Strichartz estimates and local regularity for the elastic wave equation with singular potentials
aa r X i v : . [ m a t h . A P ] A ug STRICHARTZ ESTIMATES AND LOCAL REGULARITY FOR THEELASTIC WAVE EQUATION WITH SINGULAR POTENTIALS
SEONGYEON KIM, YEHYUN KWON AND IHYEOK SEO
Abstract.
We obtain weighted L estimates for the elastic wave equation per-turbed by singular potentials including the inverse-square potential. We thendeduce the Strichartz estimates under the sole ellipticity condition for the Lam´eoperator − ∆ ∗ . This improves upon the previous result in [1] which relies on astronger condition to guarantee the self-adjointness of − ∆ ∗ . Furthermore, byestablishing local energy estimates for the elastic wave equation we also provethat the solution has local regularity. Introduction
We consider the Cauchy problem for the elastic wave equation ( ∂ t u − ∆ ∗ u + δV ( x ) u = 0 ,u (0 , x ) = f ( x ) , ∂ t u (0 , x ) = g ( x ) , (1.1)where the fields f, g : R n → R n and u : R × R n → R n take values in R n , δ ∈ R , V : R n → M n × n ( R ) is a matrix valued potential, and ∆ ∗ denotes the Lam´e operatordefined by ∆ ∗ u = µ ∆ u + ( λ + µ ) ∇ div u. (1.2)Here the Laplacian ∆ acts on each component of u and the Lam´e coefficients λ, µ ∈ R are assumed to obey the standard conditions µ > , λ + 2 µ > ∗ (see (2.3) below). The equation has been widelyused to describe wave propagation in an elastic medium, and in such case u denotesthe displacement field of the medium (e.g., [18, 21]).When λ + µ = 0, the equation (1.1) is reduced to the classical wave equation ( ∂ t u − ∆ u + δV ( x ) u = 0 ,u (0 , x ) = f ( x ) , ∂ t u (0 , x ) = g ( x ) , (1.4) Mathematics Subject Classification.
Primary:35B45,35B65; Secondary:35L05 .
Key words and phrases.
Strichartz estimates, regularity, elastic wave equation.This work was supported by a KIAS Individual Grant (MG073701) at Korea Institute for Ad-vanced Study and NRF-2020R1F1A1A01073520 (Kwon) and NRF-2019R1F1A1061316 (Seo). where f , g , u and V are now assumed scalar valued. The following space-time inte-grability of the solution to (1.4), known as Strichartz estimates , has been extensivelystudied over the past several decades: k u k L qt ˙ H σr . k f k ˙ H / + k g k ˙ H − / , (1.5)where ( q, r ) is wave-admissible, i.e., 2 ≤ q ≤ ∞ , 2 ≤ r < ∞ ,2 q + n − r ≤ n −
12 and σ = 1 q + nr − n − . (1.6)When n = 3 (1.5) fails at the endpoint ( q, r ) = (2 , ∞ ) ([17]). For the free waveequation ( δ = 0), the Strichartz estimate (1.5) with q = r was obtained in [27]in connection with the restriction theorems for the cone. See [19, 15] for the generalcase q = r . For the perturbed case ( δ = 0), (1.5) has been intensively studied to get asclose as possible to potentials that decay like | x | − for large x ([3, 10, 12, 23, 4, 5, 16]).This is because this decay is known to be critical for the validity of the Strichartzestimates (1.5). See [13] which concerns explicitly the Schr¨odinger case but can beadapted to the wave equation as well.While (1.5) is well understood for the wave equation, much less is known for theelastic case. Concerning (1.1), Barcel´o et al. [1] recently obtained (1.5) for small | δ | and the wave-admissible ( q, r ), q >
2, with a symmetric V having the critical decay | V ( x ) | . | x | − , but under the stronger condition µ > , λ + 2 µn > − ∆ ∗ and − ∆ ∗ + δV definedinitially on H . By the spectral theorem, the solution to (1.1) is now written as u = cos( t √− ∆ ∗ + δV ) f + sin( t √− ∆ ∗ + δV ) √− ∆ ∗ + δV g on which their work is based.In this abstract approach, it seems not possible to get (1.5) with the ellipticitycondition (1.3) only. To overcome this lack, we instead focus on the Fourier multiplierof √− ∆ ∗ and then represent the solution to (1.1) as the sum of the solution to thefree case plus a Duhamel term by regarding the potential term as a source term.Advantages of our approach are twofold: We do not need any assumption on λ and µ other than (1.3), and the potential V does not need to be symmetric any more.We also further relax the pointwise condition | V ( x ) | . | x | − into some integrabilitycondition at the same scale, namely, V ∈ F p where F p denotes the Fefferman-Phongclass defined, for 1 ≤ p ≤ n/
2, by F p := (cid:26) V : R n → M n × n ( R ) : k V k F p = sup x ∈ R n ,r> r − np (cid:18) Z B ( x,r ) | V ( y ) | p dy (cid:19) p < ∞ (cid:27) . As remarked in [1], by the same kind of arguments as in [13], it may be shown that the inversesquare decay is critical also for the elastic wave equation.
TRICHARTZ ESTIMATES AND LOCAL REGULARITY 3
Here, B ( x, r ) denotes the open ball in R n centered at x with radius r and | V | = (cid:0) P ni,j =1 | V ij | (cid:1) / . Note that if 1 ≤ p < n/ F p contains the weak space L n/ , ∞ , and in particular, the inverse square potential | x | − .To achieve the purpose we combine two kind of arguments; one from the theoryof maximal functions involving the A p class and the other from elliptic regularityestimates (see Section 4). As a result, we obtain the following theorem which improvesupon [1, Theorem 1.2]. Theorem 1.1.
Let n ≥ and V ∈ F p for p > ( n − / . Let u be a solution to (1.1) with ( f, g ) ∈ ˙ H / ( R n ) × ˙ H − / ( R n ) . There is an ε > such that if | δ | < ε then k u k L qt ˙ H σr . k f k ˙ H / + k g k ˙ H − / (1.7) whenever ( q, r ) is wave-admissible with q > . Here, k u k L r = k{ u j }k ℓ rj L r . In addition to the Strichartz estimates, we also obtain the following theorem whichparticularly shows a local regularity of solutions to the elastic wave equation. Let usdenote by L x,t ( w ( x )) the weighted space L ( w ( x ) dxdt ). Theorem 1.2.
Let n ≥ and V ∈ F p for p > ( n − / . There is an ε > such that if | δ | ≤ ε then there exists a unique solution u ∈ L x,t ( | V | ) to (1.1) with ( f, g ) ∈ ˙ H / ( R n ) × ˙ H − / ( R n ) satisfying u ∈ C ([0 , ∞ ); ˙ H / ( R n )) and ∂ t u ∈ C ([0 , ∞ ); ˙ H − / ( R n )) . (1.8) Furthermore, k u k L x,t ( | V | ) . k V k / F p (cid:0) k f k ˙ H / + k g k ˙ H − / (cid:1) (1.9) and for any T > x ∈ R n ,R> R Z B ( x ,R ) Z T − T | (1 − ∆) / u | + | (1 − ∆) − / ∂ t u | dtdx . C ( T )( k f k H / + k g k H − / ) . (1.10) Remark . The estimate (1.10) can be regarded as some sort of local energy estimate .Indeed, it shows that the energy in any cylinder B ( x, R ) × [ t , t ] decreases uniformlylike ∼ √ R as R → L estimate (1.9) was already shown for the wave equation (1.4)with V ( x ) ∼ | x | − and used to get the Strichartz estimates (1.5) directly withoututilizing dispersive estimates (see [4, 5]). It was also shown in [24] that the waveequation (1.4) has regularity locally as (1.10). Furthermore, as mentioned in [11], if ρ is any function such that P j ∈ Z k ρ k L ∞ ( | x |∼ j ) < ∞ , one has trivially k ρ | x | − / u j k L t ([ − T,T ]; L x ( R n )) . sup R> R Z | x | We can take, for example, ρ ( x ) = | log | x || − / − δ for any δ > 0. Combining this with(1.10) gives the following global regularity for (1.1) in terms of weighted L norm: k (1 − ∆) / u k L t ([ − T,T ]; L x ( ρ | x | − )) + k (1 − ∆) − / ∂ t u k L t ([ − T,T ]; L x ( ρ | x | − )) . C ( T )( k f k H / + k g k H − / ) . The rest of this paper is organized as follows. In Section 2, we represent thesolutions to inhomogeneous elastic wave equations utilizing the Fourier transform andprovide some useful properties of the Helmholtz decomposition by which the action ofthe Lam´e operator on vector fields is simplified. To prove the theorems in Sections5 and 6, we write the solution to (1.1) as a sum of the solution to the free case plusa Duhamel term regarding the potential term as a source term. Then we apply therelevant weighted L and local smoothing estimates obtained in Sections 3 and 4 toeach of the terms. Acknowledgment. The authors would like to thank the anonymous referee for somevaluable comments and suggestions on an issue about the theory of wave operatorsby K. Yajima [14]. 2. Preliminaries Representation of the solution. In this section, we first give the solution to thefollowing inhomogeneous elastic wave equation in terms of Fourier multipliers. ( ∂ t u ( t, x ) − ∆ ∗ u ( t, x ) = F ( t, x ) ,u (0 , x ) = f ( x ) , ∂ t u (0 , x ) = g ( x ) . (2.1)Taking the Fourier transform in the spatial variable, we convert (2.1) into thefollowing system of ODE ( ∂ t b u = − L b u + b F , b u (0) = b f , ∂ t b u (0) = b g, (2.2)where L = L ( ξ ) = µ | ξ | I n + ( λ + µ )( ξξ t ) is the (matrix) symbol of − ∆ ∗ . Since µ > λ + 2 µ > 0, for every non-zero ξ , the matrix L is positive-definite. Indeed, for z ∈ C n \ { } , if z ∗ ξ = 0 then z ∗ L ( ξ ) z = µ | ξ | | z | > z ∗ ξ = 0 then z ∗ L ( ξ ) z = µ | ξ | | z | + ( λ + µ ) | z ∗ ξ | ≥ ( λ + 2 µ ) | z ∗ ξ | > √ L . Let us denote by √− ∆ ∗ and √− ∆ ∗− the Fourier multiplier operators definedby the multipliers √ L and √ L − , respectively.By a standard method of reducing (2.2) to a first-order system and Duhamel’sprinciple, if we set v = ( b u, ∂ t b u ) t , b = (0 , b F ) t and A = (cid:18) I n − L (cid:19) , TRICHARTZ ESTIMATES AND LOCAL REGULARITY 5 we then have v ( t ) = e t A v (0) + Z t e ( t − s ) A b ( s ) ds, where e M = P ∞ k =0 M k /k ! for any square matrix M . Since e θ = cos( − iθ ) + i sin( − iθ )the first n -component of v is given by b u ( t ) = cos( t √ L ) b f + sin( t √ L ) √ L − b g + Z t sin (cid:0) ( t − s ) √ L (cid:1) √ L − b F ( s ) ds. Now taking the inverse Fourier transform, the solution to (2.1) is written as u ( t ) = cos( t √− ∆ ∗ ) f +sin( t √− ∆ ∗ ) √− ∆ ∗− g + Z t sin (cid:0) ( t − s ) √− ∆ ∗ (cid:1) √− ∆ ∗− F ( s ) ds, where ϕ ( √− ∆ ∗ ) denotes the Fourier multiplier operator defined by the symbol ϕ ( √ L ). The Helmholtz decomposition. Next we are concerned with some useful proper-ties of the so called Helmholtz decomposition and related consequences on the Lam´eoperators and Fourier multipliers.The action of the Lam´e operator ∆ ∗ on vector fields is well understood usingthe Helmholtz decomposition , which states that every f ∈ [ L ( R n )] n can be uniquelydecomposed as f = f S + f P , where div f S = 0 and f P = ∇ ϕ for some ϕ ∈ ˙ H ( R n ). The components f S and f P are orthogonal in the sense that h f S , f P i [ L ( R n )] n = 0, where h f, g i [ L ( R n )] n = n X j =1 h f j , g j i L ( R n ) is the inner product on [ L ( R n )] n . For a proof we refer the reader to [25, pp. 81–83].From now on, we will abuse notation and simply write h f, g i L for this inner product.By the decomposition it follows that∆ ∗ u = µ ∆ u S + ( λ + 2 µ )∆ u P , (2.3)which show that ∆ ∗ is elliptic under the condition (1.3). Unlike the form (1.2) thatis far from being easy to handle, this representation (2.3) gives us a possibility ofreducing the matter at first to the level of the Laplacian. Indeed, by the representationcombined with the orthogonality, the elastic wave equation (2.1) is split into thefollowing system of two wave equations involving separately the vector fields u S and u P : ( ∂ t u S − µ ∆ u S = F S ,u S (0) = f S , ∂ t u S (0) = g S , ( ∂ t u P − ( λ + 2 µ )∆ u P = F P ,u P (0) = f P , ∂ t u P (0) = g P . (2.4)In the proofs of our main results, the final steps after obtaining separated results on u S and u P will be to recombine those together in order to get desired results for u .This will be done by developing some non-trivial techniques in later sections. In fact, ϕ is a solution of the Poisson equation ∆ ϕ = div f . SEONGYEON KIM, YEHYUN KWON AND IHYEOK SEO Another favorable property of the Helmholtz decomposition is that the projectors f f S and f f P (called Leray projectors ) are in fact Fourier multipliers withmatrix valued symbols. This leads to commutativity between the Leray projectorsand certain Fourier multipliers M ( D ) defined by M ( D ) f = ( M ( ξ ) b f ) ∨ , and as suchthe L -orthogonality is available for M ( D ) f S and M ( D ) f P . Lemma 2.1. Let M ( ξ ) be an n × n matrix valued tempered function defined on R n and let f ∈ [ L ( R n )] n . Then we have ( M ( D ) f ) S = M ( D ) f S , ( M ( D ) f ) P = M ( D ) f P (2.5) if and only if M ( ξ ) = m ( ξ ) I n for some scalar function m . In this case, m ( D ) f S and m ( D ) f P are orthogonal in [ L ( R n )] n and k m ( D ) f k L = k m ( D ) f S k L + k m ( D ) f P k L . (2.6) Proof. Set f = f S + f P = ( f − ∇ ϕ ) + ∇ ϕ for ϕ ∈ H ( R n ). Taking the Fouriertransform on the equation div f = div ∇ ϕ , it is easy to see that b ϕ ( ξ ) = − i | ξ | − ξ t b f ( ξ ).Hence we get c f P ( ξ ) = d ∇ ϕ ( ξ ) = iξ b ϕ ( ξ ) = | ξ | − ξξ t b f ( ξ ) and c f S ( ξ ) = ( I n − | ξ | − ξξ t ) b f ( ξ ) , and it follows that (2.5) holds if and only if ξξ t M ( ξ ) = M ( ξ ) ξξ t for all ξ ∈ R n , whichis equivalent to M ij ( ξ ) = m ( ξ ) δ ij for some m .Once we have (2.5), the Pythagorean identity (2.6) is a direct consequence of theaforementioned L -orthogonality of the Leray projectors. (cid:3) Estimates for the free propagators Now we obtain the weighted L and local smoothing estimates for the propaga-tors e it √− ∆ ∗ and e it √− ∆ ∗ √− ∆ ∗− that constitute the solution to the homogeneousproblem (2.1) with F = 0. These estimates will be used in the next sections for theproofs of Theorems 1.1 and 1.2. The former is firstly stated as follows. Proposition 3.1. Let n ≥ and V ∈ F p for p > ( n − / . Then we have k e it √− ∆ ∗ f k L x,t ( | V | ) . k V k / F p k f k ˙ H / (3.1) and k e it √− ∆ ∗ √− ∆ ∗− g k L x,t ( | V | ) . k V k / F p k g k ˙ H − / . (3.2) Proof. Once we obtain the following norm equivalence k√− ∆ ∗ f k L ∼ k f k ˙ H (3.3)the estimates (3.1) and (3.2) are easy consequences of the estimates k cos( t √− ∆ ∗ ) f k L x,t ( | V | ) . k V k / F p k f k ˙ H / (3.4)and k sin( t √− ∆ ∗ ) √− ∆ ∗− g k L x,t ( | V | ) . k V k / F p k g k ˙ H − / (3.5) TRICHARTZ ESTIMATES AND LOCAL REGULARITY 7 which will be shown later. Indeed, assuming (3.3) for the moment, (3.5) gives k sin( t √− ∆ ∗ ) f k L x,t ( | V | ) . k V k / F p k√− ∆ ∗ f k ˙ H − / ∼ k V k / F p k f k ˙ H / . Combining this with (3.4), we get (3.1). Moreover, setting f = √− ∆ ∗− g in theinequality (3.1) and using (3.3) again, we have k e it √− ∆ ∗ √− ∆ ∗− g k L x,t ( | V | ) . k V k / F p k√− ∆ ∗− g k ˙ H / ∼ k V k / F p k g k ˙ H − / . Now we show (3.3). Since √ L is self-adjoint, Parseval’s identity gives k√− ∆ ∗ f k L = (2 π ) − n h√ L ˆ f , √ L ˆ f i L = (2 π ) − n h ˆ f , L ˆ f i L = h f, − ∆ ∗ f i L . By the Helmholtz decomposition, (2.3) and Lemma 2.1, we then have h f, − ∆ ∗ f i L = h f P + f S , − µ ∆ f S − ( λ + 2 µ )∆ f P i L = µ Z R n |∇ f S | dx + ( λ + 2 µ ) Z R n |∇ f P | dx ∼ k∇ f k L ∼ k f k H . It remains to prove (3.4) and (3.5). To show (3.4), we first set F = g = 0 in theequation (2.1). The solution is then written as the sum of the corresponding solutionsto the couple of equations (2.4):cos( t √− ∆ ∗ ) f = cos( t p − µ ∆) f S + cos( t p − ( λ + 2 µ )∆) f P . (3.6)Hence, assuming the following estimates k cos( t p − µ ∆) f S k L x,t ( | V | ) . k V k / F p k f S k ˙ H / , (3.7) k cos( t p − ( λ + 2 µ )∆) f P k L x,t ( | V | ) . k V k / F p k f P k ˙ H / (3.8)for the moment and making use of the orthogonality (2.6), we have k cos( t √− ∆ ∗ ) f k L x,t ( | V | ) . k V k / F p (cid:0) k f S k ˙ H / + k f P k ˙ H / (cid:1) . k V k / F p k f k ˙ H / , which gives (3.4). Now we need only show (3.7) since the proof of (3.8) is similar.But this follows from applying the known estimate (3.4) with ∆ ∗ replaced by theLaplacian ∆ (see (2.11) in [24]): k cos( t p − µ ∆) f S k L x,t ( | V | ) = (cid:18) n X j =1 k cos( t p − µ ∆)( f S ) j k L x,t ( | V | ) (cid:19) / . (cid:18) k V k F p n X j =1 k ( f S ) j k H / (cid:19) / = k V k / F p k f S k ˙ H / . Similarly we prove (3.5); first set F = f = 0 in (2.1), and look at (2.4) to see thatsin( t √− ∆ ∗ ) √− ∆ ∗− g = sin( t p − µ ∆) p − µ ∆ − g S + sin( t p − ( λ + 2 µ )∆) p − ( λ + 2 µ )∆ − g P . (3.9)On the right side of (3.9) we apply the known estimate (3.5) for the Laplacian ∆ (see(2.12) in [24]) and the orthogonality (2.6) to get (3.5). (cid:3) SEONGYEON KIM, YEHYUN KWON AND IHYEOK SEO Next we obtain the following local smoothing estimates using the same argumentemployed to prove Proposition 3.1. Proposition 3.2. Let n ≥ . Then we have sup x ∈ R n ,R> R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / e it √− ∆ ∗ f (cid:12)(cid:12) dtdx . k f k H / (3.10) and sup x ∈ R n ,R> R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / e it √− ∆ ∗ √− ∆ ∗− g (cid:12)(cid:12) dtdx . k g k H − / . (3.11) Proof. Making use of (3.3), the estimates (3.10) and (3.11) are easy consequences ofthe following estimatessup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / cos( t √− ∆ ∗ ) f (cid:12)(cid:12) dtdx . k f k H / (3.12)and sup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / sin( t √− ∆ ∗ ) √− ∆ ∗− g (cid:12)(cid:12) dtdx . k g k H − / . (3.13)Indeed, by using (3.3), (3.13) givessup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / sin( t √− ∆ ∗ ) f (cid:12)(cid:12) dtdx . k√− ∆ ∗ f k H − / ∼ k f k H / . Combining this with (3.12), we have (3.10). Also, using (3.10) with f = √− ∆ ∗− g and (3.3), we getsup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / e it √− ∆ ∗ √− ∆ ∗− g (cid:12)(cid:12) dtdx . k√− ∆ ∗− g k H / ∼ k g k H − / . Finally, it remains to show (3.12) and (3.13). By (3.6) and the known estimate(3.12) with ∆ ∗ replaced by ∆ (see (2.21) in [24]), we havesup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / cos( t √− ∆ ∗ ) f (cid:12)(cid:12) dtdx . k f S k H / + k f P k H / , and therefore (3.12) follows from Lemma 2.1. The second estimate (3.13) follows ina similar manner using (3.9) and the corresponding estimate (2.22) in [24] for theLaplacian ∆. (cid:3) Inhomogeneous estimates In addition to the estimates in the previous section, we need to obtain the cor-responding estimates for the inhomogeneous problem (2.1) with the zero initial data f = g = 0. Analogously to the preceding propositions, we will achieve this by re-ducing the matter at first to the level of the Laplacian, and then recombining theresulting respective estimates for u S and u P to yield the estimates for the solution u .But this recombination is more delicate in the inhomogeneous case and requires somenon-trivial techniques that consist of two kind of arguments; one from the theory of TRICHARTZ ESTIMATES AND LOCAL REGULARITY 9 maximal functions involving the A p class and the other from elliptic regularity esti-mates. Even if these are relatively well-known separately, their combination seemsnew in the present context. Elliptic regularity estimates and maximal functions. As a preliminary step,we first obtain the following elliptic regularity estimates ( cf . [1, 9]) combined withthe theory of maximal functions and A p class for weighted estimates. Lemma 4.1. Let n ≥ and V ∈ F p for < p ≤ n/ . For any n -tuple of tempereddistributions F = ( F , . . . , F n ) ∈ [ S ′ ( R n )] n , if − ∆ ψ = div F (4.1) then we have k∇ ψ k L ( W ± ) . k F k L ( W ± ) (4.2) for W = M ( | V | δ ) /δ with < δ < p . Here, M ( f ) denotes the Hardy-Littlewoodmaximal function of f and the implicit constant is independent of V . Before we prove the lemma, let us record some facts about the weights. Firstly,a nonnegative locally integrable function w : R n → [0 , ∞ ] is called an A weightwhenever there is a constant c such that M w ( x ) ≤ cw ( x )almost everywhere or, equivalently,[ w ] A := sup B balls in R n (cid:18) | B | Z B w ( x ) dx (cid:19) k w − k L ∞ ( B ) < ∞ . For 1 < p < ∞ , w is also said to be of class A p if[ w ] A p := sup B balls in R n (cid:18) | B | Z B w ( x ) dx (cid:19)(cid:18) | B | Z B w ( x ) − p − dx (cid:19) p − < ∞ . It is easy to see that if 1 ≤ p < q < ∞ then A p ⊂ A q with [ w ] A q ≤ [ w ] A p .Secondly, if 1 < p ≤ n/ V ∈ F p , then for any 1 < δ < pW = M ( | V | δ ) /δ ∈ A ∩ F p and k W k F p . k V k F p . (4.3)See [6, Lemma 1] for the proof of this useful property of the Fefferman-Phong class.Finally, the following lemma, which is taken from [26, pp. 214–215] (see also [8,Proposition 2]), enables us to make the implicit constant in (4.2) independent of V . Lemma 4.2. Let δ > . For any nonnegative function w , if M w < ∞ almosteverywhere, then ( M w ) /δ ∈ A and [ w ] A is bounded by a constant independent of w . Proof of Lemma 4.1. Taking the Fourier transform on (4.1) it is easy to see that ∂ j ψ = n X k =1 R j R k F k , (4.4)where R j is the Riesz transform defined by d R j ϕ ( ξ ) = iξ j | ξ | − b ϕ ( ξ ). We shall then usethe weighted L estimate k R j f k L ( w ) . [ w ] A k f k L ( w ) , (4.5)which holds whenever w ∈ A (see [22]).Since W ± ∈ A and [ W − ] A = [ W ] A ≤ [ W ] A , it follows now from (4.4), (4.5)and Minkowski’s inequality that k∇ ψ k L ( W ± ) ≤ n X j =1 (cid:18) n X k =1 k R j R k F k k L ( W ± ) (cid:19) . [ W ] A k F k L ( W ± ) . Since V ∈ F p and 1 < δ < p , it is easy to see that M ( | V | δ )( x ) < ∞ for every Lebesguepoint x of | V | δ . Hence, [ W ] A is uniformly bounded by Lemma 4.2. (cid:3) Inhomogeneous estimates. Now we are ready to obtain the following desired es-timates in the rest of this section. Proposition 4.3. Let n ≥ and V ∈ F p for p > ( n − / . If u is a solution to (2.1) with f = g = 0 , then we have k u k L x,t ( | V | ) . k V k F p k F k L x,t ( | V | − ) (4.6) and sup x ∈ R n ,R> R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12) |∇| / u (cid:12)(cid:12) dtdx . k V k F p k F k L x,t ( | V | − ) . (4.7) Proof. Let 1 < δ < p and W = M ( | V | δ ) /δ . Since | V | ≤ W almost everywhere, using(4.3) we may prove the proposition by replacing V with W . The motivation behindthis replacement is the availability of the elliptic regularity estimates, Lemma 4.1.By (2.4) with f = g = 0, we have u S ( t ) = Z t sin (cid:0) ( t − s ) p − µ ∆ (cid:1)p − µ ∆ − F S ( s ) ds,u P ( t ) = Z t sin (cid:0) ( t − s ) p − ( λ + 2 µ )∆) (cid:1)p − ( λ + 2 µ )∆ − F P ( s ) ds. Let us first show (4.6) for ( u S , F S ) and ( u P , F P ), respectively, in place of ( u, F ).Applying the following inequality ([24, Proposition 4.2]) for w ∈ F p , p > ( n − / (cid:13)(cid:13)(cid:13)(cid:13) Z t sin(( t − s ) √− ∆) √− ∆ ϕ ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L x,t ( | w | ) . k w k F p k ϕ k L x,t ( | w | − ) , (4.8)to each component of u S , we get k u S k L x,t ( W ) . (cid:18) k W k F p n X j =1 k ( F S ) j k L x,t ( W − ) (cid:19) / = k W k F p k F S k L x,t ( W − ) . TRICHARTZ ESTIMATES AND LOCAL REGULARITY 11 The proof of (4.6) for ( u P , F P ) is similar. Hence we have k u k L x,t ( W ) . k W k F p (cid:0) k F S k L x,t ( W − ) + k F P k L x,t ( W − ) (cid:1) . (4.9)Finally we use Lemma 4.1 to conclude k F S k L x,t ( W − ) + k F P k L x,t ( W − ) . k F k L x,t ( W − ) . Fixing t ∈ R let us write F t ( x ) = F ( x, t ) and F t = F t − ∇ ψ t + ∇ ψ t , where ψ t is a solution to − ∆ ψ t = div F t . Since div( F t + ∇ ψ t ) = 0, by the uniquenessof the Helmholtz decomposition, we conclude that( F t ) S = F t + ∇ ψ t , ( F t ) P = −∇ ψ t . Making use of Lemma 4.1, we now see k F S k L x,t ( W − ) + k F P k L x,t ( W − ) = (cid:13)(cid:13) k F t + ∇ ψ t k L x ( W − ) (cid:13)(cid:13) L t + (cid:13)(cid:13) k∇ ψ t k L x ( W − ) (cid:13)(cid:13) L t . k F k L x,t ( W − ) . Combining this with (4.9), we get k u k L x,t ( W ) . k W k F p k F k L x,t ( W − ) as desired.The proof of (4.7) is similar replacing (4.8) by the following inequality ((1.4) in[24, Theorem 1]),sup x ,R R Z B ( x ,R ) Z ∞−∞ (cid:12)(cid:12)(cid:12) |∇| Z t sin(( t − s ) √− ∆) √− ∆ G ( s ) ds (cid:12)(cid:12)(cid:12) dtdx . k w k F p k G k L x,t ( | w | − ) . So we shall omit the details. (cid:3) Proof of Theorem 1.2 This section is devoted to proving Theorem 1.2. We first write the solution to (1.1)as the sum of the solution to the free elastic wave equation ( δ = 0) plus a Duhamelterm by considering the potential term as a source term. Then we apply the estimatesobtained in the previous sections to each of the terms. Existence and uniqueness. For any V ∈ F p in Theorem 1.1 and δ > 0, define T u ( t ) := Z t sin (cid:0) ( t − s ) √− ∆ ∗ (cid:1) √− ∆ ∗− δV ( · ) u ( · , s ) ds. Regarding the potential term δV ( x ) u ( x, t ) in (1.1) as a source term ( F in (2.1)), wesee that if the solution u exists then it must satisfy u = A u ( t ) := cos( t √− ∆ ∗ ) f + sin( t √− ∆ ∗ ) √− ∆ ∗− g + T u ( t ) . (5.1)By Proposition 3.1 and the estimate (4.6) with F = δV u , the operator A : L x,t ( | V | ) → L x,t ( | V | ) is well-defined. Moreover, kT u k L x,t ( | V | ) . k V k F p k δV u k L x,t ( | V | − ) ≤ | δ |k V k F p k u k L x,t ( | V | ) . (5.2) Hence kA u − A u k L x,t ( | V | ) = kT ( u − u ) k L x,t ( | V | ) . | δ |k V k F p k u − u k L x,t ( | V | ) ,that is, A is a contraction provided that | δ | is small enough. Thus, by the contractionmapping principle, there exists a unique solution u ∈ L x,t ( | V | ) to (1.1). Proof of (1.8) . In order to prove u ∈ C ([0 , ∞ ); ˙ H / ( R n )) it is enough to showsup t ∈ R k cos( t √− ∆ ∗ ) f k ˙ H / < ∞ , (5.3)sup t ∈ R k sin( t √− ∆ ∗ ) √− ∆ ∗− g k ˙ H / < ∞ (5.4)and sup t ∈ R (cid:13)(cid:13)(cid:13)(cid:13) Z t sin (cid:0) ( t − s ) √− ∆ ∗ (cid:1) √− ∆ ∗− δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) ˙ H / < ∞ . (5.5)The estimate (5.3) follows from the decomposition (3.6) combined with Plancherel’stheorem and the orthogonality (Lemma 2.1). Indeed, for any t ∈ R , k cos( t √− ∆ ∗ ) f k ˙ H / ≤ k cos( t p − µ ∆) f S k ˙ H / + k cos( t p − ( λ + 2 µ )∆) f P k ˙ H / ≤ k|∇| / f S k L + k|∇| / f P k L . k f k ˙ H / . In a similar manner, making use of (3.9) we have that for any t ∈ R k sin( t √− ∆ ∗ ) √− ∆ ∗− g k ˙ H / ≤ k sin( t p − µ ∆) p − µ ∆ − g S k ˙ H / + k sin( t p − ( λ + 2 µ )∆) p − ( λ + 2 µ )∆ − g P k ˙ H / . k|∇| − / g S k L + k|∇| − / g P k L . k g k ˙ H − / , which gives (5.4).For (5.5), by the identity sin θ = ( e iθ − e − iθ ) / i and the norm equivalence (3.3),it is enough to show thatsup t ∈ R (cid:13)(cid:13)(cid:13)(cid:13) |∇| − / Z t e i ( t − s ) √− ∆ ∗ δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L < ∞ . Since e it √− ∆ ∗ is an isometry in [ L ( R n )] n , this follows from applying the followingestimate with F = χ [0 ,t ] δV u : (cid:13)(cid:13)(cid:13)(cid:13) Z ∞−∞ e − is √− ∆ ∗ F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) ˙ H − / . k V k / F p k F k L x,t ( | V | − ) . (5.6)Since ( e it √ L ) ∗ = e − it √ L , by duality it is easy to see that (5.6) is equivalent to (3.1).The other assertion ∂ t u ∈ C ([0 , ∞ ); ˙ H − / ( R n )) can be proved similarly. Since ∂ t u ( t ) = cos( t √− ∆ ∗ ) g − sin( t √− ∆ ∗ ) √− ∆ ∗ f + Z t cos (cid:0) ( t − s ) √− ∆ ∗ (cid:1) δV ( · ) u ( · , s ) ds (5.7)we proceed with the previous argument to see that k cos( t √− ∆ ∗ ) g k ˙ H − / . k g k ˙ H − / , k sin( t √− ∆ ∗ ) √− ∆ ∗ f k ˙ H − / . k f k ˙ H / TRICHARTZ ESTIMATES AND LOCAL REGULARITY 13 and (cid:13)(cid:13)(cid:13)(cid:13) Z t cos (cid:0) ( t − s ) √− ∆ ∗ (cid:1) δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) ˙ H − / . | δ |k V k / F p k u k L x,t ( | V | ) . Proofs of (1.9) and (1.10) . From (5.1), Proposition 3.1 and the estimate (5.2), itfollows that k u k L x,t ( | V | ) . k V k / F p (cid:0) k f k ˙ H / + k g k ˙ H − / (cid:1) + | δ |k V k F p k u k L x,t ( | V | ) . Thus, for | δ | small enough, the estimate (1.9) follows.Now we prove (1.10). We first reduce the matter to obtaining the bounds for(1 + |∇| / ) and |∇| − / in place of (1 − ∆) / and (1 − ∆) − / in (1.10), respectively.This is because (1 − ∆) / (1+ |∇| / ) − and (1 − ∆) − / |∇| / are Mikhlin multipliers.Then the reduction follows from the following boundedness of Mikhlin multipliers inMorrey Space (see, for example, Theorem 3 in [20]): Lemma 5.1. Let n ≥ and M be a Mikhlin multiplier operator. Then for < p < ∞ and ≤ λ < n M : L p,λ ( R n ) → L p,λ ( R n ) , where L p,λ denotes the Morrey space defined by k f k L p,λ = sup x ∈ R n ,r> r − λp (cid:18) Z B ( x,r ) | f ( y ) | p dy (cid:19) /p < ∞ . (Note here that L p,λ = F p particularly when λ = n − p .) Indeed, applying this lemma to each component of (1 − ∆) / u in (1.10), we havesup x ,R R Z B ( x ,R ) | (1 − ∆) / u | dx = n X j =1 (cid:13)(cid:13) (1 − ∆) / (1 + |∇| / ) − (1 + |∇| / ) u j (cid:13)(cid:13) L , . n X j =1 (cid:13)(cid:13) (1 + |∇| / ) u j (cid:13)(cid:13) L , = sup x ,R R Z B ( x ,R ) | (1 + |∇| / ) u | dx. (5.8)Similarly,sup x ,R R Z B ( x ,R ) | (1 − ∆) − / ∂ t u | dx . sup x ,R R Z B ( x ,R ) (cid:12)(cid:12) |∇| − / ∂ t u (cid:12)(cid:12) dx. (5.9)To bound the right side of (5.8), using H¨older’s inequality, we first note thatsup x ,R R Z B ( x ,R ) | u j | dx ≤ sup x ,R R k χ B k L r k u j k L q . k u j k L q provided 1 /q + 1 /r = 1 / n/r = 1. This condition n (1 / − /q ) = 1 / H / ֒ → L q . Therefore, we getsup x ,R R Z B ( x ,R ) Z T − T | u | dtdx . T n X j =1 k u j k H / . T k u k H / . But, k u k H / . k f k H / + k g k H − / which follows from the proof of (1.8) togetherwith (1.9). On the other hand, by combining (5.1) with the estimates (3.12), (3.13)and (4.7) with F = δV u , we see thatsup x ,R R Z B ( x ,R ) Z T − T (cid:12)(cid:12) |∇| / u (cid:12)(cid:12) dtdx . k f k H / + k g k H − / + | δ | k V k F p k u k L x,t ( | V | ) . k f k H / + k g k H − / , where we used (1.9) for the last inequality.It remains to bound (5.9). By (5.7), it suffices to showsup x ,R R Z B ( x ,R ) Z T − T (cid:12)(cid:12) |∇| − / √− ∆ ∗ sin( t √− ∆ ∗ ) f (cid:12)(cid:12) dxdt . k f k H / , (5.10)sup x ,R R Z B ( x ,R ) Z T − T (cid:12)(cid:12) |∇| − / cos( t √− ∆ ∗ ) g (cid:12)(cid:12) dxdt . k g k H − / (5.11)andsup x ,R R Z B ( x ,R ) Z T − T (cid:12)(cid:12)(cid:12) |∇| − / Z t cos(( t − s ) √− ∆ ∗ ) δV ( · ) u ( · , s ) ds (cid:12)(cid:12)(cid:12) dxdt . k f k H / + k g k H − / . (5.12)The left side of (5.10) is bounded by k|∇| − √− ∆ ∗ f k H / ∼ k f k H / using (3.10) with f replaced by |∇| − √− ∆ ∗ f and then the norm equivalence (3.3),while (5.11) follows from using (3.10) with f = |∇| − g . Finally, making use of (3.10),(5.6) and (1.9), the left side of (5.12) is bounded by (cid:13)(cid:13)(cid:13)(cid:13) Z t e − is √− ∆ ∗ δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) H − / . | δ | k V k F p k u k L x,t ( | V | ) . k f k H / + k g k H − / . Note added in the proof. Some estimates in Sections 3 and 4 used in the proof werealso obtained in [2] for the forward initial value problem using the spectral operationalcalculus. But our approach is based on the representation of the solution in terms ofFourier multipliers, which is simpler than the abstract one.6. Proof of Theorem 1.1 Finally we prove the Strichartz estimates in Theorem 1.1. Let us recall (5.1). Forthe homogeneous terms we use the Helmholtz decomposition ((3.6) and (3.9)), theclassical Strichartz estimate (1.5) for the free wave equation (1.4) with δ = 0, and theorthogonality (2.6) to obtain k cos( t √− ∆ ∗ ) f k L qt ˙ H σr ≤ k cos( t p − µ ∆) f S k L qt ˙ H σr + k cos( t p − ( λ + 2 µ )∆) f P k L qt ˙ H σr . k f S k ˙ H / + k f P k ˙ H / . k f k ˙ H / (6.1) TRICHARTZ ESTIMATES AND LOCAL REGULARITY 15 and k sin( t √− ∆ ∗ ) √− ∆ ∗− g k L qt ˙ H σr ≤ k sin( t p − µ ∆) p − µ ∆ − g S k L qt ˙ H σr + k sin( t p − ( λ + 2 µ )∆) p − ( λ + 2 µ )∆ − g P k L qt ˙ H σr . k g S k ˙ H − / + k g P k ˙ H − / . k g k ˙ H − / (6.2)for any wave-admissible ( q, r ) (see (1.6)).For the inhomogeneous term, by (1.9) it is enough to show kT u k L qt ˙ H σr . | δ |k V k / F p k u k L x,t ( | V | ) . (6.3)To show this, we use the identity sin θ = ( e iθ − e − iθ ) / i and the Christ-Kiselev lemma(see [7]) by which it suffices to show (cid:13)(cid:13)(cid:13)(cid:13) Z ∞−∞ e i ( t − s ) √− ∆ ∗ √− ∆ ∗− δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L qt ˙ H σr . | δ |k V k / F p k u k L x,t ( | V | ) . (6.4)since we are assuming q > 2. By the norm equivalence (3.3), the estimate (6.2) isequivalently written as k sin( t √− ∆ ∗ ) f k L qt ˙ H σr . k f k ˙ H / . Combining this with (6.1) gives k e ± it √− ∆ ∗ f k L qt ˙ H σr . k f k ˙ H / from which we see that the left side of (6.4) is bounded by (cid:13)(cid:13)(cid:13)(cid:13) Z ∞−∞ e − is √− ∆ ∗ √− ∆ ∗− δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) ˙ H / ∼ (cid:13)(cid:13)(cid:13)(cid:13) Z ∞−∞ e − is √− ∆ ∗ δV ( · ) u ( · , s ) ds (cid:13)(cid:13)(cid:13)(cid:13) ˙ H − / . Now we apply the estimate (5.6) to conclude that the above is bounded by | δ |k V k / F p k u k L x,t ( | V | ) which gives (6.4) as desired. Combining (6.1), (6.2) and (6.3) yields the Strichartzestimates (1.7), and completes the proof of Theorem 1.1. References [1] J. A. Barcel´o, L. Fanelli, A. Ruiz, M. C. Vilela and N. Visciglia, Resolvent and Strichartzestimates for elastic wave equations , Appl. Math. Lett. 49 (2015), 33–41.[2] J. A. Barcel´o, M. Folch-Gabayet, S. P´erez-Esteva, A. Ruiz and M. C. Vilela, Limiting absorptionprinciples for the Navier equation in elasticity , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012),817–842.[3] M. Beals and W. Strauss, L p estimates for the wave equation with a potential , Comm. PartialDifferential Equations 18 (1993), 1365–1397.[4] N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for thewave and Schr¨odinger equations with the inverse-square potential , J. Funct. Anal. 203 (2003),519–549.[5] N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for thewave and Schr¨odinger equations with potentials of critical decay , Indiana Univ. Math. J. 53(2004), 1665–1680. [6] F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman , Proc. Amer. Math. Soc.108 (1990), 407–409.[7] M. Christ and A. Kiselev, Maximal functions associated to filtrations , J. Funct. Anal. 179 (2001),409–425.[8] R. Coifman and R. Rochberg, Another characterization of BMO , Proc. Amer. Math. Soc. 79(1980), 249–254.[9] L. Cossetti, Bounds on eigenvalues of perturbed Lam´e operators with complex potentials ,Preprint, arXiv:1904.08445.[10] S. Cuccagna, On the wave equation with a potential , Comm. Partial Differential Equations 25(2000), 1549–1565.[11] P. D’Ancona, On large potential perturbations of the Schr¨odinger, wave and Klein-Gordonequations , Commun. Pure Appl. Anal. 19 (2020), 609–640.[12] V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential , Comm.Partial Differential Equations 28 (2003), 1325–1369.[13] M. Goldberg, L. Vega and N. Visciglia, Counterexamples of Strichartz inequalities forSchr¨odinger equations with repulsive potentials , Int. Math. Res. Not. 2006, Art. ID 13927,16pp.[14] K. Yajima, The W k,p -continuity of wave operators for Schr¨odinger operators , J. Math. Soc.Japan 47 (1995), 551–581.[15] M. Keel and T. Tao, Endpoint Strichartz estimates , Amer. J. Math. 120 (1998), 955–980.[16] S. Kim, I. Seo and J. Seok, Note on Strichartz inequalities for the wave equation with potential ,Math. Inequal. Appl. 23 (2020), 377–382.[17] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existencetheorem , Comm. Pure Appl. Math. 46 (1993), 1221–1268.[18] L. D. Landau and E. M. Lifshitz, Theory of Elasticity , Pergamon, 1970.[19] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinearwave equations , J. Funct. Anal. 130 (1995), 357–426.[20] D. Maharani, J. Widjaja and M. Wono Setya Budhi, Boundedness of Mikhlin Operator inMorrey Space , J. Phys.: Conf. Ser. 1180 (2019), 012002.[21] J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity , Prentice Hall, 1983,reprinted by Dover Publications, N.Y., 1994.[22] S. Petermichl, The sharp weighted bound for the Riesz transforms , Proc. Amer. Math. Soc. 136(2008), 1237–1249.[23] F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, L p estimates for the wave equation withthe inverse-square potential , Discrete Contin. Dyn. Syst. 9 (2003), 427–442.[24] A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent po-tentials , Duke Math. J. 76 (1994), 913–940.[25] H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach , [2013reprint of the 2001 original] Modern Birkh¨auser Classics. Birkh¨auser/Springer Basel AG, Basel,2001.[26] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals ,Princeton University Press, Princeton, NJ, 1993.[27] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutionsof wave equations , Duke Math. J. 44 (1977), 705–714. Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Ko-rea E-mail address : [email protected] School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republicof Korea E-mail address : [email protected] TRICHARTZ ESTIMATES AND LOCAL REGULARITY 17 Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Ko-rea E-mail address ::