Pseudodifferential operators with homogeneous symbols

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1 Pseudodifferential oerators with homogeneous symbols Loukas Grafakos Deartment of Mathematics University of Missouri Columbia, MO Rodolfo H. Torres Deartment of Mathematics University of Kansas Lawrence, KS Abstract We rove boundedness of seudodifferential oerators with symbols satisfying the conditions β ξ γ xa(x, ξ) C β,γ ξ m β + γ on homogeneous Besov-Lischitz and Triebel-Lizorkin saces 1 Introduction The study of seudodifferential oerators with symbols in the exotic classes S1,1 m has received a lot of attention. These are oerators of the form (T f)(x) = e ix ξ a(x, ξ) ˆf(ξ) dξ, R n where the symbol a(x, ξ) is a C (R n R n ) function satisfying β ξ γ xa(x, ξ) C β,γ (1 + ξ ) m β + γ, for all β, γ n-tules of nonnegative integers. The interest in such oerators is in art due to the role they lay in the aradifferential calculus of Bony 0 AMS subject classification numbers: Primary 42B20, 47G30. Secondary 46E35, 35B65. 0 Key words and hrases: Pseudodifferential oerators, homogeneous symbols, Besov- Lischitz saces, Triebel-Lizorkin saces Suorted in art by NSF grant DMS and the University of Missouri Research Board. Suorted in art by NSF grants DMS and DMS

2 [1]. The fact that not all such oerators of order zero are bounded on L 2 comlicates their study. Nevertheless, the exotic seudodifferential oerators do reserve saces of smooth functions. See for examle Meyer [12], Runst [15], Bourdaud [2], as well as Stein [16] and the references therein. The continuity results are often obtained by making use of the so-called singular integral realization of the oerators. This involves roving estimates on the Schwartz kernels of the seudodifferential oerators similar to those of the kernels of Calderón Zygmund oerators. There is, however, an alternative aroach working directly with the symbols of the seudodifferential oerators. This aroach has been ursued by Hörmander in [9] and [10] for L 2 based Sobolev saces. The ideas in those aers combined with wavelets techniques were latter extended to L based Sobolev saces and other more general saces of smooth functions by Torres [17]. In this note we consider C symbols a(x, ξ) in R n (R n \ {0}) that satisfy the following conditions: For all n-tules of nonnegative integers β and γ there exist ositive constants C β,γ such that β ξ γ xa(x, ξ) C β,γ ξ m β + γ, (1) for (x, ξ) R n (R n \ {0}). We call such symbols homogeneous symbols of tye (1, 1) and order m. The class of all such symbols will be denoted by Ṡ1,1 m Ȯur urose is to show boundedness for seudodifferential oerators with symbols in Ṡm 1,1 on homogeneous function saces. Our results are motivated by a theorem of Grafakos [8] who roved boundedness of seudodifferential oerators with symbols in Ṡm 1,1 on homogeneous Lischitz saces. The roof in [8] follows more or less the singular integral aroach of Stein in [16]. In this aer we use a wavelet aroach borrowing ideas from [17]. The aroriate setting for our results is in the context of the homogeneous Triebel-Lizorkin F α,q (R n ) saces and Besov-Lischitz saces Ḃ α,q (R n ) (see the definition of these saces below). As with inhomogeneous symbols, it is ossible to show that the Schwartz kernels of the oerators of order zero satisfy for x y estimates of the form γ xk(x, y) + γ y K(x, y) C γ x y n γ (2) and even better estimates for x y large. See for examle the book by Meyer [13],.294. It is then ossible to analyze boundedness roerties of the oerators using versions of the T1-Theorem of David and Journé [4]. Moreover, such tye of results are alied to seudodifferential oerators 2

3 with inhomogeneous symbols in the book of Meyer mentioned before (.329) in the context of Besov saces with smoothness α > 0 and, q > 1 (cf. also [11]). The arguments therein may be adated to seudodifferential oerators with homogeneous symbols and the same range of arameters for the Besov saces. Our aroach, however, will be based on some very simle calculations that notoriously work for both full scales of F α,q (R n ) and Ḃα,q (R n ) saces. Let S(R n ) be the sace of Schwartz test functions and denote its dual by S (R n ), the sace of temered distributions. In this aer the Fourier transform of a function f S(R n ) is given by ˆf(ξ) = f(x)e ix ξ dx and S 0 (R n ) is used to denote the subsace of S(R n ) consisting of all functions whose Fourier transform vanishes to infinite order at zero. The dual sace of S 0 (R n ), with resect to the toology inherited from S(R n ), is S /P(R n ), the set of temered distributions modulo olynomials. The Triebel-Lizorkin and Besov-Lischitz saces are defined as follows. Let ϕ be a function in S(R n ) satisfying su ˆϕ {ξ : 1/2 ξ 2} and ˆϕ C > 0 for 3/5 ξ 5/3. Define ϕ j (ξ) = 2 jn ϕ(2 j ξ). For α real, 0 <, q <, and f in S /P(R n ) define the Triebel-Lizorkin and Besov-Lischitz norms of f by and f F α,q (R n ) = ( (2 jα ϕ j f ) q ) 1/q L, j= f Ḃα,q (R n ) = ( ( 2 jα (ϕ j f) L ) q ) 1/q, j= resectively. Note the the above definitions are given modulo all olynomials, so strictly seaking an element of the saces F α,q (R n ) and Ḃα,q (R n ) is an equivalent class of distributions. It can be shown that the definition of these saces is indeendent of ϕ. The sace S 0 (R n ) is dense in all of these saces. For these and further roerties of these function saces we refer to the books [14] and [19]. We have the following results for oerators with symbols in Ṡm 1,1 acting on the saces F α,q (R n ) and Ḃα,q (R n ). Theorem 1.1 Let 0 <, q <. For α > n(max(1, 1, q 1 ) 1), every seudodifferential oerator (T f)(x) = a(x, ξ) ˆf(ξ)e ix ξ dξ, f S 0 R n 3

4 with symbol a(x, ξ) in Ṡm 1,1 extends to a bounded oerator that mas the sace F α+m,q (R n ) to F α,q (R n ). For α n(max(1, 1, q 1 ) 1), every T as above which satisfies T x γ = 0 for all γ n(max(1, 1, q 1 ) 1) α extends to a bounded oerator from F α+m,q (R n ) to F α,q (R n ). Theorem 1.2 Let 0 <, q <. For α > n(max(1, 1, q 1 ) 1), every seudodifferential oerator (T f)(x) = a(x, ξ) ˆf(ξ)e ix ξ dξ, f S 0 (3) R n with symbol a(x, ξ) in Ṡm 1,1 sace Ḃα+m,q extends to a bounded oerator that mas the (R n ) to Ḃα,q (R n ). For α n(max(1, 1, q 1 ) 1), every T as above which satisfies T x γ = 0 for all γ n(max(1, 1, q 1 ) 1) α extends to a bounded oerator from Ḃα+m,q (R n ) to Ḃα,q (R n ). We end this section with the following observation. Note that if m γ + β < 0, then γ ξ β x a(x, ξ) is singular at ξ = 0 and for a general function f in S the integral in (3) is not absolutely convergent. For this reason it is natural to define the oerator T initially on S 0. 2 The roof of the theorems As discussed above it is natural to consider T initially defined on S 0. Moreover, we have Lemma 2.1 Let a(x, ξ) be a symbol in Ṡm 1,1. Then the seudodifferential oerator with symbol a(x, ξ) mas S 0 to S. In articular, its formal transose T mas S to S /P. Let f be a function in S 0 and let ξ be the Lalace oerator in the variable ξ. Since for any ositive integer N, an integration by arts gives (T f)(x) = (I ξ ) N (e ix ξ ) = (1 + x 2 ) N e ix ξ, Rn e ix ξ (I ξ) N ( (1 + x 2 ) N a(x, ξ) ˆf(ξ) ) dξ. (4) 4

5 Since ˆf vanishes to infinity order at the origin, the conditions on the symbol a(x, ξ) and an alication of Leibnitz s rule give (T f)(x) R m,n(f) (1 + x 2 ) N, (5) where R m,n is an aroriate seminorm in S. A similar comutation alies to the derivatives γ (T f), thus roving the lemma. For an n-tule of integers k and an integer j denote by Q jk the dyadic cube {(x 1,..., x n ) R n : k i 2 j x i < k i + 1}, by x Qjk its lower left corner 2 j k and by l(q jk ) its side length 2 j. For Q dyadic let ϕ Q (x) = Q 1/2 ϕ j (x x Q ). A function ϕ as in the definition of the homogeneous saces can be chosen so that for f S, f = Q < f, ϕ Q > ϕ Q, (6) where < f, ϕ Q > simly denotes the action of the distribution f on the test function ϕ Q. For f in F α,q (R n ) or Ḃα,q (R n ) the convergence in (6) is in the (quasi)-norm of the saces and for f in S 0 is in the toology of S. See [6] and [7]. It follows from Lemma 2.1 that the action of a seudodifferential oerator on S 0 can be exressed as T f = Q < f, ϕ Q > T ϕ Q. (7) The oerator given by (7) is the one that is extended to the whole homogeneous sace in our Theorems. We now turn to the roofs. The ma S ϕ (f) = {< f, ϕ Q >} Q (8) is called the ϕ-transform (or sequence of nonorthogonal wavelet coefficients). It is well known by the work in [5], [6] that the homogeneous ϕ-transform rovides a characterization of the saces F α,q (R n ) and Ḃα,q (R n ) via the equivalence of norms f F α,q (R n ) Q α/n 1/2 < f, ϕ Q > χ q 1/q Q j l(q)=2 j L 5

6 and f Ḃα,q (R n ) j q Q α/n 1/2 < f, ϕ Q > χ Q l(q)=2 j L 1/q where χ Q is the characteristic function of Q. For α,, and q as above let J = n/ min(1,, q) and let [α] be the integer art in α. A smooth molecule for F α,q (R n ) or Ḃα,q (R n ) associated with a dyadic cube Q with side length l(q) is a function m Q satisfying:, x γ m Q (x) dx = 0 if γ [J n α] (9) m Q (x) Q 1/2 (1 + l(q) 1 x x Q ) max(j+ε,j+ε α) (10) γ m Q (x) Q 1/2 γ /n (1 + l(q) 1 x x Q ) J ε, γ [α] + 1 (11) The imortance of these functions is due to the fact that if f = Q s Q m Q in S, where {m Q } is a family of smooth molecules for F α,q then f F α,q (R n ) C Q α/n 1/2 s Q χ q 1/q Q j l(q)=2 j or f Ḃα,q (R n ) C j q Q α/n 1/2 S Q χ Q l(q)=2 j (R n ) or Ḃα,q (R n ), L 1/q L For the above results we refer again to [6] and [7]. Now, let T be a linear continuous oerator from S 0 S. Assume that T ϕ Q = C Q m/n m Q, (12) where {m Q } Q is a family of smooth molecules for F α,q (R n ) or Ḃα,q (R n ). Then using the ϕ-transform it is easy to see that T can be extended as a bounded oerator from F α+m,q (R n ) to F α,q (R n ) or from Ḃα+m,q (R n ) to Ḃ α,q (R n ). 6.

7 Suose that T is a seudodifferential oerator whose symbol a(x, ξ) is in Ṡm 1,1. By the remark (12) above it will suffice to show that for a fixed dyadic cube Q, T ϕ Q is a scaled multile of a molecule. A simle dilation shows that (T ϕ Q )(x) = e ix ξ a(x, ξ) ϕˆ Q (ξ)dξ = 2 jn/2 (T Q ϕ)(2 j x k), (13) if Q = Q jk, where we set (T Q f)(x) = e ix ξ a(2 j (x + k), 2 j ξ) ˆf(ξ)dξ. Let us fix a multi-index γ. We have ( γ T Q ϕ)(x) = e ix ξ C δ (iξ) δ R n x γ δ (a(2 j (x + k), 2 j ξ) ˆϕ(ξ) dξ (14) δ γ for certain C δ constants, where δ γ simly means that δ j γ j for all j = 1,..., n. Now fix N > max(j, J α)/2. An integration by arts gives Rn ( γ T Q ϕ)(x) = e ix ξ (I ξ) N (1 + x 2 ) N δ γ By Leibnitz s rule, there exist constants K α,β such that C δ (iξ) δ x γ δ (a(2 j (x+k), 2 j ξ) ˆϕ(ξ) dξ (I ξ ) N x γ δ (a(2 j (x + k), 2 j ξ) ˆϕ(ξ)(iξ) δ ) = K α,β β ξ γ δ x (a(2 j (x + k), 2 j ξ) ξ α ( ˆϕ(ξ)(iξ) δ )). α+β =2N Using the estimates (1) we conclude that β ξ γ δ x (a(2 j (x + k), 2 j ξ) C β,γ δ 2 j β 2 j( γ δ ) 2 j ξ m+( γ δ ) β C2 jm ξ m+( γ δ ) β. (15) Summing over all α, β, and δ as above and using that ξ 1 we conclude that the integrand in (15) is ointwise bounded by C γ 2 jm (1 + x 2 ) N and thus ( γ T Q ϕ)(x) C γ 2 jm (1 + x 2 ) N. (16) 7

8 We now dilate and translate (16) to deduce that ( γ T Q ϕ)(x) C2 jn/2 2 j γ 2 jm (1 + 2 j x k 2 ) N C Q m/n 1/2 γ /n (1 + l(q) 1 x x Q ) max(j+ε,j+ε α) We must now check the vanishing moment condition for T ϕ Q. If [J n α] < 0 this condition is vacuous. If [J n α] 0, this is an easy consequence of the hyothesis T x γ = 0, since x γ (T ϕ Q )(x)dx =< x γ, T ϕ Q >=< T (x γ ), ϕ Q >= 0. Both theorems are now roved. We conclude this section with some remarks. 1. For > 1 let denote /( 1). It can be shown that the saces F α,q (R n ) can actually be considered as saces of distributions modulo olynomials of degree less than or equal to [α n/]. (See [6] age 154.) For 1 <, q <, if T mas F α+m,q (R n ) to F α,q (R n ), then by duality T mas F α,q (R n ) to F α m,q (R n ). It follows that for T to be even well-defined on F α,q (R n ) it must annihilate olynomials of degree [ α n/ ]. The conditions on T in the second art of Theorems 1.1 and 1.2 for α < 0 become then necessary for For more general oerators T with kernels satisfying estimates (2) and the usual weak boundedness roerty assumed in the T1-Theorem (which is always satisfied by seudodifferential oerators of order zero and their transoses), the results in [13] state that the conditions T (x γ ) = 0 for all γ [α] imly that T is bounded on Ḃα,q (R n ) for α > 0 and, q 1. Let now α < 0. If T is a seudodifferential oerator in Ṡ0 1,1, then T is not necessarily a seudodifferential oerator. Nevertheless, its kernel K (x, y) is K(y, x) and, hence, it still satisfies the estimates (2) by symmetry. The results in [13] state that if T annihilates olynomials of degree less than or equal to [ α], then T is bounded on Ḃ α,q, and then by duality T = T is bounded in Ḃα,q, which agrees with our results. Such duality arguments are not available for other values of the arameters but our roof is still valid. 3. When m 0, we could have recomosed the oerator with an aroriate ower of the Lalacian and reduced our roof to the case m = 0. There are also versions of the T1-Theorem for general oerators whose kernels satisfy aroriate m-versions of the estimates (2). In rincile, such results could be alied to oerators in Ṡm 1,1 and some values of the 8

9 arameters α, and q, but they lead to weaker results than the ones we have resented here (cf. [18]). 4. Note that our aroach does not require the giving of a recise meaning on the action of an oerator T (satisfying (2)) on olynomials, as it is usually required in the T1-Theorem and its versions. We only need to know that T acts on olynomials, which is automatic by duality. 5. Oerators with homogeneous symbols of degree m in ξ satisfying estimates (1) were studied by Calderón and Zygmund [3]. For this subclass of symbols a artial calculus holds but it does not extend to the whole class Ṡ1,1 m (see [16],.268). 3 Examles and alications Symbols in Ṡm 1,1 which are indeendent of x exist in abundance. For instance, it is easy to see that the recirocal of an ellitic olynomial of n variables which is homogeneous of degree m > 0 is in Ṡ m 1,1. An examle of a homogeneous symbol in the class Ṡ0 1,1 is the following: + k= e i2k x ξ φ(2 k ξ), where φ is a smooth bum suorted away from the origin. More generally, suose that the sequence of smooth functions {m k (x)} k Z in R n satisfies β m k C α 2 α k, (17) for all α n-tules of nonnegative integers and k any integer. Then the symbol a(x, ξ) = + k= m k (x)φ(2 k ξ) is in Ṡ0 1,1. We now give an alication. Let j be the Littlewood-Paley oerators defined by j g(ξ) = ĝ(ξ)φ(2 j ξ), where φ is a smooth bum suorted away from the origin which satisfies j Z φ(2 j ξ) = 1 for all ξ 0. Suose now that f is a function on R n that satisfies j f C(f) <. (18) j Z 9

10 Let F be a C function on R n with F (0) = 0. Suose that f is in some of the homogeneous function sace discussed in the revious section with index of smoothness α > 0. Denote such sace by X α,q. We claim that F (f) lies in the same sace X α,q. For the roof of this we borrow the ideas of Bony as resented in [12]. For k integer define k f k = j f, (19) j= and write f = lim k f k, with uniform convergence because of (18). The functions f k have Fourier transforms with comact suort and they are smooth (actually analytic of exonential tye). Moreover, by (18), they are uniformly bounded. Using (18) and Bernstein s inequality we obtain the following estimates for their derivatives Write now F (f) = lim α f k C α 2 α k f k C(f)C α 2 α k. (20) N N k= N F (f k ) F (f k 1 ) = + k= F (f k ) F (f k 1 ), (21) where convergence is justified from the fact that F is continuous, F (0) = 0 and that f N f and f N 0 uniformly as N +. Next aly the mean value theorem to write (21) as where F (f)(x) = m k = k= m k (x) ( k f)(x), F (tf k + (1 t)f k 1 ) dt. (22) Using (20), the smoothness of F, and the chain rule, we see that the functions m k satisfy (17). We conclude that the symbol a(x, ξ) = + j= m j (x)φ(2 j ξ) is in Ṡ0 1,1. We have that F (f) = T af. It follows from Theorems 1.1 and 1.2 that the function F (f) is in X α,q, rovided α > 0 and, q 1, or if 10

11 α > n(max(1, 1, q 1 ) 1). Observe that because of the nonlinearity of the roblem, in the estimate F (f) X α,q C f f X α,q, (23) the constant C f deends on f. In fact, both the functions m k and the symbol a deend on f. If we assume that F (t) = t D then, after a careful examination of the arguments above and of the roof of the theorems in the revious section, we see that C f in (23) is controlled by a suitable (large) ower of the bound C(f) in (18). Finally observe that the left hand side of (18) is the Ḃ0,1 norm of f. By some well-known facts about functions of exonential tye (see [19]), j f C2 jn/ j f. From this one obtains the inequality (Sobolev-Besov embedding) j Z j f C j Z 2 jn/ j f. Then, in articular, the alication described can be used in Ḃα,q (R n ) with α = n/ and q = 1 yielding (23) with C f controlled by a ower of f Ḃn/,1. References [1] J. Bony, Calcul symbolique et roation des singularités our les equations aux dérivées artielles non linéaires, Ann. Sci. Ecole Norm. Su. 14 (1981), [2] G. Bourdaud, Une algèbre maximale d oérateurs seudodifferentils, Comm. Partial Diff. Eq. 13 (1988), [3] A. Calderón and A. Zygmund, Singular integral oerators and differential equations, Amer. J. Math. 79 (1957), [4] G. David and J-L. Journé A boundedness criterion for generalized Calderón Zygmund oerators, Ann. of Math. 120 (1984), [5] M. Frazier, B. Jawerth, Decomosition of Besov saces, Indiana Univ. Math. J. 34 (1985),

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