The Journal of Fourier Analysis and Applications Volume 2, Number 3, 1996

Approximation of Distribution Spaces by Means of Kernel Operators George C. Kyriazis ABSTRACT. We investigate conditions on kernel operators in order to provide prescribed orders of approximation in the Triebel–Lizorkin spaces. Our approach is based on the study of the boundedness of integral kernel operators and extends the Strang–Fix theory, related to the approximation orders of principal shift-invariant spaces, to a wide variety of spaces.

1. Introduction 

Let P : D → D be a continuous linear operator from D = D.Rd /, the space of C ∞ functions with compact support, to its topological dual D = D .Rd /, the space of distributions. By the Schwartz kernel theorem (see [H]) there exists a distribution K ∈ D .Rd × Rd /, referred to as the kernel of P , such that the action of P on D is described by  Þ  Þ P ; = K;  ⊗ for every ; ∈ D, where ⊗ stands for the tensor product. To each such operator P we associate a family of operators Ph , h > 0, with corresponding kernels Kh .x; y/ := h−d K.x= h; y= h/:

(1.1)

For X ; Y ⊂ D , we say that P provides order of Ð approximation ¼ > 0 for the pair .X ; Y/ if for every f ∈ Y the error of approximation I − Ph f X satisfies  Ð   I − Ph f  ≤ C h¼ f Y (1.2) X for some constant C independent of f and h. Here, and for the rest of the paper, by I we denote the identity operator on D . We shall investigate conditions on the kernel K so that the operator P provides order of apÞ;q Þ+¼;q Ð proximation ¼ > 0 for pairs of various Triebel–Lizorkin spaces F˙ p ; F˙ p , Þ ∈ R, p = ∞,  Þ+¼ Ð 0 < p, q ≤ ∞, as well as for pairs of Sobolev spaces WpÞ ; Wp , 1 ≤ p ≤ ∞, Þ, ¼ ∈ N. This type of question is primarily motivated by the recent developments of wavelet theory and by classical problems of approximation theory related to the approximation orders of shift-invariant spaces. We Math Subject Classification. 41A25, 41A35, 42B25, 42B99, 46E35, 46F12. Keywords and Phrases. Strang–Fix conditions, approximation order, Triebel–Lizorkin spaces, wavelets, kernel operators. Acknowledgements and Notes. I wish to thank Professor R. A. DeVore for his valuable comments and suggestions on the paper.

c 1996 CRC Press, Inc. ISSN 1069-5869

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G. C. Kyriazis

would like to mention at the outset that for our purposes it suffices to consider kernels that are well behaved on the main diagonal {.x; y/ : x = y}. Nevertheless, part of our subsequent analysis holds for singular kernels as well under some straight forward modifications (which will not be given). A space S ⊂ D is called shift-invariant (SI hereafter) if S is invariant under translations by integers, that is, f ∈ S ⇐⇒ −j f ∈ S;

j ∈ Zd

(recall that for f ∈ D and x ∈ Rd , −x : f → f .· − x/ is the usual translation operator that is extended to D by means of duality). The most commonly used SI spaces are the so-called principal shift-invariant spaces (PSI), S = S./, generated by a single function  on Rd . One creates such a space by taking the closure, in the topology of D (or some other stronger topology), of the space generated by the finite linear combinations of the integer translates (shifts hereafter) of . Associated to S we have the family of spaces S h S h := {f : ¦h f ∈ S}; where the action of the dilation operator ¦h on any test function f ∈ D is given by ¦h f .·/ := f .·= h/: We say that S provides order of approximation ¼ > 0 for the pair .X ; Y/ if for every f ∈ Y the error of approximation EX .f; S h / := inf f − s X

(1.3)

EX .f; S h / ≤ C h¼ f Y :

(1.4)

s∈S h

satisfies

Heuristically, the connection between (1.2) and (1.4) can be seen if one considers X K.x; y/ :=  .y − j /.x − j /

(1.5)

j ∈Zd

for an appropriate function  on Rd . Then, the operators Z Z X Kh .x; y/f .y/ dy = h−d f .y/ .y= h − k/ dy .x= h − k/ Ph f .x/ = Rd

k∈Zd

(1.6)

Rd

provide a method to approximate f by elements of S h . Thus, if P provides order of approximation ¼ > 0 for the pair .X ; Y/, and leaving aside for a moment questions of convergence, one only needs that S provides order of approximation ¼ for .X ; Y/. to justify that Ph f ∈ S h to conclude  ¼Ð The pairs .X ; Y/ = Lp ; Wp , 1 ≤ p ≤ ∞, ¼ ∈ N, with Lp := Lp .Rd / the spaces of ¼ p-integrable functions and Wp := W ¼ .Lp .Rd // the corresponding Sobolev spaces equipped with the norm X f Wp¼ := f Lp + |f |Wp¼ ; |f |Wp¼ = D Þ f Lp ; |Þ|=¼

have been studied extensively during the last forty years, motivated primarily from problems associated with finite elements and the theory of splines. Consequently, the first methods developed have

Distribution Spaces

263

their roots in approximation theory and generally exploit the fact that polynomials provide efficient means for approximating smooth functions. Schoenberg [S], in the 1940s, was the first to consider the approximation properties of PSI spaces generated by a univariate compactly supported function  and to relate the order of polynomials that can be reproduced locally by the shifts of , on one hand, with the behavior of the Fourier transform b  of , on the other. In the 1970s, a more systematic study was conducted by Strang and Fix [SF], who considered the approximation orders of PSI subspaces of L2 generated by compactly supported functions. Strang and Fix proved that if for some ¼ ∈ N i. b .0/ = 0;

(1.7) D Þb .2³¹/ = 0 for all |Þ| < ¼ and ¹ ∈ Zd \ {0};  ¼Ð then the PSI space S provides order of approximation ¼ for the pair L2 ; W2 . Conditions (1.7) are the so-called Strang–Fix conditions of order ¼ (SF¼ hereafter). In [SF] it was also shown that (1.7) are necessary if the approximation is realized in some controlled manner. The results of [SF]  ¼Ð were generalized for the pairs Lp ; Wp , 1 ≤ p ≤ ∞, ¼ ∈ N, and extended to certain directions by de Boor and Jia [BJ] and Dahmen and Micchelli [DM]. Nevertheless, the requirement that the generator of the PSI space S have compact support appeared to be a major obstacle, especially with the advent of wavelets and the recent developments in radial basis theory. Among the first to consider PSI spaces generated by a globally supported function were Jackson [J], Light and Cheney [LC], and Jia and Lei [JL]. The approximation schemes that are employed in the above references are given by what are called quasi-interpolant operators. The underlying idea for their construction is that theTfinitedimensional space 5<¼ (of polynomials of degree < ¼), which by SF¼ can be shown to lie in h S h , provides locally good approximants for smooth functions. Drawbacks of this approach are that it imposes certain a priori size assumptions on the generator  due to the use of the Poisson summation formula and, equally important, that it cannot be extended to more general spaces of distributions due to its local nature. A new impetus in the study of the approximation properties of PSI subspaces of L2 was recently given by de Boor, DeVore, and Ron in [BDR]. Avoiding the quasi-interpolants, they first described the elements of S in terms of their image under the Fourier transform and then used the L2 -projector to establish the order of approximation of S. In a similar fashion, de Boor and Ron [BR] investigated the approximation power of PSI spaces in the L∞ -norm. Motivated by their work, in [K1] we studied the approximation properties of PSI subspaces of Hp spaces, 0 < p < ∞. Using the theory of convolution operators, we were able to circumvent the Poisson summation formula and established lower bounds for the approximation order of PSI spaces. As in [BDR], the virtue of our results is that they can be applied to functions  that have relatively slow decay at ∞ and we can also characterize nonintegral approximation orders. The error analysis as well as the approximation schemes in [BDR], [BR], and [K1] were given in terms of the frequency domain, and this approach seems very well suited to the study of PSI spaces generated by radial basis functions and box splines. In this direction we mention the recent papers [R] and [K2], where, using certain extensions of [BDR] and [K1], the approximation properties of PSI subspaces generated by exponential box splines related to general rational matrices were investigated. In this article, however, we will follow a new approach based on the boundedness of linear operators on various spaces. Our main tool will be the '-transform of Frazier and Jawerth—a new powerful device for the study of the action of linear operators on various spaces. As we will prove in §2 one can reduce the problem of approximation orders to that of the boundedness of certain kernel operators (KO). Moreover, we will see that the continuity of such operators on spaces of different ii.

264

G. C. Kyriazis ¼

smoothness, say from Wp to Lp , is intrinsically connected with the SF¼ , the Strang–Fix conditions of order ¼. This last observation relates somehow the problem of polynomial reproduction, or more general of approximation orders, with deeper aspects of the theory of singular integral operators. This connection should not come as a surprise since the Strang–Fix conditions (SF1 ) are already present, in some form, in the T 1 theorem of David and Journ´e [DJ] that deals with the boundedness of the Calderon–Zygmund operators on L2 . As we have already mentioned our method is applicable to a wide variety of spaces that includes all the classical Sobolev and potential spaces, and we also extend our results to Besov spaces by means of interpolation. Moreover, questions regarding simultaneous approximation can be handled trivially in this framework. Finally we note that our approximation scheme is directly related to the wavelet theory since one can always start with an appropriate  that gives rise to a multiresolution analysis and consider the L2 -projector on S./ as a kernel operator (see [M]). Throughout this paper, we shall use standard multi-index notation. In particular, for every x = .x1 ; : : : ; xd / ∈ Rd and Þ = .Þ1 ; : : : ; Þd / ∈ Nd , we define x Þ := x1Þ1 · · · xdÞd , |Þ| := Þ1 + · · · |Þ| +Þd , and D Þ := @ Þ1 x1@···@ Þd xd . For x ∈ R we let [x] stand for the greatest integer satisfying x − 1 < [x] ≤ x, x ∗ := x − [x] while x+ := max{x; 0}. By S := S.Rd / we denote the Schwartz space of infinitely differentiable, rapidly decreasing   functions on Rd and by S := S .Rd / its dual, that is, the space of tempered distributions. The Fourier transform b f of an integrable function is defined by b f .¾ / :=

Z Rd

f .x/e¾ .−x/ dx;

where e¾ .·/ := ei.·/¾ . We shall also denote by 5 the space of all polynomials on Rd , by 5<¼ the space of polyno mials of degree < ¼, and by S  =5 the space of equivalence classes of distributions in S modulo polynomials. Now let ¹ ; ¹ ∈ Z, be a family of functions in S with the following properties. i.

supp b ¹ ⊂ {2¹−1 ≤ |¾ | ≤ 2¹+1 }:

|D þ b¹ .¾ /| ≤ C2−¹|þ| for all þ ∈ Zd : P 2 d b iii. ¹∈Z |¹ .¾ /| = 1 for every ¾ ∈ R − {0}:

(1.8)

ii.

For Þ ∈ R, p = ∞, 0 < p; q ≤ ∞, the homogeneous Triebel–Lizorkin space F˙ p to be the set of all f ∈ S  =5 such that

Þ;q

   Xð ¹Þ Łq 1=q   < ∞: f F˙pÞ;q :=  2 |¹ ∗ f | Lp

is defined

(1.9)

¹∈Z

We recall that for 1 < p < ∞, F˙ p0;2 = Lp while for 0 < p ≤ 1, F˙ p0;2 = Hp , which are the real Hardy spaces introduced by Fefferman and Stein in [FS]. Also, for Þ > 0, 1 < p < ∞; F˙ pÞ;2 = HpÞ , the Potential spaces, and for integer values of Þ, F˙ pÞ;2 are the usual Sobolev spaces WpÞ equipped with their seminorm. Þ;q At this point we also introduce the spaces of complex-valued sequences f˙p that are associated Þ;q Þ;q to F˙ p . For the same range of Þ; p; q as above, we let f˙p be the collection of all complex-valued

265

Distribution Spaces

sequences s = {sQ }Q such that f

Þ;q f˙p

X     −Þ=d Ðq 1=q   < ∞;  = |sQ |˜ Q |Q|  p

Q

where ˜ Q = |Q|−1=2 Q is the L2 -normalized characteristic function of Q.

2. The equivalence between approximation orders and the boundedness of kernel operators Let P : D → D be a continuous linear operator defined by  Þ  Þ P ; = K;  ⊗ ; ; ∈ D; where K ∈ D .Rd × Rd /. For every h > 0, we let Ph (P1 := P ) be the operator, defined from (1.1) by means of dilation Þ+¼;q , and Th := Ph − Ph=2 . Our aim is to investigate conditions on K such that for every f ∈ F˙ p Þ ∈ R, 0 < ¼, 0 < p < ∞, min{1; p} ≤ q ≤ ∞ we have  Ð I − Ph f F˙pÞ;q ≤ Ch¼ f F˙pÞ+¼;q : (2.1) One can reduce (2.1) to the study of the boundedness of T := T1 = P − P1=2 from F˙ p Indeed, we have the following theorem.

Þ+¼;q

Þ;q F˙ p .

2.1.

to

Theorem

Let Ph ; h > 0; be a family of operators with P corresponding kernels Kh , given by .1:1/. Assume Þ+¼;q also that for every f ∈ F˙ p , .P − I / f = j ≥0 T2−j f in the sense of S  =5. Then,  Ð I − Ph f F˙pÞ;q ≤ Ch¼ f F˙pÞ+¼;q (2.2) if and only if there exists a constant C such that for every f ∈ F˙ p

Þ+¼;q

Tf F˙pÞ;q ≤ C f F˙pÞ+¼;q :

(2.3)

Proof. One direction is trivial since if (2.2) holds with h = 1 and h = 1=2, then ý  Ð  Ð  Tf F˙pÞ;q = Pf − P1=2 f F˙pÞ;q ≤ C I − P f F˙pÞ;q + I − P1=2 f F˙pÞ;q ≤ C f F˙pÞ+¼;q : i.e.,

For the other direction we first recall (see [Tr]) the homogeneity of the quasi norm · F˙pÞ;q , ¦h f F˙pÞ;q ≤ Ch−.Þ−d=p/ f F˙pÞ;q ;

h > 0:

(2.4)

Ð  P min{1;p} Using that Ph −I f = j ≥0 Th2−j f and the triangle inequality for · F˙ Þ;q , min{1; p} ≤ q ≤ ∞; p (see [FJ, (7.2)]), we have ∞  Ð min{1;p} X min{1;p} I − Ph f F˙ Þ;q ≤ Th2−j f F˙ Þ;q : p

j =0

p

(2.5)

266

G. C. Kyriazis

It is easily seen from (2.4) and the boundedness of T that for every h > 0 Th f F˙pÞ;q = ¦h T ¦1= h f F˙pÞ;q ≤ C h−.Þ−d=p/ T ¦1= h f F˙pÞ;q ≤ C h−.Þ−d=p/ ¦1= h f F˙pÞ+¼;q ≤ C h¼ f F˙pÞ+¼;q :

(2.6)

Employing (2.5) in (2.6) we get ∞  Ð min{1;p} X  .−j /¼ ¼ Ðmin{1;p} I − Ph f F˙ Þ;q ≤ h f F˙pÞ+¼;q 2 p

j =0

Ðmin{1;p}  : = C h¼ f F˙pÞ+¼;q

2

Þ;q Þ+¼;q 1. The above result remains valid if one replaces the pair .F˙ p ; F˙ p / by ¼ 1 ≤ p ≤ ∞, where Wp is equipped with its seminorm (recall that L1 and L∞ do not Þ+¼;q belong in the family of Triebel–Lizorkin spaces). In addition, the F˙ p spaces, in the right-hand Þ+¼;q side of (2.2) and (2.3), could be replaced by the homogeneous Besov spaces B˙ p , since their semi-quasi norms satisfy the same homogeneity property. min{1;p} 2. The subadditivity of · F˙ Þ;q ; min{1; p} ≤ q ≤ ∞, was the reason for considering only p this range of q. 2

Remarks.

¼ .Lp ; Wp /,

In the next section we will study conditions on a general kernel R so that the associated Þ;q operator T satisfies (2.3). Our estimates actually hold for the full range of the F˙ p spaces, that is, for −∞ < Þ < ∞, p = ∞, 0 < p; q ≤ ∞.

3. Boundedness results of kernel operators Let T : D → D be a continuous linear operator with kernel R.x; y/ ∈ D .Rd × Rd /. To any such operator corresponds a transpose T ∗ : D → D defined by  ∗ Þ  Þ T ;  = T ; ; having kernel R∗ .x; y/ = R.y; x/. We are interested in kernels R that have ` ≥ 0 continuous derivatives in the first variable; the `th derivatives are in Lip.Ž/, 0 < Ž < 1; and satisfy the size conditions |D1 R.x; y/| ≤ C.1 + |x − y|/−d−¼−| | ; 

| | ≤ `;

(3.1)

and for | | = ` and |x − x  | < 1 |D1 R.x; y/ − D1 R.x  ; y/| ≤ C|x − x  |Ž .1 + |x − y|/−d−¼−| |−Ž 





(3.2)

(D1 R.x; y/ refers to  derivatives with respect to the first variable). If R satisfies (3.1) and (3.2) we say that T is a kernel operator of order ¼ and smoothness ` + Ž and denote this fact by T ∈ KO¼ .` + Ž/ (when ` = Ž = 0 we will write T ∈ KO¼ instead). It is obvious that if T ∈ KO¼ ; ¼ > 0; and f ∈ Lp ; 1 ≤ p ≤ ∞, then Z Tf .x/ = R.x; y/f .y/ dy: By Tþ; we will denote the operator associated to the kernel 

Rþ; .x; y/ := .y − x/þ D1 R.x; y/

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Distribution Spaces

and use Tþ instead of Tþ;0 and D  T instead of T0; . It is easily verifiable that for ¼ > ¹, KO¼ .`+Ž/ ⊂ KO¹ .` + Ž/. Moreover, for every f ∈ Lp , 1 ≤ p ≤ ∞, and | | ≤ `, a trivial application of the Dominated convergence theorem shows that .D  T /f = D  .Tf /. As we mentioned in the introduction one can assume that the decay and smoothness conditions of the kernel R are valid only away from the main diagonal, so that Tþ; is a Generalized Calderon– Zygmund operator. Nevertheless, for the purpose of characterizing the approximation orders of shift-invariant spaces, it suffices to consider kernels that are well behaved on the main diagonal {.x; y/ : x = y}. Our first theorem reads as follows.

3.1.

Theorem

Let 1 ≤ p ≤ ∞, ¼ ∈ N, and ž > 0. Let T ∈ KO¼+ž , and assume that T y  = 0; | | < ¼: ¼ Then there exists a constant such that for every f ∈ Wp Tf Lp ≤ C|f |Wp¼ : ¼

First we assume that 1 ≤ p < ∞. Let f ∈ Wp ∩ D; since T y  = 0 for | | < ¼,

Proof. we have

Tf .x/ =

Z Rd

R.x; y/f .y/ dy

Z

X D Þ f .x/  Ð R.x; y/ f .y/ − .y − x/Þ dy Þ! Rd |Þ|<¼ Z 1 Þ XZ .y − x/ Þ R.x; y/ = D f .y + t .x − y//t ¼−1 dt dy: d Þ! R 0 |Þ|=¼

=

It follows that with RÞ .x; y/ := .x − y/Þ R.x; y/ |Tf .x/| ≤ C ≤C

XZ d |Þ|=¼ R

X Z

|Þ|=¼

≤C

|RÞ .x; y/|

X Z

Rd

d |Þ|=¼ R

Z

þ þD Þ f .y + t .x − y//þ dt dy

1þ

0

|RÞ .x; y/| dy .1 + |x − y|/

1=p Z

−d−ž

Rd

Z

Z

þ Ð þD Þ f .y + t .x − y//þ dt p

1þ



|RÞ .x; y/|

1=p

0

þ þD Þ f .y + t .x − y//þp dt

1þ

1=p :

0

Integrating with respect to x and changing the order of integration we have R

Rd

|Tf .x/|p dx ≤ C =C

XZ

Z

d |Þ|=¼ R Z 1 X

|Þ|=¼ 0

=C

XZ

1

Z Z

p

p

Rd

Rd

Rd

|Þ|=¼ 0

≤ C|f |W ¼

Z

Z

Z Z

1

þp þ .1 + |x − y|/−d−ž þD Þ f .y + t .x − y//þ dt dy dx

0

Rd

Rd

þp þ .1 + |z|/−d−ž þD Þ f .y + tz/þ dz dy dt þp þ .1 + |z|/−d−ž þD Þ f .y + tz/þ dy dz dt

.1 + |z|/−d−ž dz ≤ C|f |W ¼ : p

Rd

p

268

G. C. Kyriazis ¼

To remove the constraint f ∈ Wp ∩ D one has to employ a standard limiting argument using ¼ the density of D in Wp . ¼ Next we will treat the case p = ∞. For this we recall that every function f ∈ W∞ can be modified on a set of measure 0 so that f has continuous partial derivatives of total order < ¼ (see [St, p. 159]). Moreover, the remainder of the Taylor expansion of f r¼ .y; x/ := f .y/ −

X D Þ f .x/ .y − x/Þ Þ! |Þ|<¼

satisfies |r¼ .y; x/| ≤ C|f |W∞¼ |x − y|¼ : Now it easily follows that Z

þ |Tf .x/| = þ

Rd

Z

þ þ R.x; y/f .y/ dy þ = þ

≤ C|f |W∞¼

Z Rd

Rd

þ R.x; y/r¼ .y; x/ dy þ

|x − y|¼ |R.x; y/| dy ≤ C|f |W∞¼ :

2

The above theorem admits the following generalization.

3.2.

Theorem

Let 1 ≤ p ≤ ∞, Þ ∈ N; ¼ ∈ N+ , and ž > 0. Assume that T : D → D satisfies 1. T ∈ KO ¼+Þ+ž ∩ KO ¼+ž .Þ/, 2. T y  = 0 for every | | < ¼ + Þ: Þ+¼

Then there exists a constant such that for every f ∈ Wp

|Tf |WpÞ ≤ C|f |WpÞ+¼ :

Proof.

Since for every | | ≤ Þ and f ∈ Lp , .D  T /f = D  .Tf /, we deduce that |Tf |WpÞ ≤ C|f |WpÞ+¼

if and only if D  Tf Lp ≤ C|f |WpÞ+¼ ;

for every | | = Þ:

Thus, one has to apply Theorem 3.1 to the operators D  T ∈ KO¼+Þ+ž and the result follows.

(3.3)

2

Our goal for the remainder of this section is to extend Theorem 3.2 to the class of Triebel– Þ+¼;q Þ;q → F˙ p , Lizorkin spaces and, in particular, to study the boundedness of kernel operators T : F˙ p Þ ∈ R, ¼ ∈ N+ , p = ∞, 0 < p, q ≤ ∞. Similar questions regarding singular kernel operators were also considered by Torres [T] for Þ ≥ 0 and 0 ≤ ¼ < 1. We will do so by employing the '-transform of Frazier and Jawerth; we start by describing the atomic and molecular structures of the Triebel–Lizorkin spaces as was given in [FJ].

269

Distribution Spaces

Let Þ ∈ R, p = ∞, 0 < p; q ≤ ∞, and J := d= min{1; p; q}. Also let Q be a dyadic cube with lower left corner xQ and sidelength `Q . We say that a function aQ ∈ D is a smooth atom for Þ;q F˙ p associated with the cube Q if supp aQ ⊂ 3Q; Z

x  aQ .x/ dx = 0

for

(3.4)

ý  | | ≤ max [J − d − Þ]; [J − d] ;

(3.5)

and |D  aQ .x/| ≤ |Q|−1=2−| |=d

| | ≤ [Þ]+ + 1:

for

(3.6)

In (3.4) by 3Q we refer to the cube concentric with Q and with sidelength `3Q = 3`Q . Let M > J and Þ ∗ < Ž < 1 (Þ = [Þ] + Þ ∗ ), we say that mQ is a smooth .Ž; M/ molecule for Þ;q ˙ F p , associated with the cube Q, if Ð− max{M;M−Þ}  ; |mQ .x/| ≤ |Q|−1=2 1 + `−1 Q |x − xQ | Z x  mQ .x/ dx = 0; for | | ≤ [J − d − Þ]; Ð−M  |D  mQ .x/| ≤ |Q|−1=2−| |=d 1 + `−1 ; Q |x − xQ |

(3.7) (3.8)

| | ≤ [Þ];

(3.9)

and for | | = [Þ], |x − y|=`Q ≤ 1 |D  mQ .x/ − D  mQ .y/| ≤ |Q|−1=2−| |=d−Ž=d |x − y|Ž

sup



|w|≤|y−x|

Ð−M

1 + `−1 Q |w − .x − xQ /|

:

(3.10)

We note that (3.8), (3.9), and (3.10) are void if the numbers bounding | | are negative. The atomic decomposition of the Triebel–Lizorkin spaces, given in [FJ] by Frazier and Jawerth, reads as follows.

3.3.

Theorem

Þ;q Let Þ ∈ R, p = ∞, 0 < p, q ≤ ∞. For each f ∈ F˙ p , there P exists a family of smooth  s = {s } such that f = atoms {aQ }Q and a sequence of coefficients Q Q Q sQ aQ (in S =5) and P s f˙pÞ;q ≤ C f F˙pÞ;q : Conversely, Q sQ aQ F˙pÞ;q ≤ C s f˙pÞ;q for any family of smooth atoms {aQ }Q .

We point out that a closer look at the proof of Theorem 3.3 reveals that the number of zero moments as well as the smoothness of the atoms can be chosen to be arbitrarily high. The following result is also due to Frazier and Jawerth.

3.4.

Theorem P Þ;q If f = Q sQ mQ (in S  =5), where {mQ }Q is a family of smooth molecules for F˙ p , then f F˙pÞ;q ≤ C s f˙pÞ;q .

Recall that F˙ p0;2 = Hp and that for p ≤ 1 the atomic decomposition for the Hardy spaces (which was first given in [L] and [C]) has been used to study the boundedness of operators on these spaces by concentrating on their action on atoms (see [TW]). More generally, in order to show that

270

G. C. Kyriazis

Þ+¼;q Þ;q T : D → D has a continuous extension T : F˙ p → F˙ p , it is enough to consider the action of Þ+¼;q atom aQ associated T on smooth atoms and prove that there is a constant such that for every F˙ p Þ;q ¼=d with a dyadic cube Q, C|Q| T aQ is a F˙ p molecule corresponding to Q. To see this we note that, Þ+¼;q can be decomposed as by Theorem 3.3, every f ∈ F˙ p X f = sQ aQ ; with s f˙pÞ+¼;q ≤ C f F˙pÞ+¼;q : (3.11) Q

P Þ+¼;q by T˜ f := Q sQ T aQ . From Theorem 3.4 it readily follows that Thus, we can extend T to F˙ p X X T˜ f F˙pÞ;q = sQ T aQ F˙pÞ;q = |Q|−¼=d sQ |Q|¼=d T aQ F˙pÞ;q Q

Q

≤ C {|Q|−¼=d sQ }Q f˙pÞ;q = C s f˙pÞ+¼;q ≤ C f F˙pÞ+¼;q : The question arises as to wheather the above extension is well defined. However, in the cases in which we are interested, that is, for T ∈ KO¼ , ¼ > 0, T is a bounded operator on L2 . Moreover, Þ+¼;q it was shown in [FJ] that for f ∈ L2 ∩ F˙ p the convergence in (3.11) holds also in the L2 sense Þ+¼;q . For more details in the case where no a and therefore T˜ agrees with T at least on L2 ∩ F˙ p priori assumptions on T are known we refer the reader to [FJ] and [T] for arguments based on the boundedness of almost diagonal operators on sequence spaces. In the sequel we are going to write Þ+¼;q T instead of T˜ with the understanding that we allude to the extension of T from L2 ∩ F˙ p to Þ+¼;q F˙ p . The rest of this section is devoted to proving the following theorems.

3.5.

Theorem

Let Þ < 0, p = ∞, 0 < p, q ≤ ∞, and ¼ ∈ N+ . Assume that for some ž > 0, T : D → D satisfies 1. 2. 3. 4.

T ∈ KO ¼+ž , T y  = 0 for every | | < ¼, T ∗ ∈ KO [J −d−Þ]+ž ; T ∗ x  ∈ 5≤max{[J −d−¼−Þ];[J −d]}

for every | | ≤ [J − d − Þ].

For J − Þ ≥ d + ¼, in addition, we assume that 5. T ∗ ∈ KO¼ .L/ where L = [J − Þ − d − ¼] + 1. Then T extends to a bounded operator from F˙ p

Þ+¼;q

3.6.

into F˙ p . Þ;q

Theorem

Let 1 ≤ min.p; q/, p = ∞, Þ ≥ 0; ¼ ∈ N+ , and 0 < Ž < 1. Assume that T : D → D satisfies 1. T ∈ KO ¼+Þ ∩ KO¼ .Þ + Ž/ if

Þ ∈ N,

∩ KO .[Þ] + 1/ if Þ ∈ R \ N; T ∈ KO 2. T y  = 0 for every | | ≤ ¼ + Þ − 1 if Þ ∈ N, ¼

¼+Þ

T y  = 0 for every | | ≤ ¼ + [Þ] if ∗

3. T 1 ∈ 5<1

if

Þ ∈ R \ N;

Þ = 0.

Þ+¹;q Þ;q into F˙ p . Then for every 0 < ¹ < ¼, T extends to a bounded operator from F˙ p

271

Distribution Spaces

3.7.

Theorem

Let min.p; q/ < 1, p = ∞, Þ ≥ 0, ¼ ∈ N+ , and 0 < Ž; ž < 1. Assume also that for some 0 ≤ ¹0 < ¼, T : D → D satisfies 1. T ∈ KO ¼+Þ ∩ KO¼ .Þ + Ž/ if T ∈ KO

¼+Þ

Þ ∈ N;

∩ KO .[Þ] + 1/ if ¼

Þ ∈ R \ N;

2. T y = 0 for every | | ≤ ¼ + Þ − 1 if

Þ ∈ N;

T y = 0 for every | | ≤ ¼ + [Þ] if

Þ ∈ R \ N;

 

∗

3. T ∈ KO ∗ 

[J −d−Þ]+ž

4. T x ∈ 5≤[J −d] if

; | | ≤ [J − d − a]:

For J ≥ d + ¹0 we additionally assume that for every | | ≤ [Þ] with d + ¹0 + | | ≤ J þ þ  þ 5. þD2 D1 R.x; y/þ≤ C.1 + |x − y|/−d−¼−|þ|−| | for every |þ| ≤ [J − d − ¹0 − | |] + 1. Finally for d + ¹0 + [Þ] ≤ J and | | = [Þ] + 1 þ þ  þ ð Ł 6. þD2 D1 R.x; y/þ ≤ C.1 + |x − y|/−d−¼−|þ|−| | for every |þ| ≤ J − d − ¹0 − [Þ] . Then T extends to a bounded operator from F˙ p

Þ+¹;q

into F˙ p

Þ;q

for every ¹0 ≤ ¹ < ¼.

þ

By D2 R.x; y/ we refer to þ derivatives with respect to the second variable. Before we continue with the proof of the theorems we mention that the only reason for assuming that T ∈ KO ¼+Þ is to guarantee that T y  is well defined for every | | ≤ ¼+Þ −1 (or | | ≤ ¼+[Þ]). However, if one defines T y  as equivalent classes of distributions, modulo polynomials of certain degree, in a similar way with the definition of T 1 of David and Journ´e [DJ], then the assumption KO¼ .Þ + Ž/ (or KO¼ .[Þ] + 1/, respectively) suffices to guarantee the existence of T y  (for details see [T]). ¼;2 ¼ As we have already mentioned, for ¼ ∈ N and 1 < p < ∞; F˙ p is the Sobolev space Wp equipped with its seminorm. A direct comparison of Theorem 3.1 with Theorem 3.6, for this range of indices, favors the former over the latter. Indeed, notice that if an operator T annihilates 5<¼ , then under some decay conditions (T ∈ KO¼+ž ) Theorem 3.1 shows that Tf Lp ≤ C|f |Wp¼ : On the other hand, Theorem 3.6 requires, in addition, smoothness on the kernel of T (T ∈ KO¼ .Ž/) and gives only that Tf Lp ≤ C|f |Wp¹ ;

0 < ¹ < ¼:

(3.12)

To shed some light on this discrepancy we note that for Theorem 3.6 we will actually prove something more, namely, that T maps atoms to smooth molecules—a condition sufficient but certainly not necessary for the establishment of (3.12). As a matter of fact, the operator T of ¼ Theorem 3.1 does not map atoms of Wp to smooth molecules for Lp but nevertheless is bounded. The basic step in proving the above theorems is the following elementary lemma. We first introduce some notation. For every f defined on Rd , z ∈ Rd , and t > 0 we define  f z;t .·/ := f t · +z/;

272

G. C. Kyriazis

while for every operator T associated with a kernel R we let T z;t be the operator related to t d Rz;t .x; y/ := t d R.tx + z; ty + z/:

(3.13)

Also by Q0 we will denote the unit cube with lower left corner at the origin.

3.8 Lemma Let T ∈ KO ¼ , ¼ ∈ R+ \ N; and assume that T y m = 0 for all |m| ≤ [¼]. Then for any  ∈ D with supp  ⊂ 3Q0 , X D m  L∞ (3.14) T z;t  L∞ ≤ Ct −¼ |m|≤[¼]+1

for some constant independent of .

Proof.

√ We first assume that x ≥ 6 d. Then Z þ þ R.tx + z; ty + z/.y/ dy þ |T z;t .x/| = t d þ 3Q0

≤ Ct d

Z

(3.15) |tx − ty|−d−¼  L∞ dy ≤ Ct −¼  L∞

3Q0

√ √ because for every y ∈ 3Q0 , |x − y| > 2 d. Next we assume that |x| < 6 d. Since T y m = 0, |m| ≤ [¼]; Z T z;t .x/ = t d R.tx + z; ty + z/f .x; y/ dy; where f .x; y/ := .y/ −

X D m .x/ .y − x/m : m! |m|≤[¼]

Using the decay of R, Z

þ |T z;t .x/| = t d þ

+t ≤ Ct d

|x−y|≤1

Z

þ

dþ

Z

−¼

|x−y|>1

|x−y|≤1

+ Ct d ≤ Ct

þ R.tx + z; ty + z/f .x; y/ dy þ þ R.tx + z; ty + z/f .x; y/ dy þ

|tx − ty|−d−¼ |x − y|[¼]+1

Z

|x−y|>1

X

sup

|m|=[¼]+1

D m  L∞ dy

|tx − ty|−d−¼ |x − y|[¼] sup D m  L∞ dy |m|=[¼]

D  L∞ : m

|m|≤[¼]+1

Now combining (3.15) and (3.16) the result follows.

2

In an almost identical way we can extend the above lemma to integer values of ¼.

(3.16)

273

Distribution Spaces

3.9

Lemma

Let T ∈ KO ¼+ž , ¼ ∈ N; ž > 0; and assume that T y m = 0 for all |m| < ¼. Then for any  ∈ D with supp  ⊂ 3Q0 . X D m  L∞ (3.17) T z;t  L∞ ≤ Ct −¼ |m|≤¼

for some constant independent of . For the proof of Theorems 3.5, 3.6, and 3.7 we will basically show that T maps atoms to molecules. We start by making some general observations. Let aQ be an atom in general position associated with a cube Q with lower left corner xQ ; it follows from the definitions that aQ = |Q|−1=2 a.−xQ =`Q ;1=`Q / ⇔ a = |Q|1=2 aQQ

x ;`Q

;

(3.18)

where a is an atom corresponding to Q0 , and also that Ð.−xQ =`Q ;1=`Q /  = |Q|1=2 T aQ = |Q|−1=2 T xQ ;`Q a

Z

RxQ ;`Q 3Q0



 x − xQ ; y a.y/ dy: (3.19) `Q

Therefore, if we replace T z;t with T xQ ;`Q in Lemma 3.8 (or Lemma 3.9), then t −¼ in (3.14) (or (3.17)) becomes |Q|−¼=d .

Proof of Theorem 3.5. It suffices to show that for each atom aQ , of F˙ pÞ+¼;q , C|Q|¼=d T aQ

Þ;q is a molecule for F˙ p , with the constant independent of aQ . In other words, assuming that

i. supp aQ ⊂ 3Q; Z  ii. x aQ .x/ dx = 0

if

iii. |D  aQ | ≤ |Q|−1=2−| |=d ;

| | ≤ max{[J − d − ¼ − Þ]; [J − d]}; | | ≤ [Þ + ¼]+ + 1;

we will prove that Ð−.M−Þ/  iv. |T aQ .x/| ≤ C|Q|−1=2−¼=d 1 + `−1 |x − xQ | ; Q Z v. x  T aQ .x/ dx = 0 if | | ≤ [J − d − Þ];

M > J;

where J = d= min{1; p; q}. We start with statement v. Since, by assumption, T ∗ x  ∈ 5≤max{[J −d−¼−Þ];[J −d]} and aQ has at least max{[J − d − ¼ − Þ]; [J − d]} zero moments, then for every | | ≤ [J − d − Þ] we have  Þ  Þ T aQ ; x  = T ∗ x  ; aQ = 0: þ x−x þ √ For statement iv we first assume that |z| = þ `Q Q þ ≤ 6 d. From (3.19) and Lemma 3.9 we have |T aQ .x/| ≤ C|Q|−1=2−¼=d

X |m|≤¼

√ When |z| ≥ 6 d we consider two different cases.

D m a L∞ :

274

G. C. Kyriazis

Case 1: d + ¼ > J − Þ. Then |z − y| ≈ |z|, which implies that Z

þ þ þT aQ .x/þ ≤ C|Q|1=2

|z`Q − y`Q |−d−¼ |a.y/| dy

(3.20)

3Q0 −1=2−¼=d

≤ C|Q|

|z|

−d−¼

−1=2−¼=d

≤ C|Q|

.1 +

`−1 Q |x

− xQ |/

−d−¼

:

Case 2: d + ¼ ≤ J − Þ. Using that T ∗ ∈ KO¼ .L/ (L = [J − Þ − d − ¼] + 1) and the moments condition for a we get Z

þ |T aQ .x/| = |Q|1=2 þ

ð

R.z`Q + xQ ; y`Q + xQ /

3Q0

−

X |þ|
þ

D2 R.z`Q + xQ ; xQ /

þ .y`Q /þ Ł a.y/ dy þ þ!

þZ Z 1 X þ þ þ .y`Q /þ L−1 þ = |Q| þ s D2 R.z`Q + xQ ; xQ + s.y`Q //a.y/ ds dy þþ þ! 3Q0 0 |þ|=L Z Z 1 X þ þ þ þD R.z`Q + xQ ; xQ + s.y`Q //þ|y`Q |L ds dy ≤ C|Q|1=2 2 1=2 þ

≤ C|Q|1=2

Z

3Q0

0 |þ|=L 1

Z

3Q0

|z`Q − s.y`Q /|−d−¼−L |y`Q |L ds dy:

0

Since |z`Q − sy`Q | ≈ |z`Q |, for every 0 ≤ s ≤ 1, it is easily seen that |T aQ .x/| ≤ C|Q|−1=2−¼=d |z|−d−¼−L −.M−Þ/ ≤ C|Q|−1=2−¼=d .1 + `−1 ; Q |x − xQ |/

with M > J . From (3.20) and (3.21) the theorem follows.

(3.21)

2

Proof of Theorem 3.6. We will consider only ¼ − 1 < ¹ < ¼ since the other choices of Þ+¹;q , then there exists ¹ follow almost verbatim. We will prove that if aQ is a smooth atom for F˙ p Þ;q ¹=d a constant C, independent of aQ , such that C|Q| T aQ is a smooth .Ž0 ; M/ molecule for F˙ p for ∗ some M > d, where Ž0 is any real number satisfying 0 < Ž0 < min{Ž; 1 − ¹ } if Þ ∈ N and Þ ∗ < Ž0 < 1 if Þ ∈ R \ N. Being more specific, assuming that i. supp aQ ⊂ 3Q; Z ii. aQ .x/ dx = 0; iii. |D  aQ | ≤ |Q|−1=2−| |=d ;

| | ≤ [Þ + ¹]+ + 1;

we will prove that Z iv.

T aQ .x/ dx = 0

if

Þ = 0;

Ð−M  ; v. |D  T aQ .x/| ≤ C|Q|−1=2−¹=d−| |=d 1 + `−1 Q |x − xQ | and for | | = [Þ] and |x − x  |=`Q < 1

| | ≤ [Þ],

275

Distribution Spaces

vi. |D  T aQ .x/ − D  T aQ .x  /| ≤ C|Q|−1=2−¹=d−| |=d−Ž0 =d |x − x  |Ž0

sup

Ð−M



|w|≤|x−x  |

1 + `−1 Q |w − .x − xQ /|

:

The moments condition iv is needed only when Þ = 0. In that case we use that T ∗ 1 = const and derive Þ  Þ  T a; 1 = T ∗ 1; a = 0: As before, for z :=

x−xQ `Q

and | | ≤ [Þ] we have

D  T aQ .x/ = |Q|1=2

Z



3Q0

D1 R.z`Q + xQ ; y`Q + xQ /a.y/ dy:

√ To prove condition v we first consider |z| ≥ 6 d. In that case, |z − y| ≈ |z| and for every | | ≤ [Þ] Z þ þ  þD T aQ .x/þ ≤ C|Q|1=2 |z`Q − y`Q |−d−¹−| | |a.y/| dy 3Q0 −1=2−¹=d−| |=d

≤ C|Q|

|z|−d−¹−| |

(3.22)

−d−¹−| | : ≤ C|Q|−1=2−¹=d−| |=d .1 + `−1 Q |x − xQ |/

√ For |z| ≤ 6 d, applying Lemma 3.8 to the operator D  T ∈ KO ¹+| | ; | | ≤ [Þ], we derive þ Ð þ |D  T aQ .x/| = |Q|−1=2 þ .D  T /xQ ;`Q a .z/þ X ≤ C|Q|−1=2−¹=d−| |=d D m a L∞ (3.23) |m|≤[¹]+| |+1

−1=2−¹=d−| |=d

≤ C|Q|

:

We note, however, that to use Lemma 3.8 we need the assumption T y m = 0, |m| ≤ ¼ + [Þ] − 1. Finally we will prove condition vi with | | = [Þ] and |x − x  |=`Q < 1. For this we will make use of the Lipschitz condition of the kernel R. First we assume that Þ ∈ R\N. Then, similarly to condition v (using T y m = 0, |m| ≤ ¼+[Þ]), one can prove that Ð−d−¹−| |  |D  T aQ .x/| ≤ C|Q|−1=2−¹=d−| |=d 1 + `−1 ; | | ≤ [Þ] + 1: (3.24) Q |x − xQ | Therefore for | | = [Þ] þ þ  þD T aQ .x/ − D  T aQ .x  /þ ≤ |x − x  |

X þ þ þD þ T aQ .w/þ |þ|=[Þ]+1

where w lies in the line segment joining x and x  . From (3.24) with | | = [Þ] + 1 it follows that þ þ  þD T aQ .x/ − D  T aQ .x  /þ −d−¹−[Þ]−1 ≤ C|Q|−1=2−¹=d−[Þ]=d−1=d |x − x  |.1 + `−1 Q |w − xQ |/ Ð−d−¹−[Þ]−1  ≤ C.|x − x  |=`Q /Ž0 |Q|−1=2−¹=d−[Þ]=d sup 1 + `−1 Q |w − .x − xQ /| |w|≤|x−x  |

for any Þ ∗ < Ž0 < 1, because |z − z  | < 1.

276

G. C. Kyriazis x  −x

Next we will establish condition vi for integer values of Þ. Let z  := `Q Q such that |z − z  | = |x − x  |=`Q ≤ 1. First we assume that |x − x  | ≥ 1. Let Ž0 be any real number with 0 < Ž0 < 1 − ¹ ∗ . Applying condition v to ¹ + Ž0 (¹ + Ž0 < ¹ + .1 − ¹∗/ = ¼) for every | | = Þ we have |D  T aQ .x/ − D  T aQ .x  /| ≤ |D  T aQ .x/| + |D  T aQ .x  /| ≤ C|Q|−1=2−¹=d−| |=d−Ž0 =d ð Ł −d−¹−Ž0 −| |  −d−¹−Ž0 −| | + .1 + `−1 × .1 + `−1 Q |x − xQ |/ Q |x − xQ |/ ≤ C.|x − x  |=`Q /Ž0 |Q|−1=2−¹=d−| |=d Ð−d−¹−| |−Ž0  : × sup 1 + `−1 Q |w − .x − xQ /| |w|≤|x−x  |

At last we prove condition vi for Þ ∈ N and |x − x  |; |z − z  | < 1. We will further consider three different cases. √ Case 1: |z|; |z  | ≥ 6 d. From the smoothness of R and the fact that |z − y| ≈ |z|, we derive |D  T aQ .x/ − D  T aQ .x  /| Z þ þ 1=2 þ = |Q| [D  R.z`Q + xQ ; y`Q + xQ / − D  R.z  `Q + xQ ; y`Q + xQ /]a.y/ dy þ 3Q0

≤ C|Q|1=2

Z

|x − x  |Ž |z`Q − y`Q |−d−¹−| |−Ž dy

(3.25)

3Q0

 ÐŽ ≤ C |x − x  |=`Q |Q|−1=2−¹=d−| |=d |z|−d−¹−| |−Ž ≤ C|x − x  |Ž0 |Q|−1=2−¹=d−| |=d−Ž0 =d ×

−d−¹−| |−Ž sup .1 + `−1 Q |w − .x − xQ /|/

|w|≤|x−x  |

 for any 0 < Ž0 ≤ Ž < √ 1 because |z − z√ | ≤ 1. √ √  Case 2: |z| ≥ 6 d and |z | ≤ 6 d or |z| ≤ 6 d and |z  | ≥ 6 d. Here the result follows as in (3.25) since |z − z  |√< 1 implies√that in both cases |z − y| ≈ |z| and |z  − y| ≈ |z  |. Case 3: |z| ≤ 6 d; |z  | ≤ 6 d. For any 0 < Ž0 < min{Ž; 1 − ¹ ∗ } we have

|D  T aQ .x/ − D  T aQ .x  /| Z þ þ ð Ł = |Q|1=2 þ D  R.z`Q + xQ ; y`Q + xQ / − D  R.z  `Q + xQ ; y`Q + xQ / a.y/ dy þ Z þ ð Ł = |Q|1=2 þ D  R.z`Q + xQ ; y`Q + xQ / − D  R.z  `Q + xQ ; y`Q + xQ / ð × a.y/ − ≤ C|Q|

1=2

Z

X

Ł þ D m a.z/ .y − z/m dy þ m! |m|≤¼+| |−1

|z−y|≥1

+ C|Q|

1=2

|x − x  |Ž |z`Q − y`Q |−d−¹−| |−Ž0 |y − z|¼+| |−1 dy

Z

|z−y|<1

|x − x  |Ž |z`Q − y`Q |−d−¹−| |−Ž0 |y − z|¼+| | dy

≤ C|x − x  |Ž0 |Q|−1=2−¹=d−| |=d−Ž0 =d :

2

277

Distribution Spaces

Proof of Theorem 3.7. Let ¹ be fixed with ¹0 ≤ ¹ < ¼. Similarly to Theorem 3.6 Þ+¹;q we will show that for any atom aQ of F˙ p , |Q|¹=d T aQ is a .Ž0 ; M/ molecule with M > J ∗ (J = 1= min{1; p; q}) and Þ < Ž0 < 1 if Þ ∈ R \ N or 0 < Ž0 < min{Ž; 1 − ¹∗} if Þ ∈ N. In other Þ+¹;q atom aQ words, for every F˙ p Z

| | ≤ [J − d − Þ]; Ð−M  ; ii. |D  T aQ .x/| ≤ C|Q|−1=2−¹=d−| |=d 1 + `−1 Q |x − xQ | i.

x  T aQ .x/ dx = 0;

for

| | ≤ [Þ],

and for | | = [Þ] and |x − x  |=`Q < 1 iii. |D  T aQ .x/ − D  T aQ .x  /| ≤ C|Q|−1=2−¹=d−| |=d−Ž0 =d |x − x  |Ž0

sup

Ð−M



|w|≤|x−x  |

1 + `−1 Q |w − .x − xQ /|

:

For the moments condition i as in Theorem 3.5 it suffices to assume that for every | | ≤ [J − d − Þ], T ∗ x  ∈ 5≤[J −d] . For the rest of the proof we will concern ourselves only with p; q’s such that J ≥ d + ¹. Otherwise the proof is identical to that of Theorem 3.6, which gives conditions ii and iii with M ≥ d + ¹ > J. Therefore, without loss √ of generality we assume that J ≥ d + ¹. As before, to prove condition ii we first consider |z| ≤ 6 d. Lemma 3.8 shows that |D  T aQ .x/| ≤ C|Q|−1=2−¹=d−| |=d ; | | ≤ [Þ]: (3.26) √ Next we assume that |z| ≥ 6 d (which implies that |z − y| ≈ |z|). If d + ¹ + | | > J , then the result follows as in Theorem 3.6. For d + ¹ + | | ≤ J though, we need some extra decay that will be provided by the smoothness assumptions of the kernel R ð with respect to Łits second variable (assumption (5) of the theorem). For every such  we let L = J − ¹ − d − | | + 1; then Z þ þ þ  ð  þD T aQ .x/þ = |Q|1=2 þ D1 R.z`Q + xQ ; y`Q + xQ / 3Q0

−

X |þ|
þ



D2 D1 R.z`Q + xQ ; xQ /

þ .y`Q /þ Ł a.y/ dy þ þ!

þZ Z 1 X þ þ þ .y`Q /þ L−1 þ  1=2 þ = |Q| þ s D2 D1 R.z`Q + xQ ; xQ + s.y`Q //a.y/ ds dy þþ þ! 3Q0 0 |þ|=L Z Z 1 X þ þ þ  þD D R.z`Q + xQ ; xQ + s.y`Q //þ|y`Q |L ds dy ≤ C|Q|1=2 2 1 3Q0

≤ C|Q|1=2

Z

3Q0

0 |þ|=L 1

Z

|z`Q − sy`Q |−d−¹−| |−L |y`Q |L ds dy:

0

Since for every 0 ≤ s ≤ 1, |z − sy| ≈ y, it is easily seen from the above inequality that þ  þ þD T aQ .x/þ ≤ C|Q|−1=2−¹=d−| |=d |z|−d−¹−| |−L −M ; ≤ C|Q|−1=2−¹=d−| |=d .1 + `−1 Q |x − xQ |/ Ł where M = d + ¹ + | | + [J − ¹ − d − | | + 1 > J .

278

G. C. Kyriazis

√ Finally we prove condition iii. We note that the case where |z|; |z  | ≤ 6 d can be proved √ verbatim, as in Theorem 3.6. Some extra care should be taken though when max{|z|; |z  |} ≥ 6 d and d + ¹ + [Þ] ≤ J (for d + ¹ + [Þ] > J , of course, D  R already has sufficient decay). Without √   such that |z|; |z | ≥ 6 d. loss of generality we will consider only values of z; z ð Ł Let ½ = J − d − ¹ − [Þ] . Employing the extra smoothness of the kernel with respect to the first variable (assumption (6) of the theorem) and the Mean Value Theorem, we have for | | = [Þ] |D  T aQ .x/ − D  T aQ .x  /| þZ ² ½ X þ  þ .y`Q /þ  1=2 þ = |Q| þ D1 R.z`Q + xQ ; y`Q + xQ / − D2 D1 R.z`Q + xQ ; xQ / þ! 3Q0 |þ|<½ þ þ Ł¦ X þ  þ ð  .y`Q / a.y/ dy þþ D2 D1 R.z  `Q + xQ ; xQ / − D1 R.z  `Q + xQ ; y`Q + xQ / − þ! |þ|<½ þZ Z 1 X þ .y`Q /þ ½−1 ð þ  = |Q|1=2 þþ s D2 D1 R.z`Q + xQ ; sy`Q + xQ / þ! 3Q0 0 |þ|=½ þ þ Ł þ   − D2 D1 R.z `Q + xQ ; sy`Q + xQ a.y/ ds dy þþ Z Z 1 1=2 |y`Q |½ |z`Q − z  `Q | ≤ |Q| ×

X

3Q0

X

0 þ

|þ|=½ | |=[Þ]+1



|D2 D1 R.wy ; sy`Q + xQ /|a.y/ ds dy;

where for each | | = [Þ] + 1, wy lies in the line segment joining z`Q + xQ with z  `Q + xQ . It follows that X X Z Z 1 |D  T aQ .x/ − D  T aQ .x  /| ≤ C|Q|1=2 |y`Q |½ |z`Q − z  `Q | 0 3Q 0 (3.27) |þ|=½ | |=[Þ]+1 × |wy − sy`Q − xQ |−d−¹−| |−½ ds dy: Since |z − z  | < 1, it is easily seen that for every 0 ≤ s ≤ 1, |wþ;y − sy`Q − xQ | ≈ |z`Q | and (3.27) shows that |D  T aQ .x/ − D  T aQ .x  /| ≤ C|Q|

1=2

Z

|y`Q |½ |z`Q − z  `Q ||z`Q |−d−¹−[Þ]−½−1 dy 3Q0

≤ C|Q|−1=2−¹=d−[Þ]=d .|x − x  |=`Q /|z|−d−¹−[Þ]−½−1 ≤ C|Q|−1=2−¹=d−[Þ]=d .|x − x  |=`Q /Ž0

−M1 sup .1 + `−1 Q |w − .x − xQ /|/

|w|≤|x−x  |

for any 0 < Ž0 < 1, where M1 = d + ¹ + [Þ] + ½ + 1 > J .

2

Using real interpolation, we can also extend Theorem 3.6 to Besov spaces.

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3.10 Corollary Let 0 < q  ≤ ∞. Under the assumptions of Theorem 3.6 the operator T extends to a bounded Þ+¹;q  Þ;q operator T : B˙ p → F˙ p , 0 < ¹ < ¼, that is, there exists a constant such that for every  Þ+¹;q f ∈ B˙ p Tf F˙pÞ;q ≤ C f B˙ Þ+¹;q  : p

s;q 

We note only that B˙ p are interpolation spaces between F˙ p0 s1 ; 1 ≤ q0 ; q1 ≤ ∞; i.e., for every 0 <  < 1  Ð  B˙ ps;q = F˙ ps0 ;q0 ; F˙ ps1 ;q1 ;q  ;

Proof.

s ;q0

where s = .1 −  /s0 +  s1 (see [Tr] for details).

and F˙ p1 1 , s0 < s < s ;q

2

4. Approximation orders of kernel operators Returning to our main theme, in this section we study the efficiency of kernel operators for approximation. In particular we investigate conditions on functions ;  on Rd so that the operator P with kernel X  .y − j /.x − j / K.x; y/ := j ∈Zd

provides positive orders of approximation in various pairs of spaces (cf. (1.2)). By Theorem 2.1 we are led to the study of the boundedness of the operator T with kernel R.x; y/ := K.x; y/ − 2d K.2x; 2y/:

(4.1)

The major assumption of all theorems in the previous section is that T should annihilate polynomials of certain degree. It turns out that this fact is equivalent with the Strang–Fix conditions that have been around in approximation theory, in some form or another, since the early 1940s (Schoenberg). To this end we recall a well-known lemma from [JL]. We first need to introduce the concept of normal functions. A function f on Rd is called normal if f is locally integrable and for any x ∈ Rd , Z 1 f .x/ = lim f .y/ dy ž→0 |Bž .x/| B .x/ ž (it is trivially seen that any continuous function is normal).

4.1

Lemma Let ¼ ∈ N and let  be a normal function such that |.x/| ≤ C.1 + |x|/−M ;

M > d + ¼:

(4.2)

Assume also that  satisfies the Strang–Fix conditions of order ¼, .1:7/, D  b .0/ = 0, 0 < | | < ¼, and b .0/ = 1. Then for any |þ| < ¼ X j þ .x − j / = x þ : j ∈Zd

280

G. C. Kyriazis

Proof.

See [JL].

2

On the basis of the above lemma we will prove the following theorem.

4.2.

Theorem Let  be a normal function, and let  be such that |.x/|; | .x/| ≤ C.1 + |x|/−M ;

M > d + ¼;

¼ ∈ N:

(4.3)

If  satisfies the Strang–Fix conditions of order ¼ and, in addition, b  .0/ = b .0/ = 1 and D Þb  .0/ = Þb D .0/ = 0 for every 1 ≤ |Þ| < ¼, then T y þ = 0;

Proof.

|þ| < ¼:

(4.4)

Since T := P − P1=2 , it suffices to show that for every |þ| < ¼  þÐ P y .x/ = x þ :

Indeed, in this case a simple change of variables shows that Z Z  Ð þ d þ −|þ| P1=2 y .x/ = 2 K.2x; 2y/y dy = 2 Rd

(4.5)

K.2x; y/y þ dy Rd

  Ð Ð = 2−|þ| P y þ .2x/ = x þ = P y þ .x/; which implies that T y þ = 0: To prove (4.5) we let |þ| < ¼. Then XZ  þÐ P y .x/ =  .y − j /y þ dy .x − j / j ∈Zd

Rd

X þ  X Z =  .y/y Þ j þ−Þ dy .x − j / d Þ Þ≤þ j ∈Zd R X þ  Z  X þ−Þ Ð =  .y/y Þ dy j .x − j / Þ Rd Þ≤þ j ∈Zd  X þ Ð  X þ−Þ = j .x − j /  .0/ .−i/−|Þ| D Þb Þ Þ≤þ j ∈Zd X þ þ = j .x − j / = x ; j ∈Zd

where the last equality holds by Lemma 4.1.

2

Before we continue with the boundedness of T , we also need to establish that for every Þ+¼;q Þ+¼;q (or f ∈ L1 ; L∞ ; B˙ p ) f ∈ F˙ p X .P − I /f = T2−j f (4.6) j ≥0

in the distributional sense (recall that this was one of our main assumptions in Theorem 2.1). For (4.6) we employ the following well-known theorem.

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Distribution Spaces

4.3.

Theorem Let T ∈ KO¼+ž ; ž > 0; and assume that

R

Rd

K.x; y/ dy = 1. Then if 1 ≤ p < ∞

P2−j f − f Lp −→ 0;

j → ∞:

(4.7)

For p = ∞, (4.7) still holds if in addition K.x; y/ is a continuous function of x for each y and f is uniformly continuous.

Proof.

See [M p. 42].

2

∩ L2 . We note that for ;  as Returning now to (4.6) we first restrict ourselves to f ∈ F˙ p in Theorem 4.2 the operator P satisfies the conditions of Theorem 4.3 (for L∞ we additionally need Þ+¼;q ∩ L2 (or f ∈ Lp ; 1 ≤ p ≤ ∞),  to be continuous). It readily follows that for every f ∈ F˙ p X T2−j f: Pf − f = .Pf − P1=2 f / + .P1=2 f − P1=4 f / + · · · = Þ+¼;q

j ≥0 Þ+¼;q Þ;q Assuming in addition that T : F˙ p → F˙ p is a continuous operator, then (see Theorem 2.1) X T2−j f F˙pÞ;q ≤ C f F˙pÞ+¼;q ; j ≥0

i.e.,

P

j ∈Zd

Þ;q Þ+¼;q T2−j f converges in the sense of F˙ p . It follows that for every f ∈ F˙ p ∩ L2

.P − I /f F˙pÞ;q ≤ C f F˙pÞ+¼;q : Þ+¼;q Þ+¼;q Finally by extending .P − I / continuously from F˙ p ∩ L2 to F˙ p we derive (4.6) for general Þ+¼;q Þ;q ˙ ˙ and convergence in the sense of F p (a similar analysis holds for Besov spaces as well). f ∈ Fp Thus, we have shown that (4.6) holds in all the cases in which we are interested.

4.4.

Theorem

Let 0 ≤ ¹ < ¼ < ∞ with ¹; ¼ ∈ N . Also let 1 ≤ p ≤ ∞ and 0 < ž < 1. Assume that  is a normal function that satisfies 1. |.x/| ≤ C.1 + |x|/−d−¼−ž ; | | ≤ ¹; 2. |D  .x/| ≤ C.1 + |x|/−d−.¼−¹/−| |−ž ; .0/ = 0 for every 3.  satisfies the Strang–Fix conditions of order ¼, b .0/ = 1 and D þ b 1 ≤ |þ| < ¼; 4. for p = ∞,  is continuous. ¼

Then there exists a constant such that for every f ∈ Wp |f − Ph f |Wp¹ ≤ Ch¼−¹ |f |Wp¼ ;

h > 0;

where Ph f .x/ :=

X

h−d

j ∈Zd

and  is any function on Rd satisfying

Z Rd

f .y/ .y= h − j / dy .x= h − j /;

(4.8)

282

G. C. Kyriazis

5. | .x/| ≤ C.1 + |x|/−d−¼−ž ; 6. b  .0/ = 1 and D þb  .0/ = 0 for every 1 ≤ |þ| < ¼.

Proof. Following our discussion above it suffices to prove that R.x; y/ = K.x; y/ − 2d K.2x; 2y/ gives rise to an operator T := P − P1=2 that satisfies the assumptions of Theorem 3.2. From Theorem 4.2 it is easily seen that conditions 3, 6 along with the decay of  and  guarantee that T y  = 0 for | | < ¼. To show that T ∈ KO ¼+ž ∩ KO ¼−¹+ž .¹/ it is sufficient to prove that P ∈ KO ¼+ž ∩ KO ¼−¹+ž .¹/, i.e., |K.x; y/| ≤ C.1 + |x − y|/−d−¼−ž ;

(4.9)

|D1 K.x; y/| ≤ C.1 + |x − y|/−d−.¼−¹/−| |−ž :

(4.10)

and that for every | | ≤ ¹ 

For (4.9) we note that by 1 and 5 X |K.x; y/| ≤ |.x − j /| | .y − j /| j ∈Zd

X

≤C

.1 + |y − j |/−d−¼−ž .1 + |x − j |/−d−¼−ž

j ∈Zd

≤ C.1 + |x − y|/−d−¼−ž : Similarly (4.10) follows from the smoothness of  (assumption 2) and the decay of  as X  |D1 K.x; y/| ≤ | .y − j /| |D  .x − j /| j ∈Zd

≤C

X

.1 + |y − j |/−d−¼−ž .1 + |x − j |/−d−.¼−¹/−| |−ž

(4.11)

j ∈Zd

≤ C.1 + |x − y|/−d−.¼−¹/−| |−ž : Finally, when p = ∞ the continuity of  guarantees that K.x; y/ is a continuous function of x for every y ∈ Rd , a fact needed for (4.6). 2 For ¹ = 0 the above theorem tells us that the principal shift-invariant space S./ provides ¼ order of approximation ¼ for the pair .Lp ; Wp /, 1 ≤ p ≤ ∞, i.e., ELp .f; S h / ≤ Ch¼ |f |Wp¼ :

(4.12)

This result is already known and was proved by Light and Cheney [LC] .p = ∞/ and Jia and Lei [JL] .1 ≤ p ≤ ∞/. Nevertheless, the approximation schemes that are employed in the above references appear to be more complex than our kernel operator and the results in [LC, JL] are not extended to pairs of Sobolev spaces.

4.5.

Theorem

Let Þ < 0, 0 < ž, 0 < p < ∞, min{1; p} ≤ q ≤ ∞, and ¼ ∈ N+ . Assume that ;  are functions on Rd with  normal and such that 1. |.x/|; | .x/| ≤ C.1 + |x|/−d−M−ž ; M = max{¼; [J − d − Þ] + 1}; 2.  satisfies the Strang–Fix conditions of order ¼;

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Distribution Spaces

3.  satisfies the Strang–Fix conditions of order [J − d − Þ] + 1;  .0/ = D þ b .0/ = 0 for every 1 ≤ |þ| < max{¼; [J −d −Þ]+1}. 4. b  .0/ = b .0/ = 1, and D þb For J − a ≥ d + ¼ we also assume that 5. |D   .x/| ≤ C.1 + |x|/−d−¼−| | ;

| | ≤ L, with L = [J − Þ − d − ¼] + 1.

Then the operator P corresponding to the kernel X K.x; y/ :=  .y − j /.x − j /

(4.13)

j ∈Zd

 Þ;q Þ+¼;q Ð provides order of approximation ¼ for the pair F˙ p ; F˙ p , that is, there exists a constant such Þ+¼;q that for every f ∈ F˙ p   .I − Ph /f  ˙ Þ;q ≤ Ch¼ f ˙ Þ+¼;q : Fp F p

Proof. It suffices to prove that the operator T with kernel R.x; y/ = K.x; y/−2d K.2x; 2y/ satisfies the assumptions of Theorem 3.5. From Theorem 4.2 we see that conditions 1, 2, 3, and 4 guarantee that T y  = 0 for | | < ¼ and that T ∗ x  = 0 for | | ≤ [J − d − Þ]. To show that T ∈ KO ¼+ž and T ∗ ∈ KO[J −d−a]+ž it is sufficient to prove that P satisfies the same assumptions, i.e., |K.x; y/| ≤ C.1 + |x − y|/−d−3−ž ;

3 = max{¼; [J − d − Þ]};

(4.14)

For this, we note that by 1 |K.x; y/| ≤

X

|.x − j /| | .y − j /|

j ∈Zd

≤C

X

.1 + |y − j |/−d−3−ž .1 + |x − j |/−d−3−ž

(4.15)

j ∈Zd

≤ C.1 + |x − y|/−d−3−ž : Finally for J − Þ ≥ d + ¼ we have to prove that T ∗ ∈ KO¼ .L/. Similarly to (4.14) from 1 and 5 we have that for every | | ≤ L X  |D2 K.x; y/| ≤ |.x − j /| |D   .y − j /| j ∈Zd

≤C

X

.1 + |x − j |/−d−¼−L .1 + |y − j |/−d−¼−| |

j ∈Zd

≤ C.1 + |x − y|/−d−¼−| | :

4.6.

2

Theorem

Let 1 ≤ min{q; p}, p = ∞, Þ ∈ N, ¼ ∈ N+ , and 0 < Ž < 1. Assume that ,  are defined on Rd with  normal and such that 1. |.x/|; | .x/| ≤ C.1 + |x|/−d−¼−Þ−Ž ; | | ≤ Þ; 2. |D  .x/| ≤ C.1 + |x|/−d−¼−| | ;

284

G. C. Kyriazis

3. for every | | = Þ and |x − x  | < 1 |D  .x/ − D  .x  /| ≤ |x − x  |Ž 4. 5. 6.

sup .1 + |w − x|/−d−¼−Þ−Ž ;

|w|≤|x−x  |

 satisfies the Strang–Fix conditions of order ¼ + Þ; b  .0/ = b .0/ = 1, and D þb  .0/ = D þ b .0/ = 0 for every 1 ≤ |þ| < ¼ + Þ; for Þ = 0,  satisfies the Strang–Fix conditions of order 1.

Then for every 0 < ¹ < ¼ the operator P corresponding to the kernel X K.x; y/ :=  .y − j /.x − j /

(4.16)

j ∈Zd

 Þ;q Þ+¹;q Ð provides order of approximation ¹ for the pair F˙ p ; F˙ p .

Proof. We will prove that T satisfies the conditions of Theorem 3.6. Assumptions 1, 4, 5, and 6 show that T y  = 0, | | ≤ ¼ + Þ − 1, and T ∗ 1 = 0 (when Þ = 0). To complete the proof we need to prove that T ∈ KO¼+Þ ∩ KO¼ .Þ + Ž/. It suffices to show that P ∈ KO¼+Þ ∩ KO¼ .Þ + Ž/, i.e., |K.x; y/| ≤ .1 + |x − y|/−d−¼−Þ ; |D1 K.x; y/| ≤ C.1 + |x − y|/−d−¼−| | ; 

(4.17) | | ≤ Þ;

(4.18)

and for | | = Þ and |x − x  | < 1 |D1 K.x; y/ − D1 K.x  ; y/| ≤ C|x − x  |Ž .1 + |x − y|/−d−¼−a−Ž : 



(4.19)

Equations (4.17) and (4.18) follow easily, while for (4.19) we have |D1 K.x; y/ − D1 K.x  ; y/| X ≤ C | .y − j /| |D  .x − j / − D  .x  − j /| 



j ∈Zd

≤ C|x − x  |Ž

X

.1 + |y − j |/−d−¼−Þ−Ž

j ∈Zd

≤ C|x − x  |Ž .1 + |x − y|/−d−¼−Þ−Ž :

sup

.1 + |x − j − z|/−d−¼−Þ−Ž

|z|≤|x−x  |<1

2

It is now routine to extend Theorem 4.6, on the basis of Corollary 3.10, to approximation  Þ;q Þ+¹;q  Ð orders for the pairs F˙ p ; B˙ p . 4.7 Corollary Let 0 < q  ≤ ∞. Under the assumptions of Theorem 4.6 we have that the operator P provides  Þ;q Þ+¹;q  Ð order of approximation ¹ for the pair F˙ p ; B˙ p for every 0 < ¹ < ¼. Finally from Theorem 3.7 we derive the following result.

4.8.

Theorem

Let 0 < p < 1, p ≤ q ≤ ∞, Þ ∈ N, and 0 < Ž < 1. Also let 0 ≤ ¹0 < ¼ and assume that ;  are functions on Rd , with  normal that satisfy

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Distribution Spaces

1. |.x/|; | .x/| ≤ C.1 + |x|/−d−3−Ž ; 3 = max{¼ + Þ; [d=p − d − Þ] + 1}; | | ≤ Þ; 2. |D  .x/| ≤ C.1 + |x|/−d−¼−| | ;     Ž 3. |D .x/ − D .x /| ≤ |x − x | sup|w|≤|x−x  | .1 + |w − x|/−d−¼−Þ−Ž for every | | = Þ, and |x − x  | < 1; 4.  satisfies the Strang–Fix conditions of order ¼ + Þ; 5. if d=p − d − Þ ≥ 0,  satisfies the Strang–Fix conditions of order [d=p − d − Þ] + 1;  .0/ = D þ b .0/ = 0 for every 1 ≤ |þ| < max{¼ + Þ; [d=p − 6. b  .0/ = b .0/ = 1, and D þb d − Þ] + 1}. In addition, for d=p ≥ d + ¹0 we assume that for every | | ≤ Þ with d + ¹0 + | | ≤ d=p 7. |D  .x/| ≤ C.1 + |x|/−d−¼−| |−[d=p−d−¹0 −| |]−1 ; |D þ  .x/| ≤ C.1 + |x|/−d−¼−| |−|þ| ;

for every |þ| ≤ [d=p − d − ¹0 − | |] + 1:

Finally we assume that for d + ¹0 + Þ ≤ d=p and | | = Þ + 1 8. |D  .x/| ≤ C.1 + |x|/−d−¼−Þ−1−[d=p−d−¹0 −Þ] ; |D þ  .x/| ≤ C.1 + |x|/−d−¼−|þ|−Þ−1 ;

for every |þ| ≤ [d=p − d − ¹0 − Þ].

Then for every ¹0 ≤ ¹ < ¼ the operator P corresponding to the kernel X K.x; y/ :=  .y − j /.x − j / j ∈Zd

 Þ;q Þ+¹;q Ð provides order of approximation ¹ for the pair F˙ p ; F˙ p .

Sketch of the Proof. To avoid being repetitive we only mention that one has to follow the guidelines of the proof of Theorem 4.6 and show that the operator T := P − P1=2 satisfies the assumptions of Theorem 3.7, which can be done using elementary arguments. 2

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Received July 29, 1994 Purdue University, Department of Mathematics, West Lafayette, Indiana 47907 Current address: University of Cyprus, Dept. of Mathematics and Statistics, P. O. Box 537, Nicosia, Cyprus E-mail address: [email protected]