Asymptotic Estimates for Some Dispersive Equations on the Alpha-modulation Space

Transcription

1 University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 06 Asymptotic Estimates for Some Dispersive Equations on the Alpha-modulation Space Justin Trulen University of Wisconsin-Milwaukee Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Trulen, Justin, "Asymptotic Estimates for Some Dispersive Equations on the Alpha-modulation Space" 06. Theses and Dissertations. Paper 5. This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact

2 Asymptotic Estimates for Some Dispersive Equations on the α-modulation Space. by Justin G. Trulen A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics at The University of Wisconsin Milwaukee May 06

3 ABSTRACT Asymptotic Estimates for Some Dispersive Equations on the α-modulation Space. by Justin G. Trulen The University of Wisconsin-Milwaukee, 06 Under the Supervision of Dr. Lijing Sun The α-modulation space, Mp,q R n, is a function space developed by Gröbner in 99. The α-modulation space is a generalization of the modulation space, M s p,qr n, and Besov space, B s p,qr n. In this thesis we obtain asymptotic estimates for the Cauchy Problem for dispersive equation, a generalized half Klein-Gordon, and the Klein-Gordon equations. The wave equations will also be considered in this thesis too. These estimates were found by using standard tools from harmonic analysis. Then we use these estimates with a multiplication algebra property of the α-modulation space to prove that there are unique solutions locally in time for a nonlinear version of these partial differential equations in the function space C[0, T ], M p,q R n. These results are obtained by using the fixed point theorem. ii

4 c Copyright by Justin G. Trulen, 06 All Rights Reserved iii

5 Table of Contents Background. Introduction Fourier Transform Fourier Multipliers Dispersive Equations Thesis Outline Function Spaces 9. Besov Spaces Modulation Spaces α-modulation Spaces Embedding Properties Asymptotic Estimates Statement of Main Results Existing Theory and New Estimates Proof for the Asymptotic Estimate for the Cauchy Problem for Dispersive Equation Proof for the Asymptotic Estimate for the Generalized Half Klein- Gordon Equation Proof for the Asymptotic Estimate for the Fourier Multiplier Θt Proof for the Asymptotic Estimate for the Fourier Multiplier Θ K t Asymptotic Estimates for Homogeneous Solutions iv

6 4 Nonlinear Dispersive Equations Solution to the Nonlinear Cauchy Problem for Dispersive Equations Solution to the Nonlinear Generalized Half Klein Gordon Equation Solution to the Nonlinear Wave Equation Solution to the Nonlinear Klein Gordon Equation Bibliography 96 Curriculum Vitae 00 v

7 List of Figures. Distribution of s c when λ > Distribution of s c when λ Distribution of s 0 for all p and q Relationship between M s p,pr n, M p,p R n, B s p,pr n, and H s R n Distribution of s 0 when p and q vi

8 ACKNOWLEDGMENTS This dissertation could not have been completed without the contributions of many individuals. In particular, I wish to express my deepest gratitude and sincerest appreciation to my advisor Professor Lijing Sun for proposing to me an exciting topic for my dissertation, for her exceptional guidance, constant support, inspiration, and endless encouragement during my graduate study at University of Wisconsin at Milwaukee. Her direction and support were crucial for the completion of the dissertation. I would like to thank my committee members: Professor Dashan Fan, Professor Gabriella Pinter, Professor Chao Zhu, and Professor Wei Lang for their valuable input and for serving in my committee. I would also like to thank Professor Kyle Swanson, Professor Peter Hinow, Dr. Bradford Schleben, Mrs. Natalie Jipson, and Mr. Daniel Yee and everyone at University of Wisconsin at Milwaukee for their full support and encouragement. Finally, but not least, I would like to thank my parents Gerry and Sue Trulen, and my sister Catrina Trulen for their unwavering support. And a special thanks goes to my wife Heather Trulen and son Benjamin Trulen for their love and support. They have been both a permanent source of strength, especially during difficult times. vii

9 Chapter Background. Introduction Nonlinear dispersive equations have been studied both in mathematics and physics due to their importance in describing how waves of different wavelength propagate at different velocities. The Schrödinger equation, Airy and Klein-Gordon equation are classic examples of dispersive equations. We will also consider the wave equation in this thesis too. An intuitive description of what dispersive equations describe would be the following: as the main wave travels through a medium it will begin to break into many smaller waves as time evolves. As time continues to evolve, the waves of different wavelengths propagate at different velocities. A quantitative understanding of this behavior gives one insight into the behavior of the solution. Since very few partial differential equations PDEs can be solved explicitly other methods need to be used to establish the existence of a solution. Thus additional techniques are required to understand when a PDE has a solution and how such a solution behaves. Some of these analytical techniques are: separation of variables, method of characteristics, change of variables, the Lie group method, and numerical methods. In this thesis we are going to focus on one analytical method that would be classified as an integral transform. Our integral transform relies on using the Fourier transform to turn the PDE into a differential equation that is easier to solve explicitly. In this process Fourier multipliers are produced. As mentioned above, Fourier multipliers arise naturally in the study of PDEs

10 and in the convergence of Fourier series. The fundamental problem in the study of Fourier multipliers is to relate the boundness properties of the Fourier multiplier on certain function spaces to properties of its symbol. In 977 Strichartz [5] found L p R n space decay estimates for the solutions to the Schrödinger equation and the Klein-Gordon equation with certain conditions on the initial data. In 995 both Lindblad and Sogge [], and Ginibre and Velo [3] found non-endpoint estimates for the wave equation. Shortly after that in 998 Keel and Tao [9] found endpoint estimates for the Schrödinger and wave equation. In general unimodular Fourier multipliers do not preserve any L p -norm except for when p =. For this reason L p R n spaces may not be an appropriate space for this study. In 983 Feichtinger [8] introduced a new function space called the modulation space, denoted M s p,qr n. The modulation space is used to measure the smoothness of a function or distribution in a different way than the L p R n space, and can be understood as a measure of the phase space concentration of a function. This has made it possible for other Strichartz estimates to be found. In 007 Benyi and Grochenig [] showed that the solutions to both the Schrödinger equation and the Wave equation are in the non-weighted modulation space M 0 p,qr n provided their initial data is in M 0 p,qr n. Wang and Hudzik [7] studied the nonlinear Schrödinger equation and the nonlinear Klein-Gordon equation. Supposing that the initial data is in the non-weighted modulation space, M 0 p,qr n, using specified nonhomogenuous functions they were able to show that both the nonlinear Schrödinger equation and the nonlinear Klein-Gordon equation have solutions in a similar non-weighted modulation space. More general results were found by Chen, Fan, and Sun [5] in 0 and by Deng, Ding, and Sun [6] in 03. Gröbner s Ph.D thesis [5], written in 99, introduced another function space called the α-modulation space, denoted by M p,q R n. This function space was not only a generalization for the modulation space M s p,qr n, but also served as an intermediate space for both the modulation space M s p,qr n and the Besov space, B s p,qr n. The parameter α [0, ] serves as a tuner in the sense that as α approaches 0 + the α-modulation space M p,q R n begins to look more like modulation space

11 M s p,qr n. As α approaches the α-modulation space, M p,q R n begins to look more like the Besov space B s p,qr n. Because of this fact, and the fact that Fourier multipliers can preserve the M p,q -norm in more general cases this makes the α- modulation space M p,q R n a more suitable function space to work on. In the following sections we will present basic equations and definitions that will be used throughout this dissertation. The goal is to provide a simple introduction and reference that will serve as a basis for the remainder of the chapters. Other important notations used throughout this dissertation are as follows: define Z = Z + {0}, for a space X let X denote the dual space of X, for p R + denote the conjugate of p as p, that is p + p =, with the modifications that if p = 0, then p = or if p =, then p = 0. For all multi-indices α define α = α + + α n, for all x, ξ R n define xξ = x ξ + +x n ξ n, we write A B if there is a positive constant C such that A CB, and define x = + x.. Fourier Transform Let L p R n denote the usually L p function space with the norm f L p R n R = fx p p dx. n where 0 < p <. When p = we will make the usual modification of f L = ess sup x R n fx. Suppose α = α,, α n and β = β,, β n, where α i, β i Z for all i n, are multi-indices. Also, suppose the conventions of x α = x α x α n n and β fx = β x β n x n fx. The Schwartz Space, denoted by S R n, is the set of all smooth functions fx, meaning functions that are infinitely differentiable, that satisfy the following x sup α β fx < C α,β <, x R n 3

12 where C α,β is a constant that depends on all multi-indices α and β. The Schartz space can be thought of as the set of all smooth functions that decay faster than the reciprocal of any polynomial. With this view point it is clear that S R n L p R n for all p. For all f S R n define the Fourier transform, denoted by ˆfξ = F fξ, by ˆfξ = fxe ixξ dx,. R n where ξ R n and xξ = x ξ + + x n ξ n. Since S R n L R n equation. makes sense for all f S R n. linear property Furthermore, the Fourier transform enjoys the af + bgξ = a ˆfξ + bĝξ, where f, g S R n and a, b C, and enjoys the following other properties as well:. ˆ f = ˆf,. τ y fξ = e iyξ ˆfξ, 3. e ixy fx ξ = τ y ˆfξ, 4. δ a f ξ = t n δ t ˆf, 5. α f ξ = iξ α ˆfξ, 6. α ˆfξ = ix α fx ξ, 7. f gξ = ˆfξĝξ, with a > 0, t > 0, and τ y fx, δ a fx, fx and f gx are defined as τ y fx = fx y,. δ a fx = fax,.3 fx = f x,.4 f gx = fx ygydy. R n.5 4

13 See [4, 0] for details of the proofs of these properties. To define the inverse of the Fourier transform, first note that if f S R n, then ˆf S R n [4]. Because of this fact, we are able to define the inverse of the Fourier transform for all f S R n. Define the inverse ˇfx = F fx by F fx = ˇfx = fξe ixξ dξ..6 π n R n With this definition we have the following Fourier inversion property: F F fξ x = F F fξ x = fx. We also have the important Plancherel s Identity. Theorem... Plancherel s Identity [4] For all f S R n the following holds f L R n = ˆf = ˇf L. L R n R n Note that Plancherel s Identity is often associated with a conservation of energy estimate. We are able to define the Fourier transform on L R n. This follows immediately because if f L R then converges and makes sense. ˆfξ = fxe ixξ dx, R n Decay and continuity of such Fourier transforms of L R n functions are characterized in the following theorem: Theorem... Riemann-Lebesgue Lemma [6, 4, 0] If f L R n, then ˆf is uniformly continuous and lim ˆfξ = 0. ξ This result can also be interpreted as a smoothing effect that the Fourier transform has on L -functions, which may not be continuous. Note that we are not always able to define the inversion. To see this, consider the following example: 5

14 Example..3. [0] Let n = and set fx = χ a,b x to be the characteristic function on the interval a, b. Clearly f L R. Then it follows that Here we see ˆf / L R. ˆfξ = = b a fxe ixξ dx e ixξ dx = e ibξ e iaξ iξ ia+bξ sinb aξ = e. ξ Because of this, the Fourier inverse is defined only when it makes sense, that is, the integral is convergent in the L -sense; or when ˆf L R n. We can also extend the definition of the Fourier transform to L R n by considering a dense subspace of L R n. Such a space would be L R n L R n. Using Theorem.., to define the Fourier transform of f L R n pick a sequence f N L R n L R n that converges to f in L R n. Since f N L L, then ˆf N will be defined for all N. Then it follows that the function ˆf such that ˆf N ˆf 0 as N, L R n can be taken as the Fourier transform of f. The existence of ˆf L R n is guaranteed since the Fourier Transform is an isometry on L R n [4]. Note that the definition of ˆf is independent from the choice of f N. See [4, 0]. The final extension we will define is the Fourier transform for all L p R n where < p <. This is simply done by decomposing an L p -function, f, into the sum of an L -function, f, and an L -function, f. One such decomposition is set f x = fxχ {x R n : fx >}x and f x = fxχ {x R n : fx }x. Thus ˆf is defined as ˆfξ = ˆf ξ + ˆf ξ. Note that the definition of ˆf is independent of the choice of f and f [4]. Such an extension leads to the following well-know theorem: 6

15 Theorem..4. Hausdorff-Young inequality [4, 0] For p and every function f L p R n we have the following estimate: ˆf f L p L R n p R n. From the Hausdorff-Young inequality we get the following well known corollary: Corollary..5. [4, 0] The following estimate holds ˆf f L L R n R n. This result gives one an upper bound for the supremum of ˆf over ξ R n, which is the L -norm of f..3 Fourier Multipliers Let C C R n be the set of all functions that are smooth and have compact support. We call C C R n the set of Test Functions. For all f C C R n we define the Fourier multiplier with symbol µ, denoted H µ fx, by H µ fx = µξ π ˆfξe ixξ dξ..7 n R n First note that C C R n S R n. From this, if f C C R n, then H µ f makes sense. Second, the Fourier multiplier is a linear operator. As we will see later, the Fourier multiplier arises naturally in the formal solution of PDEs with constant coefficients. As mentioned earlier, a fundamental problem in the study of Fourier multipliers is to relate the boundedness properties of H µ on certain function spaces to properties of the symbol µ. We will now cite a couple well-known theorems about Fourier multipliers. The next two theorems give sufficient conditions for the µ L R n and when the operator H µ is bounded. These results are known as the Hörmander-Mihlin Multiplier Theorem and Marcinkiewicz Multiplier Thoerem. Theorem.3.. The Hörmander-Mihlin Multiplier Theorem on R n [4, 3] Let µξ be a complex-valued bounded function on R n / {0} that satisfies either: 7

16 . Mihlin s condition α µξ A ξ α, [ n ] for all multi-indices α +, or,. Hörmander s condition sup R n+ α R>0 [ n ] for all multi-indices α +, R< ξ <R then for all < p < we have the following estimate α µξ dξ A <, H µ L p L p C n max p, p A + µ L. For the next theorem let I j = j+, ] j [ j+, j and R j = I j I jn for all j = j,, j n Z n. Theorem.3.. The Marcinkiewicz Multiplier Thoerem on R n [4] Let µ be a bounded function on R n that is C n, i.e. functions whose n th derivative exists, in all regions R j. Suppose there is a constant A such that for all k {,, n}, all j,, j k {,, n}, all l j,, l jk Z, and all ξ I ls for s {,, n} / {j,, j k } we have j µξ dξjk dξ j A <, I lj I ljk then for < p < there is a constant C n < such that H µ L p L C p n A + µ L max p, p 6n. Both theorems are consequences of Littlewood-Paley theory and dyadic decomposition, see [4, 4] for more details. In this thesis we will be focused on a special type of Fourier multiplier. The Unimodular Fourier multiplier is a Fourier multiplier whose symbol has the form e iνξ where ν is a real-valued function. It is important to note that unimodular Fourier multipliers in general do not preserve the L p -norm, except when p =. In the case when p =, one can simply apply Plancherel s identity, Theorem... To better understand this we first need a well known lemma. 8

17 Lemma.3.3. [0] If t 0 and p [, ], then we have e it : L p R n L p R n, where e it is the Fourier Multiplier with the symbol e it ξ, is continuous and the following estimate holds e it f n L p R n c t p p f L p R n. Now we can consider the following example: Example.3.4. [0] Consider the Fourier multiplier e it with the symbol e it ξ, then the Fourier multiplier H µ is not a bounded operator from L p R n L p R n for p and t 0. If it was a bounded operator from L p L p it would be bounded for p. So without loss of generality, suppose p >. Let t 0 and pick f L p R n L p R n L R n. Then it follows f L p R n = e it e it f L p R n c 0 e it L p R n c 0 ct f L p R n, which is a contradition. Furthermore, using the same unimodular Fourier multiplier from Example.3.4 we have the next example. Example.3.5. Suppose p > and t 0. Let g L R n such that g / L p R n. Define f to be fx = H µ gx = e it ξ ĝξe ixξ dξ. π n R n Note that f L p R n. Then the unimodular Fourier multiplier H µ0, with symbol 9

18 µ 0 ξ = e it ξ is not a bounded operator from L R n L p R n since H µ0 fx = e it ξ ˆfξe ixξ dξ π n R n e it ξ F e it ξ ĝξe ixξ dξ xe ixξ dξ π n R π n n R n = e it ξ F F e it ξ ĝξ xe ixξ dξ π n R n = e it ξ e it ξ ĝξe ixξ dξ π n R n = ĝξe ixξ dξ π n R n = gx. and this is not in L p R n. Because of these two examples, one sees that L p spaces may not be appropriate function spaces to study Fourier multipliers and formal solutions to PDEs. Next we will make the precise definition of what we mean by dispersive equations, and look at some dispersive equations that we will be considering in this dissertation..4 Dispersive Equations Consider the general PDE of the form F x, t ut, x = 0, where F is a polynomial in the partial derivatives. We seek to find a solution to this PDE that takes the form of ut, x = Ae ikx tξ. Note ut, x will be a solution if and only if F ik, iξ = 0, which is called a dispersive relation. In some cases we can write ξ = ξk. With this we can define the group velocity, denoted by c g k, as c g k = ξ k. 0

19 If c g is not constant, i.e. c gk = ξ k 0, then the waves are called dispersive. The physical meaning can be interpreted as time evolves waves of different wavelengths disperse in the medium at different velocities. That is a wave with one hump will break into several smaller waves over time. Such equations have been studied in both mathematics and physics because of this property. In this dissertation we will consider both Cauchy problem for dispersive equation, a generalized Klein-Gordon, wave, and the Klein-Gordon equation. First we must introduce the fractional Laplacian. Let = x = x + + x n the standard Laplacian. Let α o R +. Now define the fractional Laplacian with order α o of f, denoted by αo fx, by αo fx = ξ αo π ˆfξe ixξ dξ..8 n R n Such an operator is formally known as the Riesz potential, and has been studied independently. The reader is directed to [3] for more details. With this definition we can state the Cauchy problem for dispersive equation as { i t ut, x + αo ut, x = 0, for t, x R+ R n, u0, x = u 0 x, for x R n..9 We can get a formal solution to this PDE by taking the Fourier transform to get an equivalent system of { t ût, ξ = i ξ αo ût, ξ, for t, ξ R + R n, û0, ξ = û 0 ξ, for ξ R n, which is a first order differential equation. Such equation has a solution of the form û = e it ξ αo û 0 ξ. Now taking the Fourier inverse we have the formal solution to Cauchy problem for dispersive equation.9 ut, x = π n R n e it ξ αo = H e it ξ αo u 0x. û 0 ξe ixξ dξ

20 For the sake of convenience we will write the formal solution, H e it ξ αo u 0x, in the form H e it ξ αo u αo it 0x = e u 0 x..0 When α o =,, 3, the Cauchy problem for dispersive equation corresponds to the half- wave equation, the Schrödinger equation, and essentially the Airy equation. These cases will be of particular interest, and their solution s asymptotic estimates will be explored in more detail later in the dissertation. To understand how the wave equation fits into the big picture, first we have tt ut, x ut, x = 0, for t, x R + R n, u0, x = u x, for x R n,. t u0, x = u x, for x R n, and take the Fourier transform we get tt ût, ξ + ξ ût, ξ = 0, for t, ξ R + R n, û0, ξ = û ξ, for ξ R n, t û0, ξ = û ξ, for ξ R n. form Solving this second order ordinary differential equation we get a solution in the Using the initial conditions we get ût, ξ = c cost ξ + c sint ξ. û 0 ξ = c and û ξ = c ξ. Thus we get a solution of ut, x = cost ξ û π n 0 ξe ixξ dξ + R π n n = H cost ξ u 0 x + H sint ξ u x, ξ where we can take sint ξ R ξ n û ξe ixξ dξ H cost ξ u 0 x = cost u0 x and H sint ξ u x = sint u ξ x.

21 The α-modulation space estimate for cost u 0 x is equivalent to the α-modulation αo it space estimate for e u 0 x. Furthermore, to see why α o = corresponds to what is called the half- wave equation we can rewrite Wave equation. as t + i ξ t i ξ ût, ξ = 0, and if we consider the second factor of this factorized form we have Now working backwards we have t i ξ ût, ξ = 0. t i ξ ût, ξ = 0, i t ût, ξ + ξ ût, ξ = 0, i t ut, x + ut, x = 0.. The other dispersive equation we will consider in this paper is a generalized Klein-Gordon equation. To familiarize ourselves with this dispersive equation we will first consider the Klein-Gordon equation of the form tt ut, x + I ut, x = 0, for t, x R + R n, u0, x = u 0 x, for x R n, t u0, x = u x, for x R n, where Iut, x = ut, x is the identity operator. Again we can find the formal solution of this equation by taking the Fourier transform and writing this equation in the equivalent form tt ût, ξ + + ξ ût, ξ = 0, for t, ξ R + R n, û0, ξ = û 0 ξ, for ξ R n, t û0, ξ = û ξ, for ξ R n. The general solution has the form of ût, ξ = c cos t + ξ and using the initial conditions to get the relationships + c sin t + ξ, û 0 ξ = c and û ξ = c + ξ, 3

22 we obtain the particular solution sin t + ξ ût, ξ = cos t + ξ û 0 ξ + û + ξ ξ. Again, taking the Fourier inverse we get ut, x = cos t + ξ π n û 0 ξe ixξ dξ+ R n Rn sin t + ξ û π n + ξ ξe itxξ dξ. and Again we will define these Fourier multipliers as the following operators H cos t+ ξ u 0 x = cos ti u 0 x, H sin t+ ξ + ξ u x = sin ti u I x. Now we can turn our attention to the generalized Klein-Gordon equation. First, we define the Bessel Potential I αo [3] by I αo fx = + ξ αo π ˆfξe ixξ dξ..3 n R n Then the Generalized Klein-Gordon equation of order α o by tt ut, x + I αo ut, x = 0, for t, x R + R n, u0, x = u 0 x, for x R n, t u0, x = u x, for x R n..4 Using the same techniques as before for with the Klein-Gordon equation we find the formal solution ut, x = π π cos R n sin R n t + ξ αo t + ξ αo + ξ αo 4 û 0 ξe ixξ dξ+ û ξe ixξ dξ,

23 using the convention of cos ti αo u 0 x, and sin ti αo correspond to the appropriate Fourier Multipliers. I αo u x, Furthermore, like the treatment of the wave equation, items. through. we get a generalized half Klein-Gordon equation of the form i t ut, x + I αo ut, x = 0..5 Again, with the same calculations we arrive at a solution of ut, x = αo e it+ ξ û π n 0 ξd ixξ dξ R n = He it+ ξ αo u 0x. Furthermore, we will use the following convention of He it+ ξ αo u αo iti 0x = e u 0 x. We will now close this section by providing in detail some known results of dispersive equations. The first two results are the original Strichartz estimates that can be found in [5]. These are non-endpoint estimates and the first of the two deals with the Schrödinger equation. Theorem.4.. [5] Let ux, t be a solution of the inhomogeneous free Schrödinger equation { i t ut, x + λ ut, x = gt, x, for t, x R R n, u0, x = u 0 x, for x R n, where λ be a nonzero constant, u 0 L R n, and g L p R n+ for p = then u L q R n+ for q = n + n and u L q R n+ f L R n + g L q R n+. n + n + 4, 5

24 The second result deals with the Klein-Gordon equation. Theorem.4.. [5] Let ut, x be a solution to the Klein-Gordon, where m > 0 tt ut, x + ut, x m ut, x = gt, x, for t, x R R n, u0, x = u 0 x, for x R n, t u0, x = u x, for x R n, where B u0 L R n, B u L R n, where B = m, and g L q R n+, then u L q R n+ and u L q R n+ B u0 L R n + with the following restrictions on p and q B L u + g L R n q R n+, if n = n + n + 3 p if n < p 6 5, 6 q <. n + n + 4, n + n q n + n, After Strichartz results, the non-endpoint estimates for the wave equation were found by both Lindblad and Sogge [], and Ginibre and Velo [3]. The last result is one found by Tao and Keel [9]. This finds the endpoint estimates for both the Schrödinger and wave equation. Though the theorem is stated in the context of Hilbert spaces, it is easily applied to these differential equations. First, we need to start with a definition. Definition.4.3. A pair q, r that satisfies q, r, q, r, σ,,, and q + σ r σ, is called σ-admissible. A pair q, r that satisfies q, r, q, r, σ,,, and is called sharp σ-admissible. q + σ r = σ, Now we can state Tao and Keel [9] result. 6

25 Theorem.4.4. [9] Suppose t R + and let Ut : H L xx, where X, dx is a measure space, be an operator defined on a Hilbert space H. If U obeys the following estimates: Utf L x X f H for all f H, and.6 UsUt g L x X t s σ g L x X,.7 for some σ > 0, t s, and all g L xx, then for all sharp σ-admissible exponent pairs q, r and q, r U obeys the following estimates Utf L q f t Lr x H, Us F sds F L q H UtUs F sds s<t L q t Lr x t Lr x, F L q. t L r x Furthermore, for all t and s, and g L xx, if the equation.7 in the hypothesis can be strengthen to UsUt g L x X + t s σ g L x R n, then the conclusion can be extend to all σ-admissible exponent pairs q, r and q, r..5 Thesis Outline The remainder of this thesis is organized as follows: in Chapter we introduce the Besov, Modulation, and α-modulation spaces. We will discuss the construction of these function spaces, behavior of these function spaces and various embedding properties of these function spaces. We also discuss some recent developments of asymptotic estimates of Fourier Multipliers on these function spaces. In Chapter 3 we state our main results dealing with asymptotic estimates of several Fourier Multipliers on the α-modulation space. We will discuss the existing theory in harmonic analysis needed for this dissertation as well as develop some basic 7

26 estimates. We present the proofs to these main results followed by asymptotic estimates for homogeneous solutions of several well known PDEs on the α-modulation space. In Chapter 4 we conclude with presenting several results detailing when a unique local solution exists for some nonlinear PDEs. These results are obtained by using the results from Chapter 3 and using a fixed point theorem. 8

27 Chapter Function Spaces In this chapter we will look at the construction of three well know function spaces: Besov, modulation, and α-modulation. Then we will discuss some properties, embedding and multiplier theorems. First we will give the definition of some basic function spaces that will be used throughout this chapter. Let C R n be the set of all complexed-valued, bounded, and uniformly continuous functions on R n. Now define C m R n to be the set of all functions, f, such that α f C R n for all multi-indices α, where α m. See [6] for a full treatment of these function spaces. For k 0 an integer, define the Sobolev space, denoted W p,k R n, be the set of all locally integrable functions f on R n such that α f exists in the weak sense and belongs to L p R n for all multi-indices α k. See [6, 7, ] for more information about this classic space.. Besov Spaces First introduced in 959 by Besov [4] the Besov space is a function space that measures functions in a different way than the L p function spaces. The Besov space breaks the spatial domain into different regions using the dyadic decomposition, that is, on regions localized in { ξ : ξ j}. The construction of the Besov space relies on the Littlewood-Paley decomposition operators, j. Let {φ j } j=0 be a sequence of functions in S Rn and α = α,, α n with 9

28 α i Z for all i n, be a multi-index such that the following is satisfied: supp φ 0 {ξ : ξ }, supp φ j { ξ : j ξ j+}, for j {,, }, j α D α φ j ξ c α, where c α > 0 is a constant,. φ j ξ. j=0 Denote the set of all functions that satisfy condition. as X B. First note X B is not empty. A standard construction of such a function follows by letting ρ be a smooth radial bump function supported in the ball centered at the origin with radius such that ρξ = if ξ < and ρξ = 0 if ξ. Now define φ to be φξ = ρξ ρξ. Define the sequence of functions {φ j ξ} k=0 by φ j ξ = φ j ξ, if j N, φ 0 ξ = φ j ξ. j=. The sequence of functions defined in. satisfies the properties in.. For any sequence of functions {φ j } j=0 operators, j, by X B, define the Littlewood-Paley decomposition j = F φ j F for all j N {0}. The Besov norm, denoted by f B s p,q R n, is defined by f B s p,q R n = sjq j f q L p R n j=0 Then the Besov space, denoted by B s p,qr n, is a set of functions f S R n where f B s p,q R n <. The reader is directed to [6,, 8] for this and other similar constructions of the Besov space. The Besov space enjoys many properties some of which we will list below. The reader is directed to [6] for a more exhaustive discussion. First we can make a statement as to how Cauchy sequences behave in B s p,qr n. 0 q.

29 Proposition... [6] If 0 < p, q and s R, then B s p,qr n is a quasi- Banach space. If p, q and s R, then B s p,qr n is a Banach space. The construction of the Besov space is independent of the choice of {φ j } j=0. This is formalized in the next proposition. Proposition... [6] If {φ j } j=0, {ϕ j} j=0 X BR n, then they genarate equivalent quasi-norms on B s p,qr n. The Besov space enjoys several embedding properties with some of the other function spaces discussed already. Proposition..3. [6] The followings embeddings are valid:. S R n B s p,qr n S R n for 0 < p, q, s R.. S R n is dense in B s p,qr n. 3. B s p,qr n L p R n for p <, 0 < q, s > B 0 p,r n L p R n B 0 p, R n for p. 5. B 0,R n C R n B 0, R n. 6. B m,r n C m R n B s, R n for p, m =,,. 7. B s p,q 0 B s p,q for 0 < p, 0 < q q, s R. 8. B s p,minp,q B s p,maxp,qfor 0 < p, q, s R. 9. B s p,r n W p,k R n B, p, R n for p, m =,,. Note that S R n is not dense in B s,qr n or in B s p, R n. Next we can say something about the diversity of B s p,qr n. Proposition..4. [6] For 0 < p 0, p, q 0, q and s 0, s R, then B s 0 p 0,q 0 R n = B s p,q R n, if and only if s 0 = s, p 0 = p, and q 0 = q.

30 agree. This is to say that two Besov spaces coincide if and only if all of the parameters The next few propositions describe other types of properties. To start, there are certain conditions that will make the Besov space a multiplication algebra. Proposition..5. [8] For functions f, g Bp,qR s n and s > n, we have p fg B s p,q R n f B s p,qr n g B s p,qr n. For a function f B s p,qr n, there is a well known scaling property. Recall the definition of δ a from Chapter Section. equation.3. Proposition..6. [8] The following estimate is valid δ λ f B s p,q R n λ n p max {, λ s } f B s p,q R n. The dual space of B s p,qr n is well understood. It is formalized as: Proposition..7. [6, 7] For 0 < p, q < and s R, the dual space of B s p,qr n, denoted B s p,qr n, is given by B s p,qr n = B s+n min{p,} p,q. B s p,qr n has a few isomorphic properties. The next proposition deals with the Bessel Potential I σ.3. which was defined in Chapter Section.4 equation Proposition..8. [6] I σ : Bp,qR s n Bp,q s σ R n is an isomorphism for s, σ R and 0 < p, q. Recall the definition of τ y in Chapter Section., equation.. Note that F φ j F τ y fx = F φ j e iyξ F f = τ y F φ j F fx = F φ j F fx y. Thus j commutes with translation. Then we have the following proposition:

31 Proposition..9. [6] τ y : B s p,qr n B s p,qr n is an isomorphism for s R n and 0 < p, q. The next proposition describes a multiplier theorem that is associated with B s p,qr n. Define the norm N [6] by µ N = sup sup + ξ α D α µξ, α N ξ R n where α is a multi-index and N is a positive integer. Proposition..0. [6] Let 0 < p, q and s R. If µ C R n, f B s p,qr n, and if N is sufficiently large, then there exists a constant c > 0 such that H µ f B s p,q R n c µ N f B s p,q Rn. Here we see that the Besov norm is preserved. B s p,qr n enjoys a couple of real and complex interpolation properties. Let H be a linear complex Hausdorff space. A 0, A be quasi-banach spaces such that A 0 H and A H where means linear and continuous embedding. Define kt, a as kt, a = inf a0 a 0 A A + t a A. 0 a A This is known as the k-functional [3, 6]. A 0 + A is the set defined as A 0 + A = {a A : a = a 0 + a where a 0 A 0 and a A }. Define the real interpolation space, A 0, A Θ,q, by { A 0, A Θ,q = a A 0 + A : a A0,A Θ,q } <, where a A0,A Θ,q is defined as a A0,A Θ,q = 0 t Θ kt, a q dt q, t and with the usual modification made when q =. For this construction and a more in-depth treatment of real interpolation see [3]. With the above setup we state the following proposition. 3

32 Proposition... [6] For 0 < Θ <, 0 < q, q 0, q, s 0, s R where s 0 s and s = Θs 0 + Θs, then for 0 p B s 0 p,q 0 R n, B s p,q R n = Θ,q Bs p,qr n. Furthermore, if s 0, s R with s = Θs 0 + Θs and 0 < p 0, p < with then p = Θ p 0 + Θ p, B s 0 p 0,p 0 R n, B s p,p R n Θ,p = Bs p,pr n. For complex interpolation, let A = {z C : 0 < Rez < } and the closure of A, denoted by A, to be A = {z R : 0 Rez }. We say f is S R n -analytic function in A if it satisfies the following three conditions: for fixed z A, fz S R n, F φf fx, z is a uniformly continuous and bounded function in R n A for all φ S R n with compact support in R n, F φf fx, z is an analytic function in A for all φ S R n with compact support in R n and every fixed x R n. For fz, an S R n -analytic function in A, define F B s 0 p 0,p 0 R n, B s p,p R n where s 0, s R and 0 < p 0, p, q 0, q to be F B s 0 p 0,p 0 R n, B s p,p R n = { f : fit B s 0 p 0,p 0 R n, f + it B s p,p R n for } all t R, f s F B 0 p 0,p 0 R n,b s p,p R n <, and where f s F B 0 p 0,p 0 R n,b s p,p R n is defined as f F B s 0 p 0,p 0 R n,b s p,p R n = max sup l=0, t R Note F B s 0 p 0,p 0 R n, B s p,p R n is a quasi-banach space. B s 0 p 0,p 0 R n, B s p,p R n Θ by B s 0 p 0,p 0 R n, B s p,p R n Θ fl + it B s l p l,p l R n. For 0 < Θ < define = { g : there exists f F B s 0 p 0,p 0 R n, B s p,p R n with g = fθ }. 4

33 Define the norm, B s 0 p 0,p 0 R n,b s p,p R n Θ, on this space by g B s 0 p 0,p 0 R n,b s p,p R n Θ = inf f F B s 0 p 0,p 0 R n,b s p,p R n See [3] for more details of complex interpolation. following proposition. f F B s 0 p 0,p 0 R n,b s p,p R n..3 With this setup we have the Proposition... [6] Let s 0, s R and 0 < p 0, p, q 0, q. If s, p, and q satisfy s = Θs 0 + Θs, p = Θ + Θ, p 0 p q = Θ + Θ, q 0 q then B s 0 p 0,p 0 R n, B s p,p R n Θ = Bp,qR s n. A recent result for unimodular Fourier multipliers was developed by G. Zhao, J. Chen, D. Fan, and W. Guo in 05 [8]. Their result can be summed up as follows Theorem..3. [8] Let β > 0, and µ be a real-valued C R n \ {0} function which is homogeneous of degree β. Suppose that the Hessian matrix of µ is nondegenerate at R n \ {0}. Let 0 p i, q i, s i R for i =,. Then the Fourier multiplier e iµd is bounded from B s p,q R n to B s p,q R n if and only if p p, s n = s n + θβn min p p χ {0,} θ 0, q q { p, }, p.4 holds for some θ [0, ]. Moreover, we have the following asymptotic estimates for the operator norm e iµd s B p,q R n B s p,q R n BθA, θa q q, when q > q,, when q q where { θ 0, and θ [0, ] respectively if condition.4 holds, where A = n min p, } and Bp, q is the beta function defined as p q q Bp, q = 0 t x t y dt. 5

34 Precisely, for fixed p < < p and q > q, we have the following blow-up rates e iµd { s B p,q R n B s p,q R n. Modulation Spaces θ q q, as θ 0 +, θ q q, as θ. The next function space we will construct will be the modulation space. This function space was first introduced by Feichtinger in 983 [8]. There are two common constructions of the modulation space: a continuous, and a discrete construction. The continuous definition of the modulation space makes use of the short-time Fourier transform of a function f with respect to g S R n, denoted by V g fx, ξ, which is defined as V g fx, ξ = gt xfte itξ dt, R n for x, ξ R n. Here we say g is a window function. For 0 < p, q < we define the norm f M s p,q R n by q q f M s p,q R n = V g fx, ξ R R p p dx + ξ sq dξ, n n with the usual modifications if p or q are infinite. Then the modulation space, denoted by M s p,qr n is defined as the set of all functions f S R n where f M s p,q R n <. The reader is directed to [8, 9, 6] for more details about this and similar continuous construction of the modulation space. To define the modulation space in a discrete way let Q k be the unit closed cube with center k, c <, C >, and {σ k } k Z n C c be a sequence of functions such that σ k ξ c, for all ξ Q k, supp σ k {ξ : ξ k < C}, k Z n σ k ξ, for all ξ R n, α σ k ξ..5 6

35 Denote the set of all sequences of functions that satisfy conditions.5 by X M R n. Note that X M R n is not empty. To see this let ρ S R n where ρ : R n [0, ] be a smooth radial bump function with support on the ball B0, n and ρξ = n if ξ. Define ρ k by ρ k ξ = ρξ k, for k Z n. Now define σ k by σ k ξ = ρ k ξ j Z n ρ j ξ. Thus, the sequence of functions {σ k } k Z n defined above satisfy condition.5. For any sequence of functions {σ k } k Z n in X M R n and fixed k Z n define the smooth projection of f onto {ξ : ξ k < C}, denoted by k f, as k = F σ k F. For 0 < p, q define the norm f M s p,q R n by f M s p,q R n = k Z n k sq k f q p q. Now we can define the modulation space, Mp,qR s n, for all f S R n where f M s p,q R n <. Note that M p,qr s n is a quasi-banach space and when p, q, then Mp,qR s n is a Banach space [8, 7]. Furthermore, we have the exact nature of the modulation space Mp,qR s n when we pick different φ X M and a relationship between the discrete and continuous definitions. Proposition... [8] If {σ k } k Z n, {ϕ k } k Z n X M R n, they generate the same quasi-banach space Mp,qR s n. Proposition... [8] For 0 < p, q, s R, then M s p,q R n and M s p,qr n are equivalent norms on the modulation space M s p,qr n. We will see later that M s p,qr n = M s,0 p,q R n and other properties of the modulation space M s p,qr n will be an immediate consequence of the properties for the 7

36 α-modulation space M p,q R n. Thus, we will forgo all other properties of the modulation space M s p,qr n. Instead we will cover more recent results of the modulation space M s p,qr n, as it relates to the study of PDEs and Fourier multipliers. convenience let M 0 p,qr n = M p,q R n. Bényi, Gröchenig, Okoudjou, and Roders [] showed the following multiplier theorem: Theorem..3. [] If α o [0, ], then the Fourier multiplier with symbol e i ξ αo is bounded from M p,q R n into M p,q R n for all p, q and for any dimension n. As of consequence of Theorem..3 A. Bényi, K. Gröchenig, K. Okoudjou, and L. Roders were able to establish the following Fourier Multiplier Theorem: Corollary..4. [] Let ut, x be the solution of the Cauchy problem of dispersive equation.9 that takes the form of equation.0. For α o =, t > 0, and any dimension, then u Mp,qR n = e it u 0 Mp,qR n + t n u0 Mp,qR n. Wang and Hudzik [7] used an almost orthogonality argument and simple calculations to show the following multiplier theorem: Theorem..5. [7] Let s R, p, and 0 q, then e it f M s p,q R n + t n p f M s p,q Rn. Chen, Fan, and Sun [5] also use an almost orthogonality argument and techniques for oscillating integrals to show: Theorem..6. [5] If α o =, t >, s R, and p, q, then αo it e f t n p f M M s p,q R n s p,q R n, and: 8 For

37 Theorem..7. [5] If t >, s R, p, q, and α o > but α o, then e αo it f f M s M p,q R n s p,q R n + tn p f M s+γαo p,q R n, where γα o = α o n p..3 α-modulation Spaces The final function space we will look at is the α-modulation space M p,q R n. Much like the modulation space M s p,qr n, there are two ways of constructing it. One way is the continuous definition using an admissible covering, and the other way is the discrete construction. Let # denote the cardinality of a finite set. Suppose I is a set of countable intervals I R n denoted by I is called admissible covering of R n if the following is satisfied:. R n = I, I I. # {I I : x I} m 0 for all x R n and where m 0 is some positive integer. Also, if there exists a constant 0 α such that + ξ α I + ξ α for all I I α and for all ξ R n, then I α is called α-covering. For the sake of convenience, we can let m 0 =. See [0] for a more detailed treatment of admissible coverings. Without loss of generality, we can construct a bounded admissible partition of the unity, {ψ α I } I Iα S R n, that is associated with an admissible α-covering. Let f F L R n be defined by f F L R n = F fξ L R n, then {ψ α I } I Iα S R n satisfies the following:. sup I I α ψ α I F L R n <, 9

38 . supp ψ α I I for all I I α, 3. I I α ψ α I ξ for all ξ R n. Finally, define the segmentation operator, denoted by P α I, by for all I I α and for all f S R n. P α I f = F ψ α I F f, Let p, q <, s R, and 0 α. Suppose I α is an admissible α- covering of R n and {ψi α } I Iα is the associated bounded admissible partition of the unity. Define the α-modulation space, denoted by Mp,q R n, on the set of tempered distributions f S R n that satisfies: f M p,q R n α = I I α P α I f q L p R n + ω I sq q <, where ω I I for all I I α, and making the usual modification when p and q are infinite. First note that the definition of the α-modulation space does not depend on the choice of {ω I } I I. See [5, ] for more details of this construction. For the discrete construction, let c < and C > be two positive numbers which relate to the space dimension n, and 0 α <. Suppose {η α k } k Z n Schwartz functions that satisfies the following: ηk α ξ, if ξ k α α α k < c k α, { supp ηk α ξ R n : ξ k α α α k < C k ηk α ξ, for all ξ R n, k Z n k α δ α α }, D δ ηk α ξ, for all ξ R n and all multi-index δ. be a sequence of.6 Denote the set of sequences of functions that satisfies.6 by X A. Note that X A is not empty. To see this, let ρ be a smooth radial bump function supported on 30

39 the open ball of radius centered at the origin that satisfies ρξ = when ξ < and ρξ = 0 when ξ. For any k Z n define ρ α k by ξ k α Now define η α k by ρ α k ξ = ρ η α k ξ = ρξ Thus this η α k satisfies condition.6. For {η α k } k=0 X A define α k by α k C k α α l Z n ρ α l ξ.. α k = F η α k F..7 For 0 < p, q, s R, and α [0, define the norm M p,q R n by f M p,q R n = k Z n k sq α α k f q L p R n q..8 We now define the α-modulation space Mp,q R n as the set of all f S, such that f M p,q R n <. When α = it will be understood that we are using the continuous definition of Mp,q R n. We will now cover some properties of the α-modulation space Mp,q R n, and by doing so, add to our understanding of the modulation Space Mp,qR s n as well. The next few theorems are similar to the theorems we had for the Besov Space B α p,qr n. Proposition.3.. [5] For 0 < p, q, s R and α [0,, then M p,q R n and M p,q R n are equivalent norms on the modulation space M p,q R n. Proposition.3.. [8] Let {η α k } k Z n and { η α k } k Z n be in X A, then they generate equivalent quasi-norms on M p,q R n. Proposition.3.3. [8] For 0 < p, q, s R, and α [0, ], then M p,q R n is a quasi-norm. If p, q, then M p,q R n is a Banach space. 3

40 Proposition.3.4. [8] The following embedding is true S R n M p,q R n S R n. If 0 < p, q <, then S R n is dense in M p,q R n. There are conditions which we have for an embedding from M s,α p,q R n into M s,α p,q R n. These results are summed up as follows: Proposition.3.5. [8] Let 0 < p p and 0 < q, q. If q q and s s + nα, then p p If q > q and s > s + nα p p M s,α p,q R n M s,α p,q R n. + n α q q M s,α p,q R n M s,α p,q R n., then With this proposition we can say something about the modulation space M s p,qr n, which takes the form of the following corollary, shown by Wang and Hudzik [7]. Corollary.3.6. [7] If s s, 0 < p p, and 0 < q q, then M s p,q R n M s p,q R n. Furthermore, if q > q and with s s > n q q M s p,q R n M s p,q R n., then There are also conditions that guarantee an embedding from M s,α p,q R n into M s,α p,q R n and dilation property. To be able to state these results first we need to decompose R + into suitable regions. These decompositions can be found in [8]. Let 0 < p, q and α, α [0, ] [0, ]. Now define Rp, q; α, α as: { Rp, q; α, α = max 0, nα α q, nα α p p + } p..9 3

41 Now we decompose R + into three regions in two different ways. These decompositions will help us understand and simplify Rp, q; α, α. The first decomposition is defined as R + = S S S 3 where S, S, and S 3 are defined as { S = p, R + : q q p, p }, { S = p, R + : q p + q, p > }, S 3 = R +/ {S S }. See Figure. for how R + is decomposed in this case. The second decomposition is defined as R + = T T T 3 where T, T, and T 3 are defined as: { T = p, R + : q p q, p > }, { T = p, R + : q p + q, p }, T 3 = R +/ {S S }. See Figure. for how R + is decomposed in this case. With the above decomposition for R + we have the following for Rp, q; α, α. For α α, we have nα α q, when p p, S, q Rp, q; α, α = nα α p + q +, when p, S, q 0, when p, S 3. q If α < α, then we have nα α q, when p p, T, q Rp, q; α, α = nα α p + q, when p, T, q 0, when p, T 3. q With this understanding of Rp, q; α, α we can state under what condition M s,α p,q R n is embedded into M s,α p,q R n. 33

42 q S S, S 3 0, 0, 0 p Figure.: Distribution of s c when λ >. Proposition.3.7. [8] For α, α [0, [0,, then if and only if s s + Rp, q; α, α. Now define s c and s p as M s,α p,q R n M s,α p,q R n, s c = { Rp, q;, α, λ >, Rp, q; α,, λ, and s p = n min, p. Proposition.3.8. [8] Let 0 α <, λ > 0, and s + s c 0, then δ λ f { n M p,q R n λ p max max {, λ} s p, λ s+sc} f M p,q R n, 34

43 q, 0 T 3, T, T 0, 0 p Figure.: Distribution of s c when λ. for all f M p,q R n. Conversely, if δ λ f M p,q R n λ n p F λ f M p,q R n, holds for some F : 0, 0, and f M p,q R n, then F λ max { max {, λ} sp, λ s+sc}. Furthermore letting s = s c, then it follows that δ λ f M p,q R n λ n p F λ f M p,q R n, where F is the following: lnλ max{0, q p, q + } p, when λ >, p, F λ = λ n p lnλ q, when λ >, p, lnλ max{0, p q, p q}, when λ. 35

44 The next theorem gives us an understanding of the dual space of the α-modulation space M p,q R n. Proposition.3.9. [8] For 0 < p, q <, s R, and α [0, ], then: M p,q R n s+nα = M min,p max,p,max,q R n. With this proposition, a similar statement can be made about the dual space of the modulation space M s p,qr n, which can be found in Wang and Hudzik [7] and Han and Wang [8]. Proposition.3.0. [7, 8] For s R and 0 < p, q <, we have M s p,q R n = M s max,p,max,q R n. Proposition.3.. [8] For s, σ R the mapping I σ Mp,q s σ,α R n is an isomorphism. : M p,q R n This leads us straight to the following corollary: Corollary.3.. [7] For s, σ R the mapping I σ Mp,q s σ R n is an isomorphism. : M s p,qr n The next theorem deals with the understanding of the α-modulation space as an multiplication algebra, i.e. when fg M p,q R n f M p,q R n g M p,q R n. First we must decompose R + into two regions, this decomposition can be found in [8]. Define D as D = { p, R + : q q p, p }, and define D as D = R +/D. 36

45 Define s 0 as when s 0 = nα p + n α min p, D and q s 0 = nα p + n α max when p, q {, } nα α + q α q p {, p, } { nα α + max, q q α p, }, q D. See Figure.3 to see such a distribution of s 0. Now we are q D, D 0, 0 p Figure.3: Distribution of s 0 for all p and q. able to state the multiplication algebra theorem. Proposition.3.3. [8] If s > s 0, then for all f, g M p,q R n, then fg M p,q R n f M p,q R n g M p,q R n, i.e. the α-modulation space M p,q R n is a multiplication algebra. 37

46 Using this result we obtain a similar version for the modulation space M s p,qr n. Corollary.3.4. [8] Suppose that { n min, }, when q s > { n max, p, }, when q q p, D, q p, D, q then the modulation space M s p,qr n is a multiplication algebra, i.e., for all f, g M s p,qr n. fg M s p,q R n f M s p,q Rn g M s p,q Rn, There are known results for the α-modulation space Mp,q R n when it comes to complex interpolation. Recall the definition for complex interpolation spaces, equation.3 in Section.. We then have the following result: Proposition.3.5. [8] For 0 α <, 0 < θ <, p = θ + θ, p 0 p q = θ + θ, and s = θs 0 + θs, then q 0 q M s 0,α p 0,q 0 R n, M s,α p,q R n = M θ p,q R n. Next we will cover some of the recent results. First, Guo and Chen [7] developed Strichartz estimates for the nonlinear Cauchy problem for dispersive equation.9 and the nonlinear Wave equation. using a T T and duality argument. The first result is for the nonlinear Cauchy problem for dispersive equation. Theorem.3.6. [7] Suppose s R, q, α o 0, ] and β, r, q and p, q satisfies r + n p n 4, and r + n p n 4, 38

47 with r, q, n, p, q, ñ,,, then k Z n k sq α u 0 s+δr,p,α M + q ut, r L p dt x R,q R n s+δr,p+δ r, pq k α k Z n q r R q F t, r r q dt, L p x n where δr, p = α r n + β and F t, x is a nonlinear term. More p r precisely we have: k Z n k sq α R k αe it αo u 0 r L p R n q q r dt u 0 M s+δr,q,α,q R n, and k sq α k Z n R αo it s e s+δr,p+δ r, pq k α k Z n F sds r L p R n dt q r q F r L p R n dt R q r q. Along similar lines as Theorem.3.6, Guo and Chen [7] found Strichartz estimates for the nonlinear Wave equation, which are as follows: Theorem.3.7. [7] Let s R, q, 0 α <, and p, r and p, r both satisfy and n n p r > 0, n n p r ñ > 0, p r 39