Strichartz estimates for the kinetic transport equation
aa r X i v : . [ m a t h . A P ] D ec STRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORTEQUATION
EVGENI Y OVCHAROV
Abstract.
In this paper we prove Strichartz estimates for the kinetic trans-port equation and make a detailed investigation on their range of validity.In one spatial dimension we find essentially all possible estimates, while inhigher dimensions some endpoint and inhomogeneous estimates remain open.The remaining estimates are analogous to the remaining open inhomogeneousStrichartz estimates in other contexts. The Strichartz estimates that we de-rive extend the previous work by Castella and Perthame [5] (1996) and Keeland Tao [11] (1998) in the context of the kinetic transport equation, and thetechniques of Foschi [7] to the current setting. Introduction
The purpose of this paper is to study the range of validity of the Strichartzestimates for the kinetic transport (KT) equation ∂ t u ( t, x, v ) + v · ∇ x u ( t, x, v ) = F ( t, x, v ) , ( t, x, v ) ∈ (0 , ∞ ) × R n × R n , (1) u (0 , x, v ) = f ( x, v ) . (2)The solution u to (1), (2) has the form u = U ( t ) f + W ( t ) F , where U ( t ) f = f ( x − tv, v ) , W ( t ) F = Z t −∞ U ( t − s ) F ( s ) ds, and supp F ⊆ (0 , ∞ ). We want to study estimates of the form k u k L qt L rx L pv . k f k L ax,v + k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , where L qt L rx L pv stands for the mixed Lebesgue space L q ((0 , ∞ ); L r ( R n ; L p ( R n ))),and L ax,v stands for L a ( R n ). In the sequel we shall study separately the full rangeof Strichartz estimates for each operator U ( t ) and W ( t ). The Strichartz estimatesfor the KT equation of the form(3) k U ( t ) f k L qt L rx L pv . k f k L ax,v are called homogeneous, while the estimates(4) k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v are called inhomogeneous. As it is typically done, we shall prove estimates (3) and(4) under the slightly more general assumptions t ∈ R and supp F ⊆ ( −∞ , ∞ ). For Date : November 1, 2018.2010
Mathematics Subject Classification.
Primary: 35B45, Secondary: 35Q20.
Key words and phrases.
Strichartz estimates, kinetic transport, dispersive.Much of this work was done while at the University of Edinburgh. The author would like tothank Damiano Foschi for the several suggestions he made about this work. the sake of simplicity we shall again use the same notation L qt L rx L pv for the space L q ( R ; L r ( R n ; L p ( R n ))).Strichartz estimates for the KT equation appeared first in the note of Castellaand Perthame [5] (1996) where a range of homogeneous estimates and some specialinhomogeneous estimates are presented. In the seminal paper of Keel and Tao [11](1998) the authors dedicate a small paragraph to the KT equation where they ex-tend the homogeneous estimates proved in [5]. However, the endpoint homogeneousestimate proves too difficult to be resolved by the methods presented in [11], whichinitiates an ongoing mathematical investigation. The first partial answer in thatdirection is given by Guo and Peng [8] (2007) who provide counterexamples in onespatial dimension confirming the (expected) failure of the endpoint estimate k U ( t ) f k L t L ∞ x L v . k f k L x,v (5)there.Presently, we extend the work of the previously mentioned authors by makinga detailed analysis of the range of validity of the Strichartz estimates for the KTequation. The new estimates that we prove concern mostly the inhomogeneousoperator W ( t ) but we also prove new estimates for U ( t ) of the more general form(6) k U ( t ) f k L qt L rx L pv . k f k L bx L cv . In fact we prove that the latter estimates are equivalent to some special inhomo-geneous estimates (4) which explains why the investigation of the inhomogeneousestimates is to us of primary interest.As a motivation for studying Strichartz estimates for the KT equation we canpoint out the fact that they have been a very fruitful tool in the context of the waveand the Schr¨odinger equations in the analysis of the nonlinear Cauchy problem.Application of such type appeared in Bournaveas et al. [4] (2008) where the authorsprove the existence of some weak solutions to a nonlinear kinetic system modelingchemotaxis.The paper is organized as follows. In the next section we present the main resultsof the paper which we hope are given in a form convenient for referencing. In thesection immediately after it we make some additional introductory remarks thatwill be useful to those who wish to read on with our proofs. They follow in thesections after it.2.
Strichartz estimates for the KT equation
In order to formulate our results we first need several definitions.
Definition 2.1.
We say that the exponent triplet ( q, r, p ), 1 ≤ p, q, r ≤ ∞ , is KT-admissible if 1 q = n (cid:18) p − r (cid:19) , a def = HM( p, r ) , (7) 1 ≤ a ≤ ∞ , p ∗ ( a ) ≤ p ≤ a, a ≤ r ≤ r ∗ ( a ) , (8)except in the case n = 1, ( q, r, p ) = ( a, ∞ , a/ p, r ) we have denoted the harmonic mean of p and r , i.e. a = HM( p, r ) whenever1 a = 12 (cid:18) r + 1 p (cid:19) . TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 3
For convenience we give explicitly the exact lower boundary p ∗ to p and the exactupper boundary r ∗ to r which are given in ( p ∗ ( a ) = nan +1 , r ∗ ( a ) = nan − , if n +1 n ≤ a ≤ ∞ ,p ∗ ( a ) = 1 , r ∗ ( a ) = a − a , if 1 ≤ a ≤ n +1 n . (9)Note that the second line in (9) is placed to restrict the Lebesgue exponents p and r in the range [1 , ∞ ].We have used above the convention that 1 / ∞ , and thus, for example, for n = 1 r ∗ ( a ) = ∞ . Furthermore, throughout this text we shall always use theconvention 1 / ∞ = 0 and 1 / ∞ in the context of Lebesgue exponents. Tripletsof the form ( q, r, p ) = ( a, r ∗ ( a ) , p ∗ ( a )), for ( n + 1) /n ≤ a < ∞ , will be calledendpoint. The H¨older conjugate exponent will be denoted by ′ e.g. 1 /r + 1 /r ′ = 1.Conditions (8), (9) are equivalent to a ≤ q , and p ≤ r , and 1 ≤ a , p , condition (7)is equivalent to 1 q + nr = na , HM( p, r ) = a. Note that although the latter redaction of condition (7) resembles more closelythe admissability conditions for the wave and the Schr¨odinger equations, the for-mer version is more natural in the present context in view of the fact that in theinhomogeneous estimates the Lebesgue exponent a does not appear explicitly.To describe the range of the inhomogeneous estimates we shall need the nexttwo definitions. Following Foschi [7], we give the following Definition 2.2.
We say that the exponent triplet ( q, r, p ) is
KT-acceptable if1 q < n (cid:18) p − r (cid:19) , ≤ q ≤ ∞ , ≤ p < r ≤ ∞ , (10)or if q = ∞ , 1 ≤ p = r ≤ ∞ .Note that a KT-acceptable triplet is always KT-admissible. To further describethe range of validity of the inhomogeneous estimates we give the following Definition 2.3.
We say that the two KT-acceptable exponent triplets ( q, r, p ) and(˜ q, ˜ r, ˜ p ) are jointly KT-acceptable if1 q + 1˜ q = n (cid:18) − r − r (cid:19) , q + 1˜ q ≤ , (11) HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) , (12)and if the exponents satisfy further the additional restrictions(i) for r, ˜ r = ∞ (13) n − p ′ < n ˜ r , n − p ′ < nr , (ii) if r = ∞ then the point (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) ∈ Σ ∪ B ,Σ = { ( µ, , κ, ν, − κ,
1) : 0 < µ, ν < , < µ + ν < , κ = ( µ + ν ) /n } ,B = (0 , , , , , , (14)(iii) if ˜ r = ∞ then the point (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) ∈ Σ ∪ C ,Σ = { ( µ, − κ, , ν, , κ ) : 0 < µ, ν < , < µ + ν < , κ = ( µ + ν ) /n } ,C = (0 , , , , , . (15) EVGENI Y OVCHAROV
Conditions (14) and (15) are sharp which is demonstrated on counterexamplesbased on Besicovitch sets in Ovcharov [12].
Theorem 2.4.
Let u be the solution to the Cauchy problem for (1) , (2) . Then theestimate (16) k u k L qt L rx L pv . k f k L ax,v + k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , holds for all f ∈ L ax,v and all F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v if and only if ( q, r, p ) and (˜ q, ˜ r, ˜ p ) aretwo KT-admissible exponent triplets and a = HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) , apart from thecase in higher dimensions n > of ( q, r, p ) being an endpoint triplet for which thecorresponding estimates remain unresolved. Note that Theorem 2.4 allows the second triplet (˜ q, ˜ r, ˜ p ) to be endpoint andexcludes only the estimates where the first triplet ( q, r, p ) is endpoint. Below weemploy the notation L qt L rx L pv ( V ) for the Lebesgue space L q ( R ; L r ( R n ; L p ( V ))) (or L q ((0 , ∞ ); L r ( R n ; L p ( V )))) over a finite velocity domain V ⊂ R n . Theorem 2.5 (Generalized inhomogeneous estimates) . Suppose that ( q, r, p ) and (˜ q, ˜ r, ˜ p ) are two jointly KT-acceptable exponent triplets that further satisfy the fol-lowing conditions (i) 1 < q, ˜ q < ∞ , q > ˜ q ′ , then the estimate (17) k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v holds for all F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v . (ii) ˜ q = ∞ , < q < ∞ , then the estimate k W ( t ) F k L q, ∞ t L rx L pv . k F k L t L ˜ r ′ x L ˜ p ′ v , (18) (cid:0) k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , q ≥ ˜ p ′ (cid:1) holds for all F ∈ L t L ˜ r ′ x L ˜ p ′ v . (iii) q = ∞ , < ˜ q < ∞ , then the estimate k W ( t ) F k L ∞ t L rx L pv . k F k L ˜ q ′ , t L ˜ r ′ x L ˜ p ′ v , (19) (cid:0) k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ˜ q ′ ≤ p (cid:1) holds for all F ∈ L ˜ q ′ , t L ˜ r ′ x L ˜ p ′ v ( F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v ) . (iv) 1 < q, ˜ q < ∞ , q = ˜ q ′ . Under the assumption of a finite velocity space V ⊂ R n we have that the estimate k W ( t ) F k L qt L r,qx L Pv ( V ) . V k F k L ˜ q ′ t L ˜ r ′ , ˜ q ′ x L ˜ P ′ v ( V ) (20) ( k W ( t ) F k L qt L rx L Pv ( V ) . V k F k L ˜ q ′ t L ˜ r ′ x L ˜ P ′ v ( V ) , q ≤ r and ˜ q ≤ ˜ r ) holds for all F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v ( V ) , whenever P, ˜ P are such that ≤ P < p and ≤ ˜ P < ˜ p .Conversely, if estimate (17) holds for all F ∈ L q ′ t L r ′ x L p ′ v , then ( q, r, p ) and (˜ q, ˜ r, ˜ p ) must be two jointly KT-acceptable exponent triplets, apart from condition (13) whose necessity is not fully verified. TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 5
Remark . As indicated, in the range q ≥ ˜ p ′ estimate (18) can be strengthenby replacing the Lorentz norm L q, ∞ by the Lebesgue norm L q . Analogously, theLorentz norm L ˜ q ′ , in estimate (19) can be replaced by the Lebesgue norm L ˜ q ′ inthe range ˜ q ′ ≤ p . This is proved in Lemma 8.3. By the same token, in the range q ≤ r and ˜ q ≤ ˜ r , estimate (20) implies its analogue in Lebesgue norms. Remark . If we restrict ourselves to finite time intervals [0 , T ], we have thecontinuous embeddings L q,r ([0 , T ]) ֒ → L p ([0 , T ]) , q > p, ≤ q, p, r ≤ ∞ ,L p ([0 , T ]) ֒ → L q,r ([0 , T ]) , p > q, ≤ q, p, r ≤ ∞ , see [2, p. 217]. For example, let ( ∞ , r, p ) and (˜ q, ˜ r, ˜ p ) be such that estimate (19)holds and let 1 ≤ e Q < ˜ q . Then we have the local inhomogeneous estimate k W ( t ) F k L ∞ t ([0 ,T ]; L qx L rv ) . T k F k L e Q ′ t ([0 ,T ]; L ˜ r ′ x L ˜ p ′ v ) for any 0 < T < ∞ and any F ∈ L ˜ Q ′ t ([0 , T ]; L ˜ r ′ x L ˜ p ′ v ). Theorem 2.8 (The Equivalence Theorem) .A.
The following three estimates k U ( t ) f k L qt L rx L pv . k f k L bx L cv , ∀ f ∈ L bx L cv , (21) k W ( t ) F k L qt L rx L pv . k F k L t L bx L cv , ∀ F ∈ L t L bx L cv , (22) k W ( t ) F k L ∞ t L b ′ x L c ′ v . k F k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L q ′ t L r ′ x L p ′ v . (23) are equivalent whenever ≤ q, r, p, b, c ≤ ∞ . B. Whenever b = c = 2 estimate (21) is equivalent to k W ( t ) F k L qt L rx L r ′ v . k F k L q ′ t L r ′ x L rv , ∀ F ∈ L q ′ t L r ′ x L rv . (24)As a direct consequence of Theorem 2.5 and the Equivalence Theorem we obtain Corollary 2.9.
We have the estimate (25) k U ( t ) f k L q, ∞ t L rx L pv . k f k L bx L cv for all f ∈ L bx L cv whenever the exponent 5-vector ( q, r, p, b, c ) satisfies the followingconditions q + nr = nb , HM( p, r ) = HM( b, c ) def = a, (26) p b = c .The Equivalence Theorem, part B, together with Theorem 2.5, part (iv), implythe following weaker substitute for the endpoint homogeneous Strichartz estimateover finite velocity spaces. Corollary 2.11.
Let ≤ P < p ∗ ( a ) , ( n + 1) /n ≤ a < ∞ , and let V ⊂ R n bebounded. Then, the following estimate k U ( t ) f k L at L r ∗ ( a ) x L Pv ( V ) . V k f k L ax,v , (31) holds for all f ∈ L ax,v . General introductory remarks
We owe the reader an explanation why our results do not follow from earlierworks on Strichartz estimates. As it is well-known these estimates follow from twomain ingredients, the decay and the energy estimates. Besides these, in the contextof the KT equation, it is also necessary to assume a further structure condition thatgreatly increases the range of estimates we may prove. Consider the decay estimate(32) k U ( t ) f k L ∞ x L v . | t | n k f k L x L ∞ v , ∀ t ∈ R , for the KT equation. Note that the mixed Lebesgue norm in (32) creates difficultiesin interpolation if one uses the real method. Moreover, the general results of Keeland Tao [11] and Taggart [16] are based on the real method and if applied to thepresent context produce Strichartz estimates in non-Lebesgue norms. Additionalcomplication arises from the fact that the KT propagator U ( t ) preserves a wholefamily of Lebesgue norms(33) k U ( t ) f k L ax,v = k f k L ax,v , ∀ t ∈ R , < a ≤ ∞ , and not just the L -norm (corresponding to the energy estimate in other contexts).To mark the different nature of estimate (33) we shall call it the transport estimateand any class L ax,v for 1 ≤ a ≤ ∞ we shall call a transport class. The transport esti-mate is a consequence of the special case a = 2 in (33) and the following invarianceof the homogeneous KT equation(34) f → f α , U ( t ) f → ( U ( t ) f ) α , < α < ∞ . Furthermore, this invariance allows us to prove new homogeneous Strichartz esti-mates from already proven ones. In fact, the exponents in k U ( t ) f k L qt L rx L pv . k f k L bx L cv , ∀ f ∈ L bx L cv , transform according to the rule(35) ( q, r, p, b, c ) → ( αq, αr, αp, αb, αc ) TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 7 and any two estimates whose exponents are related in such a way are equivalent.Note that there is no such convenient tool in the inhomogeneous setting.To summarize, the Strichartz estimates that we shall prove in the sequel are con-sequences of the decay estimate (32), the transport estimate (33), and the structuralassumption (34).We would like next to highlight some special advances that we make in thepresent work. Most of all, we study the equivalence between different types ofStrichartz estimates. One such result is the fact that in the context of the KTequation the Strichartz estimates for the operator W ( t ) and that of T T ∗ , T T ∗ F = Z ∞−∞ U ( t − s ) F ( s ) ds, are equivalent. This greatly simplifies the use of duality arguments and we do notany longer need the Christ-Kiselev lemma in order to deduce the inhomogeneousStrichartz estimates via the corresponding estimates for T T ∗ .We also show that any homogeneous estimate has corresponding inhomogeneousestimates to which it is equivalent e.g. k U ( t ) f k L qt L rx L pv . k f k L bx L cv , ∀ f ∈ L bx L cv , (36) k W ( t ) F k L qt L rx L pv . k F k L t L bx L cv ∀ F ∈ L t L bx L cv . (37)are equivalent, and more generally see Theorem 2.8.One possible application of this equivalence is in the study of the range of esti-mates (36). Since the proof of these estimates in the present context is an entirelynew result, we shall give an example from the context of the Schr¨odinger equation.The estimate k U s ( t ) f k L qt L rx . k f k L px , ∀ f ∈ L px , (38)where by U s ( t ) we have denoted the Schr¨odinger propagator, was investigated by T.Kato [10] (1994) for 1 < p ≤
2. We obtain a larger range of such estimates in higherdimensions n >
2, see our PhD Thesis [13]. This improvement is essentially due tothe fact that our approach benefits from the more recent advances introduced byKeel and Tao [11] and Foschi [7] in the inhomogeneous setting and, of course, theimplication of (36) by (37).The last special result to be considered here is concerned with the endpointStrichartz estimates for the KT equation in higher dimensions as in Theorem 2.5,part (iv). We prove there a class of estimates with a loss of integrability that canbe made arbitrary small compared to the original endpoint estimates. However,our estimates are given entirely in terms of Lebesgue norms which is useful inapplications. Furthermore, we give a counterexample showing that there does notexist a family of perturbed local estimates in a “full neighborhood” around anygiven endpoint estimate. The existence of the latter is required by the methods ofKeel and Tao [11] and Foschi [7], and hence why it has not been yet possible toresolve in the positive the endpoint estimates of the considered type.The different cases in Theorem 2.5 can be visualized quite easily. Let us firstremember that the Lebesgue space L p is best seen as a “function” of 1 /p rather than p in the context of interpolation. Therefore, the range of validity of the estimate (17)corresponds to a region in R of points with coordinates (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ).The projection of that region over the 1 /q -1 / ˜ q -plane is visualized in Figure 1. EVGENI Y OVCHAROV /q / ˜ q q + q = 1 11 O A B Figure 1.
Acceptable range of (1 /q, / ˜ q ).The inner part of ∆ OAB corresponds to the non-endpoint inhomogeneous esti-mates, while its three sides correspond to the endpoint inhomogeneous estimates.In the context of Theorem 2.5, the inner part of ∆
OAB corresponds to part (i),the cathetus OA - to part (ii), the cathetus OB - to part (iii), and the hypotenuse AB - to part (iv). The inhomogeneous estimates can be put into three groups inrising order of difficulty: the inner part of ∆ OAB , the two catheti OA and OB ,and the hypotenuse AB .Note that in one spatial dimension condition (13) is void and thus there thecomplete range of validity of the Strichartz estimates for the KT equation is nowknown. In higher dimensions, however, the necessity (sharpness) of this conditionis open. A similar question one encounters in other contexts, see e.g. Foschi [7] inthe context of the Schr¨odinger equation.Before we end this section we remark that the estimates that we prove remainvalid for more general domains than those considered in the definition of equation(1). For example, the domain of t may be any interval I ⊆ R , and the domain of v may be any measurable set V ⊆ R n . The claims follow from the fact that thetransport and the dispersive estimate for the KT equation remain valid for thesedomains, as can one easily see by a simple modification of the proofs of Lemmas4.1 and 4.2.The remainder of the paper is organized as follows. In the next section we givesome auxiliary facts about the KT equation and in Section 5 we present the T T ∗ method and some other duality arguments including the proof of the EquivalenceTheorem 2.8. The proof of the Strichartz estimates for the Cauchy problem (The-orem 2.4) is given in Section 6. The local inhomogeneous Strichartz estimates arederived in Section 7. The generalized Strichartz estimates are proved in Section8. In Section 9 we show sharpness of the estimates that we prove by means ofcounterexamples. We finish the paper with Section 10 where we list some stillunanswered questions regarding the Strichartz estimates for the KT equation.4. Some properties of the kinetic transport equation
Lemma 4.1 (The dispersive estimate [14]) . The kinetic transport evolution group U ( t ) obeys the estimate (39) k U ( t ) f k L ∞ x L v ≤ | t | n k f k L x L ∞ v , TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 9 for all f ∈ L x L ∞ v .Proof. Z R n | U ( t ) f | dv = Z R n | f ( x − tv, v ) | dv ≤ Z R n sup y ∈ R n | f ( x − tv, y ) | dv ≤ | t | n Z R n sup y ∈ R n | f ( z, y ) | dz = 1 | t | n k f k L x L ∞ v . (cid:3) Lemma 4.2 (The transport estimate) . The kinetic transport evolution group U ( t ) obeys the estimate (40) k U ( t ) f k L ∞ t L ax L av ≤ k f k L ax,v , < a ≤ ∞ , for all f ∈ L ax,v .Proof. Trivial. (cid:3)
Corollary 4.3 (The decay estimate) . The kinetic transport evolution group U ( t ) obeys the estimate (41) k U ( t ) f k L rx L pv ≤ | t | n ( p − r ) k f k L px L rv , ≤ p ≤ r ≤ ∞ , for all f ∈ L px L rv .Proof. Complex interpolation between the decay estimate (39) and the two trans-port estimates (40) with a = 1 and a = ∞ . (cid:3) Lemma 4.4.
The formal adjoint to U ( t ) is the operator U ∗ ( t ) = U ( − t ) .Proof. We denote by h· , ·i the scalar product on L ( R n ). Thus, h U ( t ) f, g i = Z ∞−∞ f ( x − tv, v ) g ( x, v ) dxdv = Z ∞−∞ f ( y, v ) g ( y + tv, v ) dydv = h f, U ( − t ) g i , where we have made the substitution y = x − tv . (cid:3) Lemma 4.5 (Scaling properties of U ( t ) and W ( t )) . The evolution operators U ( t ) and W ( t ) enjoy the following scaling properties U ( t ) f λ = f ( x/λ − tv/λ, v ) = { U ( · ) f } ( t/λ, x/λ, v ) , where f λ ( x, v ) = f ( x/λ, v ) ,U ( t ) f λ = f ( x/λ − tv/λ, v/λ ) = { U ( · ) f } ( t, x/λ, v/λ ) , where f λ ( x, v ) = f ( x/λ, v/λ ) ,W ( t ) F λ = λ Z t/λ −∞ F ( s, x/λ − ( t/λ − s ) v, v ) ds = λ { W ( · ) F } ( t/λ, x/λ, v ) , where F λ ( t, x, v ) = F ( t/λ, x/λ, v ) ,W ( t ) F λ = Z t −∞ F ( s, x/λ − ( t − s ) v/λ, v/λ ) ds = { W ( · ) F } ( t, x/λ, v/λ ) , where F λ ( t, x, v ) = F ( t, x/λ, v/λ ) . Proof.
Direct inspection. (cid:3) Duality and the
T T ∗ -principle At the heart of the proof of Strichartz estimates lie duality arguments. In thissection we introduce the main elements of all duality constructions in later proofs.5.1.
Basics.
Let us consider the operator T : L x,v → L qt L rx L r ′ v , { T f } ( t, x, v ) = f ( x − tv, v ) = U ( t ) f, for some Lebesgue exponents 2 ≤ q, r ≤ ∞ . Its formal adjoint is the L -valuedintegral T ∗ : L q ′ t L r ′ x L rv → L x,v , { T ∗ F } ( x, v ) = Z ∞−∞ F ( s, x + sv, v ) ds. The composition of the two has the form
T T ∗ : L q ′ t L r ′ x L rv → L qt L rx L r ′ v , { T T ∗ F } ( t, x, v ) = Z ∞−∞ F ( s, x − ( t − s ) v, v ) ds = Z ∞−∞ U ( t − s ) F ( s ) ds. In view of the
T T ∗ -principle, see e.g. [15, p. 113], T and T T ∗ are equally boundedwith k T k = k T T ∗ k . Thus, the following two estimates are equivalent k T f k L qt L rx L r ′ v ≤ C k f k L x,v , ∀ f ∈ L x,v , (42) k T T ∗ F k L qt L rx L r ′ v ≤ C k F k L q ′ t L r ′ x L rv , ∀ F ∈ L q ′ t L r ′ x L p ′ v , (43)where C = k T k . We shall call (43) the symmetric T T ∗ -estimate. By h· , ·i we denotethe duality pairing on R n h f, g i = Z R n f ( x, v ) g ( x, v ) dxdv for the (mixed) Lebesgue spaces. Thus, in bilinear formulation (43) reads (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ h{ T T ∗ F } ( t ) , G ( t ) i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k F k L q ′ t L r ′ x L rv k G k L q ′ t L r ′ x L rv . In view of Lemma 4.4, this is equivalent to (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ Z ∞−∞ h U ( s ) ∗ F, U ( t ) ∗ G i dsdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k F k L q ′ t L r ′ x L rv k G k L q ′ t L r ′ x L rv , ∀ F, ∀ G ∈ L q ′ t L r ′ x L rv . In [11] Keel and Tao noted that by symmetry, i.e. by changingthe roles of F and G , the latter estimate is always implied by the estimate | B ( F, G ) | ≤ C k F k L q ′ t L r ′ x L rv k G k L q ′ t L r ′ x L rv , ∀ F, ∀ G ∈ L q ′ t L r ′ x L rv , (44)for the bilinear form B ( F, G ) =
Z Z s TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 11 We now consider the inhomogeneous estimates. Suppose that ( q, r, p ) and (˜ q, ˜ r, ˜ p )are two exponent triplets such that k T f k L qt L rx L pv ≤ C k f k L ax,v , ∀ f ∈ L ax,v , k T f k L ˜ qt L ˜ rx L ˜ pv ≤ C k f k L a ′ x,v , ∀ f ∈ L a ′ x,v , for some 1 ≤ a ≤ ∞ . The composition L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v T ∗ → L ax,v T → L qt L rx L pv is bounded and thus k T T ∗ F k L qt L rx L pv ≤ C k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v . (45)This is the general non-symmetric T T ∗ -estimate for the KT equation. It does notany longer imply boundedness for the operator T , but as we shall see next, it impliesboundedness for the operator W ( t ) F . Lemma 5.1. In the context of the KT equation the non-symmetric T T ∗ -estimateand the inhomogeneous Strichartz estimates are equivalent, i.e (45) is equivalent to k W ( t ) F k L qt L rx L pv ≤ C k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v . (46) Proof. (i) In one direction the claim follows immediately from | W ( t ) F | ≤ T T ∗ | F | . (ii) In the other direction we have the following argument. It is easy to see thatby duality (46) is equivalent to | B ( F, G ) | . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v k G k L q ′ t L r ′ x L p ′ v , (47) ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ G ∈ L q ′ t L r ′ x L p ′ v . By making the substitution σ = − s, τ = − t in the definition of B ( F, G ) we get B ( F, G ) = Z Z τ<σ h U ( − σ ) ∗ F, U ( − τ ) ∗ G i dτ dσ. The integral in the line above can be written as( − n Z Z τ<σ h U ( σ ) ∗ F ′ , U ( τ ) ∗ G ′ i dτ dσ def = ( − n B ′ ( F ′ , G ′ ) , by making the substitution x → − x and setting F ′ ( t, x, v ) = F ( − t, − x, v ), G ′ ( t, x, v )= G ( − t, − x, v ). Thus the boundedness of the bilinear form B ( · , · ) implies theboundedness of the bilinear form B ′ ( · , · ) on the same spaces. The claim followsfrom the fact that the boundedness of T T ∗ is equivalent to that of the bilinearform B + B ′ . (cid:3) For convenience we summarize the bilinear formulation of the duality argumentsof this paragraph in Lemma 5.2. (i) The boundedness of the operator T : L x,v → L qt L rx L r ′ v of the form T f = U ( t ) f is equivalent to the boundedness of the bilinear mapping B : L q ′ t L r ′ x L rv × L q ′ t L r ′ x L rv → C . (ii) The boundedness of the operators W ( t ) : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v → L qt L rx L pv and T T ∗ : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v → L qt L rx L pv is equivalent to that of the bilinear mapping B : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → C . Equivalence of Strichartz estimates.Lemma 5.3 (The Duality lemma) . The following two estimates for W ( t ) are equiv-alent k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , k W ( t ) F k L ˜ qt L ˜ rx L ˜ pv . k F k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L q ′ t L r ′ x L p ′ v , for ≤ p, q ≤ ∞ .Proof. It follows immediately from the fact the boundedness of W ( t ) is equivalentto that of the symmetric operator T T ∗ on the considered spaces. (cid:3) Theorem 5.4 (The Equivalence Theorem) . (i) The following three estimates k U ( t ) f k L qt L rx L pv . k f k L bx L cv , ∀ f ∈ L bx L cv , (48) k W ( t ) F k L qt L rx L pv . k F k L t L bx L cv , ∀ F ∈ L t L bx L cv , (49) k W ( t ) F k L ∞ t L b ′ x L c ′ v . k F k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L q ′ t L r ′ x L p ′ v . (50) are equivalent whenever ≤ q, r, p, b, c ≤ ∞ . (ii) Whenever b = c = 2 estimate (48) is equivalent to k W ( t ) F k L qt L rx L r ′ v . k F k L q ′ t L r ′ x L rv , ∀ F ∈ L q ′ t L r ′ x L rv . (51) Proof. (i) The homogeneous estimate (48) trivially implies the first inhomogeneousestimate (49). In view of the Duality lemma 5.3, the two inhomogeneous estimates(49) and (50) are equivalent. All it remains to show is that (49) implies (48).Let us first give a short formal proof. We choose F ( t ) = δ ( t ) f where δ ( t ) is thedelta function on R and f ∈ L bx L cv . Consequently, W ( t )[ δ ( · ) f ] = U ( t ) f and thus k U ( t ) f k L qt L rx L pv . k δ ( t ) f k L t L bx L cv = k f k L bx L cv . To make that rigorous instead of δ ( t ) we consider a smooth approximation ofthe identity δ ǫ ( t ), for ǫ > 0. So we are given the estimate | B ( F, G ) | . k F k L t L bx L cv k G k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L t L bx L cv , ∀ G ∈ L q ′ t L r ′ x L rv . It would be enough to prove that B ( δ ǫ ∗ f, G ) → Z ∞−∞ h U ( t ) f, G i dt since we have that k δ ǫ ∗ f k L t L bx L cv = k f k L bx L cv , for any ǫ > 0. To prove the limit itwould be enough to consider only nonnegative functions f and smooth nonnegativefunctions G of compact support in t ≥ 0. For t > B ( δ ǫ f, G ) = Z Z t −∞ h δ ǫ ( s ) ∗ f, U ( s − t ) G i dsdt = Z ∞−∞ Z ∞−∞ δ ǫ ( t − s ) h f, U ( − s ) G i dsdt → Z h U ( t ) f, G i dt. TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 13 The last limit is justified by the fact that the function h ( s ) = h f, U ( − s ) G i iscontinuous and thus δ ǫ ∗ h ( t ) → h ( t ) as ǫ → 0. Then, in view of Fatou’s lemma, Z h U ( t ) f, G i dt ≤ lim inf ǫ → B ( δ ǫ f, G ) . k f k L bx L cv k G k L q ′ t L r ′ x L p ′ v , ∀ f ∈ L bx L cv . (ii) This follows directly from Lemma 5.2. (cid:3) Remark . We shall need a slightly more general form of the the EquivalenceTheorem in the sequel, where the temporal norm is the Lorentz L q,s -norm in time.In fact, we shall be only interested in the case when s = q and thus L q,q is equivalentto L q , and in the cases when s = 1 or s = ∞ . The proof is almost identical andwill be omitted.5.3. Local in time decompositions and scaling. We shall further refine ourmain tool which is Lemma 5.2 by introducing a temporal localization of the bilinearform B . More precisely, B shall be decomposed into a sum of scaling invariantdyadic pieces induced by a Whitney’s dyadic decomposition applied on the domainΩ = { ( t, s ) | s < t } used to define B . Definition 5.6. We call any positive integer that is a power of two a dyadic number.Furthermore, we call a square Q in R dyadic if its side length is a dyadic numberand the coordinates of its vertices are integer multiples of dyadic numbers.We apply Whitney’s dyadic decomposition on Ω and obtain the family O ofessentially disjoint dyadic squares Q (by that we mean that overlapping on thesides is still possible) such that the distance between any square Q ∈ O and theboundary of Ω ( { ( t, s ) | t = s } ) is approximately proportional to the diameter of Q .This is immediately obvious in Figure 2. By O λ we denote the collection of all Figure 2. Whitney’s decomposition for the region s < t squares in O whose side length is λ . Thus we obtain the representation B ( F, G ) = X λ X Q ∈O λ B Q ( F, G ) , where Ω = [ λ [ Q ∈O λ , and(52) B Q ( F, G ) = Z Z Q h U ∗ ( s ) F ( s ) , U ∗ ( t ) G ( t ) i dsdt. Furthermore, whenever Q = J × I and Q ∈ O λ we have(53) λ = | I | = | J | ∼ dist(Ω , ∂ Ω) ∼ dist( I, J ) . The localized bilinear operator B Q scales in the following way(54) | B Q ( F, G ) | . λ β ( q,r, ˜ q, ˜ r ) k F k L ˜ q ′ t ( J ; L ˜ r ′ x L ˜ p ′ v ) k G k L q ′ t ( I ; L r ′ x L p ′ v ) , where β ( q, r, ˜ q, ˜ r ) = 1 q + 1˜ q − n (cid:18) − r − r (cid:19) . (55)The range of the Lebesgue exponents (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) for which we canprove that (54) holds for every Q ∈ O λ will be presented in Lemma 7.4. Property(54) implies Lemma 5.7. If q + q ≤ , then (56) X Q ∈O λ | B Q ( F, G ) | . λ β ( q,r, ˜ q, ˜ r ) k F k L ˜ q ′ t ( R ; L ˜ r ′ x L ˜ p ′ v ) k G k L q ′ t ( R ; L r ′ x L p ′ v ) . for every F ∈ L ˜ q ′ t ( R , L ˜ r ′ x L ˜ p ′ v ) , and every G ∈ L q ′ t ( R ; L r ′ x L p ′ v ) .Proof. In view of (54) X Q ∈O λ | B Q ( F, G ) | . λ β ( q,r, ˜ q, ˜ r ) X Q ∈O λ , Q = J × I k F k L ˜ q ′ t ( J ; L ˜ r ′ x L ˜ p ′ v ) k G k L q ′ t ( I ; L r ′ x L p ′ v ) . The claim now follows immediately from Lemma 5.8 below. (cid:3) Lemma 5.8. Suppose p + p ≥ . Then X Q ∈O λ , Q = J × I k f k L ˜ p ( J ) k g k L p ( I ) ≤ k f k L ˜ p ( R ) k g k L p ( R ) . Proof. The lemma follows directly from the inequality X j | a j b j | ≤ X j | a j | ˜ p p X j | b j | p p , which holds in the range p + p ≥ 1, and the fact that for each dyadic interval I there are at most two dyadic squares in O λ with side I . (cid:3) We now introduce the bilinear operator A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ s , (for adefinition of l ∞ s see below), defined by the formula A ( F, G ) = { b λ } λ ∈ Z = X Q ∈O λ | B Q ( F, G ) | λ ∈ Z . For instance, this operator is bounded whenever we have property (54) (see Lemma7.4), q ≥ ˜ q ′ , and s = − β ( q, r, ˜ q, ˜ r ). Clearly, the boundedness of A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 15 L q ′ t L r ′ x L p ′ v → l implies the boundedness of B : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → C . Thus,in view of Lemma 5.2, the estimate k{ b λ }k l . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v k G k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ G ∈ L qt L rx L pv , implies the boundedness of W ( t ) : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v → L qt L rx L pv . We summarize this factin Lemma 5.9. The boundedness of the bilinear operator A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l implies the inhomogeneous Strichartz estimate k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v . In our proofs of the inhomogeneous Strichartz estimates for the KT equationwe shall be making a repeated use of this lemma. We shall also need a numberof standard results from the theory of Interpolation Spaces which are given inthe remaining part of this paragraph. By L p = L p ( X ; B ) and L p,q = L p,q ( X ; B )we denote the Lebesgue space and the Lorentz space respectively of vector-valuedfunctions that map a fixed measure space ( X, dµ ) to a fixed Banach space B . Lemma 5.10 (see [3, p. 113]) . Suppose that < p , p , q , q ≤ ∞ , < θ < ,and p = p . Then ( L p ,q , L p ,q ) θ,q = L p,q , where /p = (1 − θ ) /p + θ/p . Suppose that B and B are two Banach spaces that are compatible for interpo-lation. Lemma 5.11 (see the Appendix of [6]) . For every ≤ p , p < ∞ , < θ < , /p = (1 − θ ) /p + θ/p and p ≤ q we have L p ( X ; ( B , B ) θ,q ) ֒ → ( L p ( X ; B ) , L p ( X ; B )) θ,q . Denote by l ps the space of number sequences with a norm k{ a } j ∈ Z k l ps = (cid:0) js | a j | p (cid:1) /p , ≤ p < ∞ , k{ a } j ∈ Z k l ∞ s = sup j ∈ Z js | a j | , p = ∞ . Lemma 5.12 (See Theorem 5.6.1 in [3]) . We have the identity (cid:0) l ∞ s , l ∞ s (cid:1) θ, = l s , where s , s ∈ R , s = s and s = (1 − θ ) s + θs . Lemma 5.13 (See p. 76 in [3]) . Suppose that ( A , A ) , ( B , B ) , and ( C , C ) areinterpolation couples and that the bilinear operator T acts as a bounded transfor-mation as indicated below: T : A × B → C ,T : A × B → C . If θ ∈ (0 , , p, q, r ∈ [1 , ∞ ] , and /r = 1 /p +1 /q , then T also acts as a boundedtransformation in the following way: T : ( A , A ) θ,p × ( B , B ) θ,q → ( C , C ) θ,r . Lemma 5.14 (See pp. 76-77 in [3]) . Suppose that ( A , A ) , ( B , B ) , and ( C , C ) are interpolation couples and that the bilinear operator T acts as a bounded trans-formation as indicated below: T : A × B → C ,T : A × B → C ,T : A × B → C . If θ , θ ∈ (0 , and p, q, r ∈ [1 , ∞ ] are such that /p + 1 /q ≥ , then T also actsas a bounded transformation in the following way: T : ( A , A ) θ ,pr × ( B , B ) θ ,qr → ( C , C ) θ + θ ,r . Proof of Strichartz estimates for admissible exponents In this paragraph we prove only the validity of the estimates in Theorem 2.4.The investigation of their sharpness shall be made in Section 9 by means of coun-terexamples.Our plan is the following one. We shall first prove the homogeneous estimate k U ( t ) f k L qt L rx L r ′ v . k f k L x , via the corresponding estimate for the T T ∗ -operator for non-endpoint exponenttriplets. Then in view of the invariance (34) we obtain k U ( t ) f k L qt L rx L pv . k f k L ax , ≤ a ≤ ∞ , q > a. By duality and composition this implies the estimate k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v whenever a = HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) and (˜ q, ˜ r, ˜ p ) is another non-endpoint KT-admissible exponent triplet. Proof. In view of the decay estimate k U ( t ) f k L rx L r ′ v . | t | β ( r ) k f k L r ′ x L rv , ≤ r ≤ ∞ , where β ( r ) = n (1 − /r ), we have k T T ∗ F k L rx L r ′ v . Z ∞−∞ k U ( t − s ) F ( s ) k L rx L r ′ v ds . Z ∞−∞ k F ( s ) k L r ′ x L rv | t − s | β ( r ) ds. We take the L q -norm in t and in view of the Hardy-Littlewood-Sobolev (HLS)theorem of fractional integration, see [2, pp. 228-229], [15], we obtain k T T ∗ F k L qt L rx L r ′ v . k F k L q ′ t L r ′ x L rv , whenever 0 < β ( r ) < 1, 1 + 1 /q = 1 /q ′ + β ( r ). The latter conditions are equivalentto 2 < r < r ∗ (2), 1 /q + n/r = n/ 2. The left endpoint r = 2 follows trivially fromthe transport estimate (40). (cid:3) The right endpoint r = r ∗ (2) remains unresolved in the context of the KTequation, unlike that of the wave and the Schr¨odinger equations, where it hasbeen resolved (in the positive) by Keel and Tao [11] (1997). In the setting of theinhomogeneous estimates, the “double endpoint” for which both exponent tripletsare endpoint remains unresolved. However, if ( q, r, p ) is non-endpoint and (˜ q, ˜ r, ˜ p ) TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 17 is endpoint the corresponding inhomogeneous estimate is still non-endpoint (sincein such case q > ˜ q ′ ) and thus holds true in view of Theorem 2.5 whose proof willbe given in Section 8.7. Local inhomogeneous estimates In order to go beyond the “standard” Strichartz estimates for the KT equationproved in the previous section we shall adapt and apply techniques pioneered byFoschi [7], and Keel and Tao [11]. We have also considered the works by Vilela [17]and Taggart [16].Our goal is to find the maximal range of estimates where we have the scalingproperty k W ( t )[ χ λJ F ] k L q ( λI ; L rx L pv ) . λ q + q − n ( − r − r ) k F k L ˜ q ′ ( λJ ; L ˜ r ′ x L ˜ p ′ v ) , ∀ λ > , (57)for any two unit intervals I and J separated by a unit distance and any F ∈ L ˜ q ′ t ( R , L ˜ r ′ x L ˜ p ′ v ), where χ λJ denotes the characteristic function of the rescaled interval λJ . Note that (57) is equivalent to (54). Lemma 7.1. Estimate (57) holds for any two non-endpoint KT-admissible triplets ( q, r, p ) and (˜ q, ˜ r, ˜ p ) with a = ˜ a ′ .Proof. The proof follows trivially from Theorem 2.4 due to the fact that β ( q, r, ˜ q, ˜ r )= 0 under the hypothesis of the lemma. (cid:3) Lemma 7.2. Estimate (57) holds with ( q, r, p ) = ( ∞ , r, p ) and (˜ q, ˜ r, ˜ p ) = ( ∞ , p ′ , r ′ ) ,where ≤ p ≤ r ≤ ∞ .Proof. Due to the decay estimate (41) we have thatsup t ∈ λI k W ( t )[ χ λJ F ] k L rx L pv . sup t ∈ λI Z λJ k F ( τ ) k L px L rv | t − τ | n ( p − r ) dτ . λ β ( ∞ ,r, ∞ ,p ′ ) k F k L ( λJ ; L px L rv ) . (cid:3) Lemma 7.3. Whenever ( q, r, p ) and (˜ q, ˜ r, ˜ p ) are exponent triplets for which esti-mate (57) holds, we have that (57) also holds with ( Q, r, p ) and ( ˜ Q, ˜ r, ˜ p ) , where ≤ Q ≤ q , ≤ ˜ Q ≤ ˜ q .Proof. A trivial application of H¨older’s inequality k W ( t )[ χ λJ F ] k L Q ( λI ; L rx L pv ) . λ Q − q k W ( t )[ χ λJ F ] k L q ( λI ; L rx L pv ) . λ β ( Q,r, ˜ q, ˜ r ) k F k L ˜ q ′ ( λJ ; L ˜ r ′ x L ˜ p ′ v ) . λ β ( Q,r, ˜ Q, ˜ r ) k F k L ˜ Q ′ ( λJ ; L ˜ r ′ x L ˜ p ′ v ) . (cid:3) Let us define the range of validity of the local estimates (57) as the set E ⊂ R consisting of exponent vectors (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) in R that correspond tovalid estimates (57). We shall only find the convex hull E ⊆ E of the points in R that correspond to the estimates in the three lemmas above. The question whether E = E remains open. Lemma 7.4 (Local inhomogeneous estimates) . Estimate (57) holds whenever theexponent triplets ( q, r, p ) , (˜ q, ˜ r, ˜ p ) satisfy the following conditions ≤ q , q ≤ , < p , p , r , r ≤ , (58) 1 r ≤ p , r ≤ p , HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) , (59) 1˜ r − p − r + 1 p ≤ nq , r − p − r + 1˜ p ≤ n ˜ q , (60) n − p ′ < n ˜ r , n − p ′ < nr , (61) or if the point (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) lies inside one of the “cubic” sets in R below ( κ, , µ, ν, − µ, , ≤ κ, µ, ν ≤ , ( κ, − µ, , ν, , µ ) , ≤ κ, µ, ν ≤ . (62) Proof. We apply the Riesz-Thorin convexity theorem to interpolate between thealready proven local estimates. To that end we need to find the convex hull of thesets in R associated with Lemmas 7.1 and 7.2 and then expand that set by therule given in Lemma 7.3.The range of validity S of the local estimates in Lemma 7.1 is given by thesystem 0 < r , r ≤ , ≤ q , q , p , p ≤ , (63) 1 q = n (cid:18) p − r (cid:19) , q = n (cid:18) p − r (cid:19) , (64) 1 r + 1 p + 1˜ r + 1˜ p = 2 , (65) n − p < n + 1 r , n − p < n + 1˜ r , (66)or if (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) ∈ { B = (0 , , , , , , C = (0 , , , , , } .Note that S is a convex polyhedron in R and the two points B and C lie onits boundary. The range of validity S of the local estimates in Lemma 7.2 is theconvex hull, in fact a triangle, of the three points(67) A = (0 , , , , , , B = (0 , , , , , , C = (0 , , , , , . The vertices B and C are already included in S and thus it would suffice to takeonly the vertex A . Hence, we obtain the following set1 Q = θq , R = θr , P = 1 − θ + θp , Q = θ ˜ q , R = θ ˜ r , P = 1 − θ + θ ˜ p , ≤ θ ≤ , where (1 /Q, /R, /P, / ˜ Q, / ˜ R, / ˜ P ) are the coordinates of the new set S writtenin terms of (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) and θ . Of course, we must also add to S the line segments [ A, B ] and [ A, C ]. We shall treat this case separately at the end. TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 19 Finally, we apply the rule given in Lemma 7.3 and thus we replace the equationsfor Q and ˜ Q above with the following inequalities1 ≥ Q ≥ θq , ≥ Q ≥ θ ˜ q , plus the restrictions 1 r ≤ p , r ≤ p , (68)which were implicitly assumed in (64). We first eliminate q and ˜ q from the system for S to obtain1 Q ≥ n (cid:18) θp − θr (cid:19) , ⇔ Q ≥ n (cid:18) θ − P − R (cid:19) , ⇔ θ ≤ P ′ + 1 R + 2 nQ . Similarly, θ ≤ P ′ + 1˜ R + 2 n ˜ Q , Q , Q ≤ . As expected, condition (65) is invariant1 R + 1 P + 1˜ R + 1˜ P = 2 . Reworking condition (66), we obtain θ < n + 1 n − R + 1 P ′ , θ < n + 1 n − R + 1˜ P ′ . Condition (68) is replaced by1 P ′ + 1 R ≤ θ, P ′ + 1˜ R ≤ θ. Finally, conditions (63) are transformed into1 P ′ , P ′ , R , R ≤ θ, ≤ Q , Q , P , P , R , R ≤ . We group all conditions obtained in the previous 5 steps according to theirtype 0 , P ′ , P ′ , R , R , P ′ + 1 R , P ′ + 1˜ R ≤ θ. (69) θ ≤ R + 1 P ′ + 2 nQ , R + 1˜ P ′ + 2 n ˜ Q , n + 1 n − R + 1 P ′ , n + 1 n − R + 1˜ P ′ , . (70) 0 ≤ Q , Q , P , P , R , R ≤ , P + 1 R + 1˜ R + 1˜ P = 2 . (71) We discard the redundant conditions like0 , P ′ , P ′ , R , R ≤ θ, which are all weaker than the other two in (69).There exists θ solving all inequalities in (69), (70), if and only if every quantityin (69) is bounded from above by any quantity in (70). Thus we form all possible combinations between the quantities in the two types of (reduced) inequalities toobtain the following set of conditions1 R + 1 P ′ ≤ R + 1˜ P ′ + 2 n ˜ Q , R + 1˜ P ′ ≤ R + 1 P ′ + 2 nQ , R + 1 P ′ < n + 1 n − R + 1˜ P ′ , ⇔ n − P ′ < n ˜ R , R + 1˜ P ′ < n + 1 n − R + 1 P ′ , ⇔ n − P ′ < nR , R ≤ P , R ≤ P , describing the region S . We apply the rule given in Lemma 7.3 to the two line segments [ A, B ] and[ A, C ] to obtain the following two “cubic” regions in R ( µ, , κ, ν, − κ, , ≤ µ, ν, κ ≤ , ( µ, − κ, , ν, , κ ) , ≤ µ, ν, κ ≤ . (72)Hence, the computation of the set E is finished. (cid:3) Proof of the generalized inhomogeneous Strichartz estimates Generalized inhomogeneous non-endpoint estimates. In this paragraphwe prove the inhomogeneous Strichartz estimate k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , in the range q > ˜ q ′ . Thanks to Lemma 5.9, we have reduced this problem toshowing the estimate k{ b λ }k l . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v k G k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ G ∈ L q ′ t L r ′ x L p ′ v , where { b λ } λ ∈ Z = X Q ∈O λ | B Q ( F, G ) | λ ∈ Z . We shall next specify the range of validity of the above estimates in terms of thevector P = (1 /q, /r, /p, / ˜ q, / ˜ r, / ˜ p ) ∈ E for which 1 /q + 1 / ˜ q = n (1 − /r − / ˜ r ) . (73)Let us denote by ∆ the set in R { ( x, y ) | x > , y > , x + y < , ( x, /r, /p, y, / ˜ r, / ˜ p ) ∈ E } and its interior (largest open subset) by int(∆). In this paragraph we prove that forany such vector P ∈ int(∆) the corresponding inhomogeneous Strichartz estimateholds true.Under the assumption of the latter condition and in view of Corollary 5.7 wehave the estimate | b λ | . λ β ( q,r, ˜ q, ˜ r ) k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v k G k L q ′ t L r ′ x L p ′ v , TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 21 or equivalently, { b λ } ∈ l ∞ s with s = − β ( q, r, ˜ q, ˜ r ). Let us set1 /q = 1 /q + ǫ, / ˜ q = 1 / ˜ q + ǫ, /q = 1 /q − ǫ, / ˜ q = 1 / ˜ q − ǫ, for some small enough ǫ > 0, whose existence is guaranteed by our assumptions,such that the perturbed exponent vectors do not leave int(∆). Then we have that β ( q , r, ˜ q , ˜ r ) = 2 ǫ , and β ( q , r, ˜ q , ˜ r ) = β ( q , r, ˜ q , ˜ r ) = − ǫ . The following bilinearmaps A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞− ǫ ,A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ ǫ ,A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ ǫ , are bounded. In virtue of Lemma 5.14, we have that the map A : ( L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v ) / , ˜ q ′ × ( L q ′ t L r ′ x L p ′ v , L q ′ t L r ′ x L p ′ v ) / ,q ′ → ( l ∞ ǫ , l ∞− ǫ ) / , is also bounded. Finally, in view of Lemma 5.12 and the embeddings of the Lorentzspaces, we obtain A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l . All assumption made in this paragraph are explicitly stated in Theorem 2.5,part (i). We remark that condition (60) together with (73) is equivalent to ( q, r, p )and (˜ q, ˜ r, ˜ p ) being KT-acceptable. Furthermore, in this case the inequalities in (60)have to be taken as strict inequalities so that P ∈ int(∆). Let us also note that thetwo locally acceptable “cubic” sets in (72) give rise to the two globally acceptable“cubic cross section” sets Σ and Σ in Definition 2.3.8.2. Global inhomogeneous endpoint estimates with ˜ q = ∞ . In this para-graph we prove the inhomogeneous Strichartz estimates with P lying on either ofthe two catheti of ∆ OAB in Figure 1. Since by duality both type of estimatesare equivalent, it is enough to consider only the case ˜ q = ∞ . We exclude the twoendpoints (0 , 0) and (1 , 0) from our considerations. We suppose that P (1 /q, /r, /p, , / ˜ r, / ˜ p ) ∈ { < /q < } ∩ E and is such that q satisfies every inequality of E as a strict inequality. We alsoassume the scaling condition (73) from the previous paragraph. Then we have A : L t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ ǫ ,A : L t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞− ǫ , where 1 q = 1 q − ǫ , q = 1 q + 1 ǫ . The real method with parameters ( θ, q ) = (1 / , A : L t L ˜ r ′ x L ˜ p ′ v × L q ′ , t L r ′ x L p ′ v → l . Equivalently, in view of the T T ∗ -principle, k W ( t ) F k L q, ∞ t L rx L pv . k F k L t L ˜ r ′ x L ˜ p ′ v , (74) for all F ∈ L t L ˜ r ′ x L ˜ p ′ v . The explicit restrictions on the Lebesgue exponents ( q, r, p )and ( ∞ , ˜ r, ˜ p ) are stated in Theorem 2.5, part (ii). Analogously, the dual case isstated as part (iii) of that theorem.In the remainder of this paragraph we address (i) the corresponding homogeneousestimates to (74) via the Equivalence Theorem 2.8 in its stronger form for Lorentzspaces and (ii) the sharpening of (74) to the Lebesgue norm L qt . Lemma 8.1. The estimate k U ( t ) f k L q, ∞ t L rx L pv . k f k L bx L cv , (75) holds for all f ∈ L bx L cv , whenever q + nr = nb , HM( r, p ) = HM( b, c ) def = a, r < ncn − ,p < b ≤ a ≤ c < r, < q < ∞ , ≤ p, ˜ p, r, ˜ r < ∞ . Proof. The range of validity of estimate (75) is determined in the following way.We first write the conditions defining the set E , however, any inequality where q appears is taken as a strict inequality. To that system we add the scaling condition(73). Thus, we have that 1 /q = n (1 − /r − / ˜ r ), and0 < q , q < , < p , p , r , r ≤ , r ≤ p , r ≤ p , HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) , q < n (cid:18) p − r (cid:19) , ≤ n (cid:18) p − r (cid:19) ,n − p ′ < n ˜ r , n − p ′ < nr , or that the point (1 /q, /r, /p, , / ˜ r, / ˜ p ) belongs to the set( κ, , µ, , − µ, , < κ, µ < , κ = nµ. The latter set of exponents does not give us anything new as it essentially expressesa special case of the decay estimate k U ( t ) f k L q, ∞ t L ∞ x L nqv . k f k L nqx L ∞ v . Let us use the more natural notation for the exponents b = ˜ r ′ and c = ˜ p ′ . Thus weget the following system of conditions1 q + nr = nb , HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) ,r < ncn − , p < b ≤ c < r, < q < ∞ , ≤ p, ˜ p, r, ˜ r < ∞ . (cid:3) Corollary 8.2. The estimate k U ( t ) f k L qt L rx L pv . k f k L bx L cv , b = c, (76) TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 23 holds for all f ∈ L bx L cv whenever q + nr = nb , p < b < a < c < r, r < nn − c, q ≥ c, HM( r, p ) = HM( b, c ) def = a, < q, b, c, r < ∞ , ≤ p < ∞ . The L qt -norm in (76) can be replaced by the L q,ct -norm. In such case the assumption q ≥ c can be removed.Proof. Each estimate in the statement of this corollary can be proved by interpo-lating two estimates (75) with the real method. Indeed, let us perturb slightly theexponents q , b , and c , keeping r and p fixed, in such a way that they remain in therange of validity of the estimates (75). For example, the perturbed exponents canbe taken as follows1 /q = 1 /q + n/ǫ, /b = 1 /b + 1 /ǫ, /c = 1 /c − n/ǫ, /q = 1 /q − n/ǫ, /b = 1 /b − /ǫ, /c = 1 /c + n/ǫ. We then interpolate by the real method with ( θ, q ) = (1 / , c ), and make use ofLemma 5.11. (cid:3) Let us remark that the case of b = c , excluded in Corollary 8.2, is not new andis considered in Theorem 2.4. Lemma 8.3. Suppose that ( q, r, p ) and ( ∞ , ˜ r, ˜ p ) are two jointly KT-acceptable ex-ponent triplets and < ˜ p ′ ≤ q < ∞ . Then the estimate k W ( t ) F k L qt L rx L pv . k F k L t L ˜ r ′ x L ˜ p ′ v , holds for all F ∈ L t L ˜ r ′ x L ˜ p ′ v . Similarly, if ( ∞ , r, p ) and (˜ q, ˜ r, ˜ p ) are two jointly KT-acceptable exponent triplets and < ˜ q ′ ≤ p < ∞ , then the estimate k W ( t ) F k L ∞ t L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v holds for all F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v .Proof. The lemma follows directly from Lemma 8.1, Corollary 8.2, and the Equiv-alence Theorem 2.8. The range of validity of these estimates is identical to that ofthe generalized homogeneous estimates except for the usual change of notation.Let us verify that the range of the exponents is the same as that assumed inTheorem 2.5. The assumption there is that ( q, r, p ) and ( ∞ , ˜ r, ˜ p ) are jointly KT-acceptable and that 1 < q < ∞ . This immediately implies the following range1 < q < ∞ , ≤ p, r, ˜ p, ˜ r < ∞ . Next, the requirement that ( q, r, p ) is KT-acceptable and that q < ∞ leads to p < r .Therefore r > 1. The scaling condition (73) together with the fact that ( q, r, p ) isKT-acceptable implies that p < ˜ r ′ . The identity HM( p, r ) = HM(˜ p, ˜ r ) togetherwith p < r , and ˜ p ≤ ˜ r , and p < ˜ r ′ , leads to p < ˜ r ′ ≤ ˜ p ′ < r. The latter implies that 1 < ˜ p ≤ ˜ r < ∞ . Thus we obtain the following range1 < q, r, ˜ p, ˜ r < ∞ , ≤ p < ∞ , which is the range for which the estimates in this lemma are proven. Analogouslyfor the dual case. (cid:3) Global inhomogeneous endpoint estimates with q = ˜ q ′ . In this para-graph we assume that the L pv -norms are given over a bounded velocity space V ⊂ R n and prove the inhomogeneous estimates (20).We suppose now that P lies on the hypotenuse of ∆ OAB in Figure 1 and that italso belongs to E . The 4-vector (1 /r, /p, / ˜ r, / ˜ p ) should satisfy every inequalityin E as a strict inequality. Of course, we cannot remove the restriction HM( p, r ) =HM(˜ p ′ , ˜ r ′ ), but we shall perturb these exponents in such a way that they alwayssatisfy the latter condition. The exponents (1 /q, / ˜ q ) will remain fixed throughoutthis paragraph. We consider the following perturbations1 r = 1 r + ǫ, r = 1˜ r + ǫ, p = 1 p − ǫ, p = 1˜ p − ǫ, r = 1 r − ǫ, r = 1˜ r − ǫ, p = 1 p + 3 ǫ, p = 1˜ p + 3 ǫ. We have that β ( q, r , ˜ q, ˜ r ) = 2 nǫ and β ( q, r , ˜ q, ˜ r ) = β ( q, r , ˜ q, ˜ r ) = − nǫ .Hence the maps A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞− ǫ ,A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ ǫ ,A : L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v × L q ′ t L r ′ x L p ′ v → l ∞ ǫ , are bounded. In virtue of Lemma 5.14 and the well-known interpolation identity( L p ( R ; A ) , L p ( R ; A )) θ,p = L p ( R ; ( A , A ) θ,p ) , < p < ∞ , (77)see [3], the map A : ( L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v ,L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v ) / , ˜ q ′ × ( L q ′ t L r ′ x L p ′ v , L q ′ t L r ′ x L p ′ v ) / ,q ′ → ( l ∞ ǫ , l ∞− ǫ ) / , is also bounded. In view of the fact that V is bounded we have that L ˜ P ′ ( V ) ֒ → L ˜ p ′ ( V ) and L ˜ P ( V ) ֒ → L ˜ p ′ ( V ) whenever 1 ≤ ˜ P ≤ min(˜ p , ˜ p ). Analogously, L P ′ ( V ) ֒ → L p ′ ( V ) and L P ′ ( V ) ֒ → L p ′ ( V ) whenever 1 ≤ P ≤ min( p , p ). Thus wealso have that the map A : ( L ˜ q ′ t L ˜ r ′ x L ˜ P ′ v ,L ˜ q ′ t L ˜ r ′ x L ˜ P ′ v ) / , ˜ q ′ × ( L q ′ t L r ′ x L P ′ v , L q ′ t L r ′ x L P ′ v ) / ,q ′ → ( l ∞ ǫ , l ∞− ǫ ) / , is bounded. Finally, in view of the interpolation identity (77), it follows that A : L ˜ q ′ t L ˜ r ′ , ˜ q ′ x L ˜ P ′ v × L q ′ t L r ′ ,q ′ x L P ′ v → l . In view of Lemma 5.9, this implies the estimate k W ( t ) F k L qt L r,qx L Pv ( V ) . V k F k L ˜ q ′ t L ˜ r ′ , ˜ q ′ x L ˜ P ′ v ( V ) , (78)for any P, ˜ P , such that 1 ≤ P < p and 1 ≤ ˜ P < ˜ p , and any two jointly KT-acceptable exponent triplets ( q, r, p ) and (˜ q, ˜ r, ˜ p ) whose exponents further satisfythe following conditions 1 < q , ˜ q < ∞ , q = ˜ q ′ . TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 25 Counterexamples In this section we give necessary conditions for the range of validity of theStrichartz estimates for the KT equation by means of counterexamples.We first make the general remark that the validity of Strichartz estimates withexponents r = ∞ in the homogeneous setting, and with r = ∞ or ˜ r = ∞ in theinhomogeneous setting, is completely solved in [12]. There it is proved that theonly valid estimate of the form k U ( t ) f k L qt L ∞ x L pv . k f k L bx L cv , ∀ f ∈ L bx L cv in any spatial dimension is for q = p = b = c = ∞ . Also, the only valid inhomoge-neous estimates of the form k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , ∀ F ∈ L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , in any spatial dimension with either r = ∞ or ˜ r = ∞ are only those whose exponentsare explicitly stated in Definition 2.3.9.1. Homogeneous estimates. By scaling, that is Lemma 4.5, estimate k U ( t ) f k L qt L rx L pv . k f k L ax,v , ∀ f ∈ L ax,v , holds only if 1 q + nr = na , a = HM( p, r ) . Let us next find the upper bound r ≤ r ∗ ( a ). It is enough to consider only thespecial case a = 2. We shall prove the equivalent condition q ≥ 2. (In general r ≤ r ∗ ( a ) and q ≥ a are equivalent.) The claim follows directly by the translationinvariance in t of the T T ∗ -operator. Indeed, first recall that the above estimatewith a = 2 is equivalent to k T T ∗ F k L qt L rx L r ′ v . k F k L q ′ t L r ′ x L rv , ∀ F ∈ L q ′ t L rx L r ′ v . Then, in view of the famous H¨ormander’s lemma 9.1, we have that q ≥ q ′ , orequivalently q ≥ r we use the translation invariance in x of T T ∗ and thus we get that r ≥ r ′ , or equivalently, r ≥ 2. As usual, the condition r ≥ a in the general case 0 < a < ∞ follows by the power invariance (34).Let us verify the translation invariance in t of T T ∗ . Consider F τ ( t ) = F ( t − τ ).For T T ∗ F τ we have Z ∞−∞ U ( t − s ) F ( s − τ ) ds = Z ∞−∞ U ( t − τ − σ ) F ( σ ) dσ, or in other words { T T ∗ F τ } ( t ) = { T T ∗ F } ( t − τ ). Lemma 9.1 (H¨ormander [9]) . Whenever a (non-trivial) linear and bounded op-erator maps L p ( R n ) to L q ( R n ) , ≤ p, q < ∞ , and additionally this operator istranslation invariant, then we must have that p ≤ q .Remark . H¨ormander’s lemma remains true in a more general setting. For ex-ample, the space L p and L q can be vector-valued, i.e. L p ( X ; B ) and L q ( X ; B ) re-spectively, where X ⊆ R n is the set { x = ( x , . . . , x n ) | a i < x i < ∞ , i = 1 , . . . n } forsome fixed a i ∈ R ∪{−∞} , and B and B are some Banach spaces. Furthermore, the spaces L p and L q may be mixed Lebesgue spaces (or Bochner spaces in the vector-valued setting). Suppose for example that p = ( p , . . . , p k ) and q = ( q , . . . , q l )and L p and L q are the corresponding mixed Lebesgue spaces with the usual nota-tion. Consider the bounded linear operator T : L p → L q . Let u ( x , . . . , x k ) ∈ L p , v ( y , . . . , y l ) ∈ L q , τ h be the operator defined by τ h u ( x , . . . , x i , . . . , x k ) = u ( x , · · · , x i + h, . . . , x k ) , and similarly let σ h be the operator defined by σ h v ( y , . . . , y j , . . . , y l ) = v ( y , . . . , y j + h, . . . , y l ) . Then, if we have that T τ h u = σ h T u, ∀ h ≥ , it follows that either q j ≥ p i , or T = 0. The proof of that statement is virtually thesame as that of Lemma 9.1.9.2. Generalized homogeneous estimates. Let us consider the homogeneousStrichartz estimate k U ( t ) f k L qt L rx L pv . k f k L bx L cv , ∀ f ∈ L bx L cv , for data outside the transport class. Most of the arguments from the precedingparagraph apply to this case as well. By scaling, we have that the conditions1 q + nr = nb , HM( p, r ) = HM( b, c ) def = a, (79)are necessary. The following conditions r ≥ p, q < n (cid:18) p − r (cid:19) , or q = ∞ , ≤ p = r ≤ ∞ , (80)are also necessary. To that end, let us consider the equivalent estimate k T T ∗ F k L qt L rx L pv . k F k L t L bx L cv , ∀ F ∈ L t L bx L cv . The claim is proved for it in the next paragraph. Analogously, we obtain that b ≤ c .Indeed, the latter estimate is equivalent to k T T ∗ F k L ∞ t L b ′ x L c ′ v . k F k L q ′ t L r ′ x L p ′ v , ∀ F ∈ L q ′ t L r ′ x L p ′ v . The exponent triplet must be KT-acceptable (proved in the next paragraph) andthus b ′ ≥ c ′ . In fact, conditions (80) and (79) imply that either p < b ≤ a ≤ c < r ( p < b ), or a = b = c = p = r and q = ∞ .We do not have a suitable counterexample showing the necessity of the upperbound r ∗ ( c ) in Theorem 2.10 for the validity of the generalized homogeneous esti-mates (in the case when b = c , n > Generalized inhomogeneous estimates. Let us consider now the inhomo-geneous Strichartz estimate k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v . By scaling, see Lemma 4.5, we obtain that the restrictions1 q + 1˜ q = n (cid:18) − r − r (cid:19) , HM( p, r ) = HM(˜ p ′ , ˜ r ′ ) def = a, are necessary. TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 27 Consider F ( t, x, v ) = χ (0 ≤ t ≤ , | x | ≤ , | v | ≤ . When t ≫ { T T ∗ F } ( t ) = W ( t ) F ≈ χ (cid:18)(cid:12)(cid:12)(cid:12) v − xt (cid:12)(cid:12)(cid:12) ≤ t , | v | ≤ (cid:19) ≈ χ (cid:26) v ∼ t , x ∼ t (cid:27) . Hence, k W ( t ) F k L rx L pv ∼ t nr − np , t ≫ . It follows that k W ( t ) F k L qt L rx L pv < ∞ only if(81) (cid:18) nr − np (cid:19) q < − , or if q = ∞ , r = p. By the duality Lemma 5.3, the dual exponents (˜ q, ˜ r, ˜ p ) must also satisfy (81). Thuswe have that the conditions p ≤ r and ˜ p ≤ ˜ r are necessary for the validity of theconsidered estimate. The same conclusion applies for the T T ∗ -operator.We now show that conditions1 q + 1˜ q ≤ , r + 1˜ r ≤ , are necessary for the validity of the considered estimate. Indeed, the claim followsfrom the translation invariance of T T ∗ in t and x , H¨ormander’s lemma 9.1, and theequivalence of the considered estimates for T T ∗ and W ( t ). Note also that the caseswhen ˜ q = 1 or ˜ r = 1 are trivial and for example by duality can always be replacedby the cases q = 1 or r = 1. Thus we have verified that ( q, r, p ) and (˜ q, ˜ r, ˜ p ) must betwo jointly KT-acceptable exponent triplets, apart from the necessity of condition n − p ′ < n ˜ r , n − p ′ < nr , n > np ′ < q + n ˜ r , n ˜ p ′ < q + nr , is sharp. Indeed, the latter is a direct consequence of (10) and (11). The lattercondition implies the former whenever p ′ ≤ ˜ q and ˜ p ′ ≤ q . Thus, if there are someother global inhomogeneous estimates for W ( t ) not included in Theorem 2.5, theymust belong to the range ˜ q < p ′ or q < ˜ p ′ .9.4. Local inhomogeneous estimates. In this paragraph we show the fact thatin the context of the KT equation the local inhomogeneous estimates do not existin a “full neighborhood” around a given local inhomogeneous Strichartz estimate.This presents an obstruction for the application of the perturbation techniques ofKeel and Tao [11] and their extension by Foschi [7]. The endpoint estimates (of thetype that lie on the hypothenuse AB in Figure 1) remain unresolved.For example, consider the estimate(82) k W ( t ) F k L qt ([2 , L rx L pv ) . k F k L ˜ q ′ t ([1 , L ˜ r ′ x L ˜ p ′ v ) . Take F ( t, x, v ) = χ ( t ∈ [0 , , ( x, v ) ∈ Q R ), where by Q R we denote the squareof side length 2 R centered at the origin of R n . If we denote k x k ∞ = sup ≤ i ≤ n | x i | ,for x = ( x , .., x n ), we can write the latter as Q R = { ( x, v ) : k x k ∞ ≤ R, k v k ∞ ≤ R } . Hence, k F k L ˜ q ′ t ([1 , L ˜ r ′ x L ˜ p ′ v ) ∼ R n ˜ p ′ + n ˜ r ′ . We now set τ = t − s , and consider the set Q R ( τ ) given by k x − τ v k ∞ ≤ R, k v k ∞ ≤ R. Then, for t ∈ [2 , s ∈ [0 , τ ∈ [1 , Q R/ ⊂ Q R ( τ ) ⊂ Q R . Hence, k W ( t ) F k L qt ([2 , L rx L pv ) ∼ R np + nr . We conclude that condition 1 r + 1 p = 1˜ r ′ + 1˜ p ′ is necessary for the validity of the local estimates (82).10. Remaining unresolved Strichartz estimates Here we collect some of the remaining estimates for the KT equation that needto be resolved in order the full range of validity of Strichartz estimates to be known.(1) The endpoint homogeneous estimate in higher dimensions n > k U ( t ) f k L at L r ∗ ( a ) x L p ∗ ( a ) v . k f k L ax,v . (2) The full range of the non-endpoint inhomogeneous estimates k W ( t ) F k L qt L rx L pv . k F k L ˜ q ′ t L ˜ r ′ x L ˜ p ′ v , q > ˜ q ′ . In particular, one needs to either show that the condition n − p ′ < n ˜ r , n − p ′ < nr , is necessary, or otherwise find and prove the remaining estimates.(3) The endpoint inhomogeneous estimates with either q = ˜ q ′ or q = ∞ or˜ q = ∞ .(4) The full range of the local inhomogeneous estimates. Equivalently, eithershow that E = E , or otherwise find and prove the remaining estimates. References 1. A. Benedek and R. Panzone, The space L p , with mixed norm. , Duke Math. J. (1961),no. 2, 301–324.2. Colin Bennett and Robert Sharpley, Interpolation of operators , Pure and Applied Mathemat-ics, 129, Academic Press, Inc., Boston, MA, 1988.3. J¨oran Bergh and J¨orgen L¨ofstr¨om, Interpolation spaces. an introduction. , Grundlehren derMathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.4. Nikolaos Bournaveas, Vincent Calvez, Susana Guti´errez, and Benoˆıt Perthame, Global ex-istence for a kinetic model of chemotaxis via dispersion and Strichartz estimates , Comm.Partial Differential Equations (2008), no. 1-3, 79–95.5. F. Castella and B. Perthame, Estimations de Strichartz pour les ´equations de transportcin´etique , C. R. Acad. Sci. Paris, Ser. I (1996), no. 332, 535–540.6. Elena Cordero and Fabio Nicola, Some new Strichartz estimates for the Schr¨odinger equation. ,Preprint (2007), available at http://arxiv.org.7. Damiano Foschi, Inhomogeneous Strichartz estimates , J. Hyperbolic Differ. Equ. (2005),no. 1, 1–24.8. Zihua Guo and Lizhong Peng, Endpoint Strichartz estimate for the kinetic transport equationin one dimension , C. R. Math. Acad. Sci. Paris (2007), no. 5, 253–256.9. L. H¨ormander, Estimates for translation invariant operators in L p spaces. , Acta Math. (1960), no. 5, 93–140. TRICHARTZ ESTIMATES FOR THE KINETIC TRANSPORT EQUATION 29 10. Tosio Kato, An L q,r -theory for nonlinear Schr¨odinger equations. , Spectral and ScatteringTheory and Applications, Adv. Stud. Pure Math. (1994), 223–238.11. Markus Keel and Terence Tao, Endpoint Strichartz estimates , Amer. J. Math. (1998),no. 5, 955–980.12. Evgeni Y Ovcharov, Counterexamples to Strichartz Estimates for the Kinetic Transport Equa-tion based on Besicovitch sets , Preprint (2010).13. , Global regularity of dispersive equations and Strichartz estimates. , Ph.D. Thesis(2009), the University of Edinburgh.14. Benoˆıt Perthame, Mathematical tools for kinetic equations. , Bull. Amer. Math. Soc. (N.S.) (2004), no. 2, 205–244.15. Sigmund Selberg, Lecture notes on nonlinear wave equations Inhomogeneous Strichartz estimates. , Preprint (2008), http://arxiv.org.17. M.C. Vilela, Inhomogeneous Strichartz estimates for the Schr¨odinger equation , Trans. Amer.Math. Soc. (2007), no. 5, 2123–2136. Angewandte Mathematik und Bioquant, Universit¨at Heidelberg, INF 267, Heidelberg69120, Germany E-mail address ::