existence of Banach spacesB(Rⁿ⁾such that the estimate

(("^∗+zI)⁻¹f(B^∗ ≤c(z)(f(B (1.13) holds forf∈L²(Rⁿ⁾∩B(R^{n)withc(z) >}0 a constant independent offso that it does not blow up whenzapproaches the spectrum of"^∗.

Furthermore, given any interval[a,b]⊂(0,∞)we look for the existence of a constantc,such that

sup

γ∈[a,b] sup

ε∈(0,1)(("^∗+(γ ±iε)I)⁻¹(B−→B^∗ <c. (1.14) Under these conditions, if L²(Rⁿ⁾∩B(Rⁿ⁾is dense inB(R^n), we may consider R(z)the extension toB(Rⁿ⁾of the operator("^∗+zI)⁻¹defined inL²(Rⁿ⁾∩B(R^n).

Following standard techniques (see [1, Theorem 4.1], and see also [20]), from (1.13) one should be able to prove that forf∈B(R^n),

R(ω²+i0)f:=weak−lim

z→ω²,-z>0 R(z)f (1.15) exists inB^∗(Rⁿ⁾and is a weak solution of equation (1.8), which satisfies

(R(ω²+i0)f(B^∗ ≤c(ω²)(f(B.

We say that theweak limiting absorption principleholds for the operator"^∗in the spaceB(Rⁿ⁾for Kupradze radiation conditions if (1.14) and (1.15) are satisfied and, furthermore, under the extra assumptionf∈C^∞₀ (Rⁿ⁾∩B(Rⁿ⁾one has that out of the support off,R(ω²+i0)fis the sum, as in (1.5), ofusandup, satisfying (1.6)

and (1.7). We assume also the similar statement for the incoming conditions (1.9) and (1.10) when one takes

R(ω²−i0)f:=weak−lim

z→ω²,-z<0 R(z)f. (1.16) These principles have interesting consequences for the behavior of solution of wave equation and they are the first step to be accomplished, if one wants to treat some inverse scattering problems for the elasticity equations (see for instance [16] and [28]).

Function Spaces. Let us start by introducing the spacesB(Rⁿ⁾in which we study the limiting absorption principle.

We will use the scalar valued space introduced in [17], that we will denote by X^∗(Rⁿ⁾and which is given by the following norm:

(v(²_X∗ = sup

R>0

1 R

!

B(0,R)|v(x)|²dx,

wherevis a function defined onRⁿwith values inC^.This space is a homogeneous version of the space considered in [1] and [2]. In fact, if we consider v_λ(x) = v(λx),withλ>0,we have that

(v_λ(²_X∗ =λ¹⁻ⁿ(v(²_X∗.

We will replace the norm in its predual space by the equivalent expression:

(g(X ="

j∈Z

# 2^j+1

!

Cj

|g(x)|²dx

$1/2

,

wheregis a function defined onRⁿ with values inC^,and

Cj = {x∈Rⁿ :2^j <|x|≤2^j+1}, j ∈Z^{. (1.17)} Given a nonnegative function V,we will also use the weighted−L² space L²(V) defined as the Lebesgue space L²(Rⁿ⁾ with respect to the measure V(x)dx. We consider weights V in several spaces: Morrey–Campanato classes, certain homo- geneous Herz spaces and the space of functions with everywhere bounded X-ray transform.

Forα>0 and 1≤r ≤n/α, the Morrey-Campanato classes (see [6] and [29]) are given by

L^α,r(Rⁿ⁾=%

V ∈L^r_(oc(Rⁿ⁾:(V(L^α,r <∞&

, where

(V(L^α,r = sup

x∈R^n,ρ>0 ρ^α '

ρ⁻ⁿ

!

B(x,ρ)|V(y)|^rdy (1/r

.

Notice that some Lebesque spaces are included in these classes: L^α,n/α(Rⁿ⁾ = L^n/α(Rⁿ⁾We also remark that forr <n/αthe classL^α,r(Rⁿ⁾contains the Lorentz spaceL^n/α,∞(R^n).

For 1≤ p≤ ∞,we define the spaces Dp(Rⁿ⁾=%

V ∈L_(oc^p (Rⁿ\ {0}):(V(Dp <∞&

, where

(V(Dp = "

j∈Z

2^j(p−n)/p(Vχ_j(L^p, if 1≤ p<∞, and (V(D_∞ = "

j∈Z

2^j(Vχ_j(L^∞,

withχ_j =χ_Cj,whereχ_E denotes the characteristic function of the setEandCj is given by (1.17).

From the definition, it is easy to prove that if p₁≥ p₂,then

Dp1(Rⁿ⁾⊆ Dp2(R^{n). (1.18)} Besides, if V is a radial function, then (V(Dp(Rⁿ⁾ = (V(Dp(R⁾ where, abusing notation, V = V(x) = V(r),withr = |x|. In such a case, we will simply write (V(Dp to denote both,(V(Dp(Rⁿ⁾and(V(Dp(R⁾.

We want to note that the spaces X(R^{n)and D}p(Rⁿ⁾are homogeneous Herz spaces, in fact, following the notation used in [13] and [21], we have thatX(Rⁿ⁾= K˙₂^1/2,1(R^n),Dp(Rⁿ⁾= ˙K^(pp ^−n)/p,1(R^{n)if 1}≤ p<∞,andD_∞(Rⁿ⁾= ˙K_∞^1,1(R^n).

We denote byT⁽Rⁿ⁾(see [4]) the class of nonnegative radial functionsV such that

|||V||| :=sup

µ>0

! _∞

µ

r V(r)

(r²−µ²)^1/2dr <∞,

where as before, abusing notation,V = V(x)=V(r),withr = |x|.This is to say that theX−ray transform of the functionV is bounded everywhere.

In general, the limit

δ→lim0+

1 δ

!

Cδ,L

g(y)dy,

defines theX−ray transform of a functiong∈L¹_(oc(Rⁿ⁾on the set of all linesLin R^n.HereCδ,L = {y∈Rⁿ : d(y,L) <δand|x−y|<δ⁻¹},L= {x+tω/t ∈R}, x ∈R^n,ω∈Sⁿ⁻¹andd(y,L)denotes the Euclidean distance betweenyandL.

Note that ifV is a radial function, then there exists a positive constantCinde- pendent ofV such that

|||V|||≤C(V(Dp (1.19)

if and only if p>2 (see [4, Remark 1] and Remark 4.6 below).

We will use the vector-valued version of some of the spaces introduced above, which will be denoted byX(R^n),X∗(R^n)andL2(V).

The following theorems extend known limiting absorption principles for the Helmholtz equation to the Navier equation. The theorems are extensions in the sense that the Helmholtz equation can be viewed as the particular case of the Navier equation when we takeµ+λ=0.

Theorem 1.1. Letz = γ +iεwithε )= 0, ¹_p + _q¹ =1with _n+²₁ ≤ ¹_p −_q¹ ≤ ²_n if n>2,or²₃ ≤ ¹_p−¹_q <1ifn=2.LetV1be a nonnegative real valued function in L^2,r(R^n)with(n−1)/2<r ≤n/2andn>2,and letV2be a nonnegative radial function in D_r˜(R^{n)with2 <} r˜ ≤ ∞. Iff ∈ L²(Rⁿ⁾then, there exists a constant c>0independent ofz,f,V1andV2such that the following a priori estimates hold: (("^∗+zI)⁻¹f(X^∗ ≤ c|z|^−1/2(f(X, (1.20) (("^∗+zI)⁻¹f(L^q ≤ c|z|ⁿ²

)₁ p−¹q

*−1

(f(L^p, (1.21)

(("^∗+zI)⁻¹f(L²(V1) ≤ c(V1(_L^2,r(f(_L²_(V⁻¹

1 ), (1.22)

(("^∗+zI)⁻¹f(_L²_(V2) ≤ c|z|^−1/2(V2(Dr_˜(f(_L2(V₂⁻¹). (1.23) Besides, the weak limiting absorption principle for"^∗holds in the spacesX(R^n), L^q(R^{n),L2(V1)andL2(V2)}for Kupradze radiation conditions.

Theorem 1.2. Let psuch that _n+¹₁ ≤ ¹_p − ¹₂ ≤ ¹_n, and letV1 be a nonnegative real valued function inL^2,r(R^n)with(n−1)/2<r ≤n/2andn >2,and letV2

be a nonnegative radial function in D_r˜(R^n)with2<r˜ ≤ ∞.Iff∈ C^∞₀ (R^n)and we consider the unique solutionu=(u1, . . . ,un)of equation(1.8)satisfying(1.6) and(1.7), then, there exists a constantc >0independent ofω,f,V1andV2such that the following a priori estimates hold:

sup

1≤j≤n(∇uj(X^∗ ≤ c(f(X, (1.24) sup

x0,R>0

1 R

!

B(x0,R)|u(x)|²dx ≤ cω²ⁿ

)₁ p−¹2

*−3

(f(²L^p, (1.25) sup

x0,R>0

1 R

!

B(x0,R)|D^1/2u(x)|²dx ≤ c(V1(_L^2,r(f(²_L2(V₁⁻¹), (1.26) sup

1≤j≤n(∇uj(_L²_(V2) ≤ c(V2(Dr_˜(f(_L2(V₂⁻¹), (1.27) whereD^1/2u=(D^1/2u1, . . . ,D^1/2un).

Remark 1.3. Estimates (1.23) and (1.27) are weaker than those known for the Helmholtz equation (see (3.10), (3.14) and (1.19)). We will explain later why in this case we could not extend the known results for the Helmholtz equation to the

Navier equation (see Remark 3.11). A related open problem in harmonic analysis is the behavior of the spaceT⁽Rⁿ⁾with respect to the Hardy-Littlewood maximal operator.

Remark 1.4. The result given in (1.23) is sharp in the sense that the estimate is false forr˜ ≤2.We give the details in the last section.

Remark 1.5. It would be interesting to give an alternative proof of estimates (1.24) and (1.20) with the multiplier method in [22].

We recall certain well-known notions from the work of Kato [15] and Kato a Yajima [16]. Let H be a self adjoint operator in a Hilbert spaceH, so that the resolventRH(z)=(H−z)⁻¹is defined at least for-z)=0.

We say that a densely defined closed linear operator T fromH into itself is H-supersmooth if

|/RH(z)T^∗f,T^∗f0H|≤c(f(²_H f ∈D(T^∗), (1.28) uniformly in-z )=0. IfT satisfies the weaker condition

|-/RH(z)T^∗f,T^∗f0H|≤c(f(²_H f ∈D(T^∗), (1.29) uniformly in-z )= 0, we say thatT is H-smooth. HereT^∗is the adjoint ofT and D(T^∗)is the domain ofT^∗.

From the point of view of the Schr¨odinger evolution, (1.29) is equivalent to the

estimate ! _∞

−∞(T e^{it H} f(²_Hdt ≤2πc(f(²_H f ∈ H. (1.30) IfR(z)=("^∗+zI)⁻¹andVis a nonnegative real function inL^2,α(R^)n,(n−1)/2<

r ≤n/2, estimate (22) of Theorem 1.1, which is equivalent to the uniform bound

|/R(z)(V^1/2f),V^1/2f0L²|≤c(f(²_L2, (1.31) is thus equivalent to the"^∗-supersmoothing of the multiplication operator onR^n, n≥3, with multiplierV^1/2. We also have, by (1.30),

! _∞

−∞(e^it"∗f(²_L2(V)dt ≤C(f(²_L2 f∈L². (1.32) Similar results can be stated for the evolution wave equation, which in the case of the Navier operator is the natural to consider.

Consider the following forward initial value problem:





∂_ttu−"^∗u=F(x,t), (x,t)∈Rⁿ×R^{+, n}≥2, u(x,0)=f(x),

ut(x,0)=g(x).

(1.33)

Theorem 1.6. Letu(x,t)be a solution of (1.33)withf=g=0and, letV(x)be a nonnegative function inL^2,r(R^n)with(n−1)/2<r ≤n/2andn>2. Then there exists a positive constantC,only depending onn,such that the following a priori estimate holds:

sup

x0∈R^n,R>0

1 R

!

B(x0,R)

! _∞

0

..

.D^1/2x u(x,t) .. .²dtdx

≤C (V(_L^2,r

! _∞

0 (F(·,t)(²_L2(V⁻¹)dt.

(1.34)

Theorem 1.7. Letu(x,t)be a solution of (1.33)withF=0. IfV(x)satisfies the conditions of Theorem1.6then there exists a positive constantC,only depending onn,such that for anyγ ∈Rthe following a priori estimates hold:

! _∞

0 (Dx^γu(·,t)(²_L2(V)dt ≤C(V(_L^2,r)

(D^γ+1/2f(²_L2+(D^γ−1/2g(²_L2

* , n>2,

(1.35)

sup

x0∈R^n,R>0

1 R

!

B(x0,R)

! _∞

0

..Dx^γu(x,t)..²dtdx≤C)

(D^γf(²_L2+(D^γ−1g(²_L2

* , n>1.

(1.36) Remark 1.8. As above it would be of interest to obtain the above estimates by the multiplier method, this would allow to prove lower estimates (see ([30])).

ACKNOWLEDGEMENTS. We would like to thank A. Carbery, E. Hern´andez, F.J.

Martin-Reyes and L. Vega for enlightening conversations and for suggesting proper references.

2. The resolvent operator and the spectral measure

We introduce the Leray’s projection operator I−R^,where I denotes the identity matrix of ordern,andRis defined via the Fourier transform by

(R^f)/(ξ)= '

/f(ξ)· ξ

|ξ| ( ξ

|ξ|, ξ ∈R^{n, (2.1)}

withf∈L²(R^n)and/f(ξ)=(/f1(ξ), . . . ,/fn(ξ)).Observe thatR^fcan be written as multiplication by the operator matrix(R_iR_j):

R^f=−

# R1

# _n

"

j=1

Rjfj

$

, . . . ,Rn

# _n

"

j=1

Rjfj

$$

, (2.2)

where Rj for j =1, . . . ,n,are the Riesz transforms defined for anyg∈ L²(R^n), via the Fourier transform, by

(Rjg)/(ξ)=−i ξ_j

|ξ|/g(ξ).

The following lemma is easy to prove using the Fourier transform.

Lemma 2.1. The following identities hold for functionshin the appropriate space:

R^2h = R^{h, (2.3)}

(I−R^)2h = (I−R^{)h, (2.4)}

(I−R⁾R^h = R^(I−R^)h=0, (2.5)

"R^h = ∇ divh, (2.6)

whereRis defined by(2.1)andIdenotes the identity matrix of ordern.

The following lemma relates resolvent operators of Navier equations and re- solvent operators of Helmholtz equations using the Riesz transforms.

Lemma 2.2. Letz = γ +iεwithε )= 0andf ∈ L²(R^n).The following identity holds:

("^∗+zI)⁻¹f= 1 2µ+λ

'

"+ z 2µ+λ

(₋1

R^f+1 µ

'

"+ z µ

(₋1

(I−R^{)f, (2.7)} whereRis given by(2.1)and

("+z)⁻¹f=(("+z)⁻¹f1, ("+z)⁻¹f2,· · ·, ("+z)⁻¹fn).

Proof. Givenf∈L²(Rⁿ⁾we consider the resolvent operator("^∗+zI)⁻¹,and we write

u=("^∗+zI)⁻¹f, (2.8)

or equivalently

("^∗+zI)u=f. (2.9)

Note that applying the projectionRto both sides of this equation and, taking into account that the operators R^and "I commute, from (2.6) and (2.3), we get the following vectorial equation:

)

"+z(2µ+λ)⁻¹

*R^u=(2µ+λ)⁻¹R^f,

and therefore,

R^u=(2µ+λ)⁻¹ )

"+z(2µ+λ)⁻¹

*₋1

R^{f. (2.10)}

Arguing in a similar way with the operator I−R^,from (2.6) and (2.5), we get ("+zµ⁻¹)(I−R^)u=µ⁻¹(I−R^)f,

and therefore,

(I−R^)u=µ⁻¹ )

"+zµ⁻¹*₋1

(I−R^{)f. (2.11)}

Sinceu=R^u+(I−R^)u,the result follows from (2.8), (2.10) and (2.11).

As a consequence we give a precise description of the spectral measure of the operator"^∗.

Lemma 2.3. If the weak limiting absorption principle holds for the operator "^∗ in a Banach space B(R^{n), forf,g} ∈ S⁽Rⁿ⁾∩B(R^n),whereS⁽Rⁿ⁾denotes the Schwartz class, we have that

−

!

Rⁿ"^∗f(x)g(x)dx =

! _∞

0

ω/d Pω(f),g0. /d Pω(f),g0is given by the density function,

/d Pω(f),g0=

0 1

√λ+2µdσ1² ω

λ+2µ ∗R^f+ 1

√µdσ1²ω

µ ∗(f−R^f),g

3 dω 2√

ω, where/·,·0denotes the scalar product inC^n,dσkdenotes the measure on the sphere of radius k induced by the Lebesgue measure of R^n, Ris defined by (2.1) and, h∗f=(h∗ f1, . . . ,h∗ fn).

Proof. By polarization we may reduce to the caseg=f.

Recall Stone’s formula for the distribution function of the spectral projection measures of any selfadjoint extension of−"^∗. Let[a,b]⊂(0,∞),then

/(P(b)−P(a))f,f0

= lim

δ→0⁺ lim

/→0⁺

1 2πi

! _b+δ

a+δ

4)

(−"^∗−(ω+i/)I)⁻¹−(−"^∗−(ω−i/)I)⁻¹* f,f5

dω.

From (1.14), we have that .. .4

(−"^∗−(ω±i/)I)⁻¹f,f5...≤c(f(²_B,

and hence, using bounded convergence and the continuity of the spectrum, /(P(b)−P(a))f,f0

= 1 2πi

! _b

a lim

/→0⁺

4)

(−"^∗−(ω+i/)I)⁻¹−(−"^∗−(ω−i/)I)⁻¹* f,f5

dω.

Then, using (2.7), together with the following distributional identity fork∈R lim

/→0⁺(−"−k±i/)⁻¹h

= p.v.

!

Rⁿ

1

|ξ|²−k/h(ξ)e^{i x·ξ}dξ∓ iπ 2√

kχ_{k>0}d!σ^√_k∗h(x), (2.12) whereχ_{k>0}takes the value 1 ifk>0 and 0 otherwise, we get that

/(P(b)− P(a))f,f0

=

! _b

a

0' 1

√λ+2µdσ1² ω

λ+2µI∗R+ 1

√µdσ1²ω µ

I∗(I−R⁾ (

f,f 3 dω

2√ ω. Now we can reach the point a = 0 by monotone convergence, since the quantity inside the previous integral is nonnegative.

Using the spectral operational calculus, the above description of the spectral resolution allows us to define extensions of the operators given by spectral functions f(ω)as, for instance, solutions of the initial value problem of the wave equation (see Subsection 3.2) #

∂²

∂t² −"^∗

$ u=0,

or of the Schr¨odinger equation. This formula reduces the operational calculus to the study of Stein-Tomas operator given by convolution with √¹

kdσ!^√_k. Boundedness properties of this operator suggest the appropriate domains for selfadjoint exten- sions of the operator"^∗itself.

3. Proofs

3.1. Spectral Navier equation

It is natural, after Lemma 2.2, to study the behavior of singular integrals in the spaces considered in the statements of the theorems. We start by recalling some definitions.

Definition 3.1. Letwbe a measurable nonnegative function inL¹_(oc(R^{n).For 1<}

p<∞,we will say thatwis a weight in theApclass, and we will writew∈ Ap, if and only if for any cubeQinRⁿwe have that

' 1

|Q|

!

Q

w ( ' 1

|Q|

!

Q

w^−1/(p−1) (p−1

≤C withCa constant independent ofQ.

We will say thatwis a weight in the A1class, and we will writew ∈ A1,if and only if Mw(x)≤ Cw(x),for almost allx ∈R^n,whereM denotes the usual Hardy-Littlewood maximal operator, defined for anyg∈L¹_(oc(R^n)by

Mg(x)=sup

r>0

1 B_r

!

Br

|g(x−y)|dy, where Br is the Euclidean ball centered at the origin with radiusr.

For 1≤ p<∞,the smallest constantCsatisfying the corresponding previous condition is called the Apconstant forw.

From this definition, it is easy to prove that Ap ⊂ Aq if p < q. For more information about the Apclass see [11].

Lemma 3.2. Let 1 < p < ∞ andw ∈ Ap. For j = 1, . . . ,n, the following inequalities hold:

(Rjg(X ≤ c(g(X, (3.1)

(Rjg(L^p ≤ c(g(L^p, (3.2)

(Rjg(L^p(w) ≤ cw(g(L^p(w), (3.3) where R_j are the Riesz transforms, cis a constant independent ofg,andcw is a constant depending on the Apconstant forwbut independent ofg.

Proof. This lemma gathers several known estimates. The proofs of (3.2) and (3.3) can be found in [5] and [9] respectively.

Let us prove (3.1) (see [21], where a very general class of spaces is considered).

Fork ∈Z^,letχk =χ_Ck,whereC_kis given by (1.17). With this notation, since R_j is a linear operator, we can write

g="

k∈Z

gχk and Rjg="

k∈Z

Rj(gχk),

and therefore,

(R_jg(X ="

m∈Z

2^(m+1)/2(χ_mR_jg(L²

≤"

m∈Z

2^(m+1)/2"

k∈Z(χ_mRj(gχk)(L². (3.4) On the other hand, taking into account that

R_jg(x)=c_np.v.

!

Rⁿ

xj −yj

|x−y|ⁿ⁺¹g(y)dy,

wherecn = 0((n+1)/2)π^−(n+1)/2,form ∈Zfixed, using the Cauchy-Schwarz inequality, it is easy to prove that ifx∈Cm,then

|Rj(gχk)(x)|≤

6c2^−mn2^kn/2(gχk(_L² ifk<m−1, c2^−kn/2(gχ_k(L² ifk>m+1.

Thus, if we split the second sum in (3.4) into three parts, for k < m −1, for m−1≤k ≤m+1 and fork>m+1,we have that

(Rjg(X ≤c "

k∈Z

2^kn/2(gχk(_L² "

m>k+1

2^{−m(n−1)/2} +"

m∈Z

2^(m+1)/2

m"+1 k=m−1

(R_j(gχ_k)(L²

+c "

k∈Z

2^−kn/2(gχk(L²

"

m<k−1

2^m(n+1)/2. The result follows from here using (3.2) with p=2.

Given a nonnegative function in a Morrey-Campanato space, the following result (see [7, Lemma 1]) gives a method to construct a majorizing function in the A₁ class, and consequently in the A₂ class, within the same Morrey-Campanato space.

Lemma 3.3. Let V be a nonnegative function inL^α,r(R^{n),with0 < α < nand} 1<r ≤n/α.Ifr1is such that1<r1<r,thenW =(M V^r1)^1/r1 ∈ A1∩L^α,r(R^n), whereM denotes the usual Hardy-Littlewood maximal operator.

Furthermore, there exists a constantc>0independent ofV,such that theA1

constant forW is less thancand

(W(L^α,r ≤c(V(L^α,r. (3.5) Remark 3.4. Observe thatV(x)≤W(x)for almost allx ∈R^n.

In order to construct a majorizing function in the A1 class for a nonnegative function in a D_p(Rⁿ⁾space, which is in the sameD_p(Rⁿ⁾space, we will need the following results, that can be found in [12] and [14] respectively (see [12, Lemma 5.1 in Chapter IV] and [14, Corollary 2.1]).

Lemma 3.5 (Rubio de Francia algorithm). LetEbe a normed space of complex functions with norm(·(E,and letSbe a sublinear operator(i.e.|S(a F+bG)|≤

|a||S(F)| + |b||S(G)|, ∀a,b∈C^, ∀F,G ∈ E)bounded inEwith norm(S(.If S F ≥0for allF∈ E,then for all nonnegativeV inE,the function

W ="^∞

j=0

(2(S()^−jS^jV

is also a nonnegative function inEsuch that (i) V(x)≤W(x)for almost allx∈R^n, (ii) (W(E ≤2(V(E,

(iii) SW(x)≤C W(x)for almost allx ∈R^n,withC =2(S(.

Lemma 3.6. If1< p≤ ∞,then the usual Hardy-Littlewood maximal operator is bounded onDp(R^n).

Proof. (See [14] for general Herz spaces).

Consider 1< p<∞(the proof for p=∞is easier). Arguing as in the proof of (3.1), taking into account thatMis a sublinear operator, we have that

(Mg(Dp ≤ "

m∈Z

2^m(p−n)/p"

k∈Z

(χ_mM(gχk)(L^p. (3.6)

On the other hand, form ∈Zfixed, using the Holder inequality, it is easy to prove that ifx ∈Cm,then

|M(gχk)(x)|≤

6c2^−mn2^kn(p−1)/p(gχk(L^p ifk<m−1, c2^−kn/p(gχk(L^p ifk>m+1.

Thus, if we split the second sum in (3.6) into three parts, for k < m −1, for m−1≤k ≤m+1 and fork>m+1,we have that

(Mg(Dp ≤ c "

k∈Z

2^kn(p−1)/p(gχk(L^p

"

m>k+1

2^−m(n−1) +"

m∈Z

2^m(p−n)/p

m"+1

k=m−1

(M(gχk)(L^p

+c "

k∈Z

2^−kn/p(gχ_k(L^p

"

m<k−1

2^m.

The result follows from here using the fact thatMis a bounded operator onL^p(Rⁿ⁾ for 1< p≤ ∞.

Remark 3.7. From Lemma 3.6 we know that if 1 < p ≤ ∞ then, the usual Hardy-Littlewood maximal operator, M, is bounded on D_p(R^n). Thus, we can apply Lemma 3.5 to E = D_p(R^{n)with 1 < p} ≤ ∞and S = M,to conclude that given a nonnegative function V in Dp(Rⁿ⁾ there exists a nonnegative W in Dp(Rⁿ⁾∩A1,and therefore in Dp(Rⁿ⁾∩A2,such thatV(x)≤W(x)for almost allx ∈R^n.

Furthermore, there exists a constantc>0 independent ofV,such that theA₁ constant forW is less thancand

(W(Dp ≤c(V(Dp, 1< p≤ ∞.

Moreover, since the Maximal function of a radial function is also a radial function, ifV is a radial function, thenW is also a radial function.

We will use the following estimates for the resolvent of the Laplacian:

Theorem 3.8. Let z = γ +iεwithε )= 0, ¹_p + ¹_q = 1with _n+²₁ ≤ ¹_p − ¹_q ≤ ²_n ifn > 2,or ²₃ ≤ ¹_p − ¹_q < 1ifn = 2, and let V₁ be a nonnegative function in L^2,r(R^n)with(n−1)/2<r ≤n/2andn> 2,andV2 ∈T⁽R^n).Ifg∈ L²(Rⁿ⁾ then, there exists a constant c > 0independent ofz, g, V1 andV2 such that the following a priori estimates hold:

(("+z)⁻¹g(X^∗ ≤ c|z|^−1/2(g(X, (3.7) (("+z)⁻¹g(L^q ≤ c|z|ⁿ²

)₁ p−¹q

*−1

(g(L^p, (3.8)

(("+z)⁻¹g(L²(V1) ≤ c(V1(_L^2,r(g(_L²_(V⁻¹

1 ), (3.9)

(("+z)⁻¹g(_L²_(V2) ≤ c|z|^−1/2|||V2|||(g(_L2(V₂⁻¹). (3.10) Besides, the weak limiting absorption principle holds for"in the spaces X(R^n), L^q(R^{n),L2(V1)andL2(V2)}for the Sommerfeld radiation condition(1.12).

Theorem 3.9. Let psuch that _n+¹₁ ≤ ¹_p − ¹₂ ≤ ¹_n, and letV1 be a nonnegative function inL^2,r(R^n)with(n−1)/2< r ≤ n/2andn > 2,andV₂ ∈T⁽R^n).If g ∈ C₀^∞(Rⁿ⁾and we consider the unique solutionvof equation(1.11)satisfying (1.12), then, there exists a constantc>0independent ofk,g,V1andV2such that the following a priori estimates hold:

(∇v(X^∗ ≤ c(g(X, (3.11)

sup

x0,R>0

1 R

!

B(x0,R)|v(x)|²dx ≤ c k²ⁿ

)1 p−¹2

*−3

(g(²L^p, (3.12) sup

x0,R>0

1 R

!

B(x0,R)|D^1/2v(x)|²dx ≤ c(V1(_L^2,r(g(²_L2(V₁⁻¹), (3.13) (∇v(L²(V2) ≤ c|||V2|||(g(_L²_(V⁻¹

2 ), (3.14) whereD^1/2vis defined via the Fourier transform as(D^1/2v)/(ξ)= |ξ|^1/2/v(ξ), ξ ∈ R^n.

The results given in (3.7) and (3.11) can be found essentially in [17] (see [17, Lemma 2.4]). For a complete proof see [24] (see also [1] and [2]). The result given in (3.8) can be found in [18] and, (3.9) and (3.13) in [28]. The result given in (3.12) can be found in [27] (see also [23]). Finally, the results given in (3.10) and (3.14) can be found in [4].

The estimates (3.10) and (3.14) are a generalization of (3.7) and (3.11) respec- tively. Estimates related to (3.9) can be found in [8].

Remark 3.10. From (1.19), the estimates (3.10) and (3.14) also hold if we replace

|||V2|||with(V2(Dr_˜,withr˜>2 andV2being a radial function.