Boas-type formulas in Banach spaces with applications to analysis on manifolds
aa r X i v : . [ m a t h . F A ] A p r BOAS-TYPE FORMULAS IN BANACH SPACES WITHAPPLICATIONS TO ANALYSIS ON MANIFOLDS
Dedicated to 85th Birthday of my teacher Paul Butzer
Isaac Z. Pesenson Keywords:
Exponential and Bernstein vectors, Boas interpolation formula,sampling, compact homogeneous manifolds, Heisenberg group, Schr¨odinger repre-sentation.
Subject classifications: [2000] Primary: 47D03, 44A15; Secondary: 4705
Abstract.
The paper contains Boas-type formulas for trajectories of one-parameter groups of operators in Banach spaces. The results are illustratedusing one-parameter groups of operators which appear in representations ofLie groups. Preface
My theachers were Vladimir Abramovich Rokhlin (my Master Thesis Advisor)and Selim Grigorievich Krein (my PhD Thesis Advisor). I first met Paul Butzerwhen I was about 50 years old but I also consider him as my teacher since his workhad an enormous influence on my carrier and ultimately on my life.After I graduated from university it was almost impossible for me to go straight tograduate school because of the Soviet discrimination towards Jews. However, I wastrying to do some mathematics on my own. One day I came across a reference to thebook by P. Butzer and H. Berens ”Semi-Groups of operators and approximation”,Springer, 1967. Since I had some background in Lie groups and Lie semi-groupsand knew nothing about approximation theory the title sounded very intriguingto me. Unfortunately this excellent book had not been translated into Russian.Nevertheless I was lucky to get a microfilm of the book. Every day during a fewmonths I was visiting a local library which had a special device to read microfilms.By the time I finished reading the book I already knew what I was going to do: Idecided to develop a similar ”constructive theory of Interpolation Spaces” throughreplacing a single one-parameter semi-group of operators by a representation of ageneral Lie group in a Banach space.I have to say that the book by P. Butzer and H. Berens is an excellent in-troduction to a number of topics in classical harmonic analysis. In particular itcontains the first systematic treatment of the theory of Intermediate Spaces and avery detailed and application oriented treatment of the theory of Semi-Groups ofOperators. Both these subjects were considered as ”hot” topics at the end of 60s(see for example [13], [14]). In many ways this book is still up to date and I alwaysrecommend it to my younger colleagues. Department of Mathematics, Temple University, Philadelphia, PA 19122; [email protected]. The author was supported in part by the National Geospatial-IntelligenceAgency University Research Initiative (NURI), grant HM1582-08-1-0019.
Some time later I was influenced by the classical work of Paul with K. Scherer[5], [6] in which they greatly clarified the relationships between the Interpolationand Approximation spaces and by Paul’s pioneering paper with H. Berens and S.Pawelke [1] about approximation on spheres.Mathematics I learned from Paul Butzer helped me to become a graduate studentof Selim Krein another world level expert in Interpolation spaces and applicationsof semigroups to differential equations [13]-[15].Many years later when I came to US and became interested in sampling theoryI found out that Paul had already been working in this field for a number of yearsand I learned a lot from insightful and stimulating work written by P. Butzer, W.Splettst¨o¨ser and R. Stens [7].My interactions with Paul Butzer’s work shaped my entire mathematical life andthe list of some of my papers [18]-[31] is the best evidence of it.
This is what I mean when I say that Paul Butzer is my teacher.
In conclusion I would like to mention that it was my discussions with PaulButzer and Gerhard Schmeisser of their beautiful work with Rudolf Stens [9], thatstimulated my interest in the topic of the present paper. I am very grateful to themfor this. 2.
Introduction
Consider a trigonometric polynomial P ( t ) of one variable t of order n as a func-tion on a unit circle T . For its derivative P ′ ( t ) the so-called Bernstein inequalityholds true(2.1) (cid:13)(cid:13)(cid:13) P ′ (cid:13)(cid:13)(cid:13) L p ( T ) ≤ n k P k L p ( T ) , ≤ p ≤ ∞ . M. Riesz [33], [34] states that the Bernstein inequality is equivalent to what isknown today as the Riesz interpolation formula(2.2) P ′ ( t ) = 14 π n X k =1 ( − k +1 t k P ( t + t k ) , t ∈ S , t k = 2 k − n π. The next formula holds true for functions in the Bernstein space B pσ , ≤ p ≤ ∞ which is comprised of all entire functions of exponential type σ which belong to L p ( R ) on the real line.(2.3) f ′ ( t ) = σπ X k ∈ Z ( − k − ( k − / f (cid:16) t + πσ ( k − / (cid:17) , t ∈ R . This formula was obtained by R.P. Boas [2], [3] and is known as Boas or generalizedRiesz formula. Again, like in periodic case this formula is equivalent to the Bernsteininequality in L p ( R ) (cid:13)(cid:13)(cid:13) f ′ (cid:13)(cid:13)(cid:13) L p ( R ) ≤ σ k f k L p ( R ) . Recently, in the interesting papers [35] and [9] among other important resultsthe Boas-type formula (2.3) was generalized to higher order.
OAS-TYPE FORMULAS IN BANACH SPACES 3
In particular it was shown that for f ∈ B ∞ σ , σ > , the following formulas hold f (2 m − ( t ) = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k f (cid:18) t + πσ ( k −
12 ) (cid:19) , m ∈ N ,f (2 m ) ( t ) = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k f (cid:16) t + πσ k (cid:17) , m ∈ N , where A m,k = ( − k +1 sinc (2m − (cid:18) − k (cid:19) =(2.4) (2 m − π ( k − ) m m − X j =0 ( − j (2 j )! (cid:18) π ( k −
12 ) (cid:19) j , m ∈ N , for k ∈ Z and(2.5) B m,k = ( − k +1 sinc (2m) ( − k) = (2m)! π k − X j=0 ( − j ( π k) (2j + 1)! , m ∈ N , k ∈ Z \ , and(2.6) B m, = ( − m +1 π m m + 1 , m ∈ N . Let us remind that sinc(t) is defined as sin πtπt , if t = 0, and 1, if t = 0.To illustrate our results let us assume that we are given an operator D thatgenerates a strongly continuous group of isometries e tD in a Banach space E . Definition 2.1.
The subspace of exponential vectors E σ ( D ) , σ ≥ , is defined as aset of all vectors f in E which belong to D ∞ = T k ∈ N D k , where D k is the domainof D k , and for which there exists a constant C ( f ) > such that (2.7) k D k f k ≤ C ( f ) σ k , k ∈ N . Note, that every E σ ( D ) is clearly a linear subspace of E . What is really impor-tant is the fact that union of all E σ ( D ) is dense in E (Theorem 3.7). Remark 2.2.
It is worth to stress that if D generates a strongly continuous boundedsemigroup then the set S σ ≥ E σ ( D ) may not be dense in E .Indeed, (see [17] ) consider a strongly continuous bounded semigroup T ( t ) in L (0 , ∞ ) defined for every f ∈ L (0 , ∞ ) as T ( t ) f ( x ) = f ( x − t ) , if x ≥ t and T ( t ) f ( x ) = 0 , if ≤ x < t . If f ∈ E σ ( D ) then for any g ∈ L (0 , ∞ ) the function h T ( t ) f, g i is analytic in t (see below section 3). Thus if g has compact support then h T ( t ) f, g i is zero for all t which implies that f is zero. In other words in this caseevery space E σ ( D ) is trivial. One of our results is that a vector f belongs to E σ ( D ) if and only if the followingsampling-type formulas hold(2.8) e tD D m − f = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e ( t + πσ ( k − / ) D f, m ∈ N , (2.9) e tD D m f = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k e ( t + πkσ ) D f, m ∈ N , BOAS-TYPE FORMULAS IN BANACH SPACES
Which are equivalent to the following Boas-type formulas(2.10) D m − f = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e ( πσ ( k − / ) D f, m ∈ N , f ∈ E σ ( D ) , and(2.11) D m f = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k e πkσ D f, m ∈ N ∪ , f ∈ E σ ( D ) . The formulas (2.8) and (2.9) are a sampling-type formulas in the sense that theyprovide explicit expressions for a trajectory e tD D k f with f ∈ E σ ( D ) in terms of acountable number of equally spaced samples of trajectory of f .Note, that since e tD , t ∈ R , is a group of operators any trajectory e tD f, f ∈ E, is completely determined by any (single) sample e t D f, because for any t ∈ R e tD f = e ( t − t ) D (cid:0) e t D f (cid:1) . The formulas (2.8) and (2.9) have, however, a different nature: they represent atrajectory as a ”linear combination” of a countable number of samples.It seems to be very interesting that an operator and the group can be rathersophisticated (think, for example, about a Schr¨odinger operator D = − ∆ + V ( x )and the corresponding group e itD in L ( R d )). However, formulas (2.8)-(2.11) areuniversal in the sense that they contain the same coefficients and the same sets ofsampling points.We are making a list of some important properties of the Boas-type interpolationformulas (compare to [9]):(1) The formulas hold for vectors f in the set S σ ≥ E σ ( D ) which is dense in E (see Theorem 3.7).(2) The sample points πσ ( k − /
2) are uniformly spaced according to the Nyquistrate and are independent on f .(3) The coefficients do not depend on f .(4) The coefficients decay like O ( k − ) as k goes to infinity.(5) In formulas (2.10) and (2.11) one has unbounded operators (in general case)on the left-hand side and bounded operators on the right-hand side.(6) There is a number of interesting relations between Boas-type formulas, seebelow (3.9), (3.10).Our main objective is to obtain a set of new formulas for one-parameter groupswhich appear when one considers representations of Lie groups (see section 5). Note,that generalizations of (2.3) with applications to compact homogeneous manifoldswere initiated in [22].In our applications we deal with a set of non-commuting generators D , ..., D d .In subsection 5.1 these operators come from a representation of a compact Liegroup and we are able to show that ∪ σ ≥ ∪ ≤ j ≤ d E σ ( D j ) is dense in all appropriateLebesgue spaces. We cannot prove a similar fact in the next subsection 5.2 in whichwe consider a non-compact Heisenberg group. Moreover, in subsection 5.3 in whichthe Schr¨odinger representation is discussed we note that this property does not holdin general. OAS-TYPE FORMULAS IN BANACH SPACES 5 Boas-type formulas for exponential vectors
We assume that D is a generator of one-parameter group of isometries e tD in aBanach space E with the norm k · k . Definition 3.1.
The Bernstein subspace B σ ( D ) , σ ≥ , is defined as a set of allvectors f in E which belong to D ∞ = T k ∈ N D k , where D k is the domain of D k andfor which (3.1) k D k f k ≤ σ k k f k , k ∈ N . It is obvious that B σ ( D ) ⊂ E σ ( D ) , σ ≥ , However, it is not even clear that B σ ( D ) , σ ≥ , is a linear subspace. It follows from the following interesting fact. Theorem 3.2.
Let D be a generator of an one parameter group of operators e tD in a Banach space E and k e tD f k = k f k . Then for every σ ≥ B σ ( D ) = E σ ( D ) , σ ≥ , Proof. If f ∈ E σ ( D ) then for any complex number z we have (cid:13)(cid:13) e zD f (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X r =0 ( z r D r f ) /r ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ( f ) ∞ X r =0 | z | r σ r /r ! = C ( f ) e | z | σ . It implies that for any functional ψ ∗ ∈ E ∗ the scalar function (cid:10) e zD f, ψ ∗ (cid:11) is an entirefunction of exponential type σ which is bounded on the real axis by the constant k ψ ∗ kk f k . An application of the classical Bernstein inequality gives (cid:13)(cid:13)(cid:10) e tD D k f, ψ ∗ (cid:11)(cid:13)(cid:13) C ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ddt (cid:19) k (cid:10) e tD f, ψ ∗ (cid:11)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ( R ) ≤ σ k k ψ ∗ kk f k . From here for t = 0 we obtain (cid:12)(cid:12)(cid:10) D k f, ψ ∗ (cid:11)(cid:12)(cid:12) ≤ σ k k ψ ∗ kk f k . Choice of ψ ∗ ∈ E ∗ such that k ψ ∗ k = 1 and (cid:10) D k f, ψ ∗ (cid:11) = k D k f k gives the inequality k D k f k ≤ σ k k f k , k ∈ N , which implies Theorem. (cid:3) Remark 3.3.
We just mention that in the important case of a self-adjoint operator D in a Hilbert space E there is a way to describe Bernstein vectors in terms of aspectral Fourier transform or in terms of the spectral measure associated with D (see [16] , [22] - [31] for more details). Let’s introduce bounded operators(3.2) B (2 m − D ( σ ) f = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e πσ ( k − / D f, f ∈ E, σ > , m ∈ N , (3.3) B (2 m ) D ( σ ) f = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k e πkσ D f, f ∈ E, σ > , m ∈ N , where A m,k and B m,k are defined in (2)-(2.6). Both series converge in E due to thefollowing formulas (see [9]) BOAS-TYPE FORMULAS IN BANACH SPACES (3.4) (cid:16) σπ (cid:17) m − X k ∈ Z | A m,k | = σ m − , (cid:16) σπ (cid:17) m X k ∈ Z | B m,k | = σ m . Since k e tD f k = k f k it implies that(3.5) kB (2 m − D ( σ ) f k ≤ σ m − k f k , kB (2 m ) D ( σ ) f k ≤ σ m k f k , f ∈ E. Theorem 3.4. If D generates a one-parameter strongly continuous bounded groupof operators e tD in a Banach space E then the following conditions are equivalent: (1) f belongs to B σ ( D ) . (2) The abstract-valued function e tD f is an entire function of exponential type σ which is bounded on the real line. (3) The following Boas-type interpolation formulas hold true for r ∈ N (3.6) D r f = B ( r ) D ( σ ) f, f ∈ B σ ( D ) . Proof.
The proof of Theorem 3.2 shows that 1) → ψ ∗ ∈ E ∗ the function F ( t ) = (cid:10) e tD f, ψ ∗ (cid:11) is of exponential type σ and bounded on R . Thus by [9] we have F (2 m − ( t ) = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k F (cid:16) t + πσ ( k − / (cid:17) , m ∈ N ,F (2 m ) ( t ) = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k F (cid:18) t + πkσ (cid:19) , m ∈ N . Together with (cid:18) ddt (cid:19) k F ( t ) = (cid:10) D k e tD f, ψ ∗ (cid:11) , it shows (cid:10) e tD D m − f, ψ ∗ (cid:11) = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k D e ( t + πσ ( k − / ) D f, ψ ∗ E , m ∈ N , and also (cid:10) e tD D m f, ψ ∗ (cid:11) = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k D e ( t + πkσ ) D f, ψ ∗ E , m ∈ N . Since both series (3.2) and (3.3) converge in E and the last two equalities hold forany ψ ∗ ∈ E we obtain the next two formulas(3.7) e tD D m − f = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e ( t + πσ ( k − / ) D f, m ∈ N , (3.8) e tD D m f = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k e ( t + πkσ ) D f, m ∈ N . In turn when t = 0 these formulas become formulas (3.6).The fact that 3) →
1) easily follows from the formulas (3.6) and (3.5). Theoremis proved. (cid:3)
OAS-TYPE FORMULAS IN BANACH SPACES 7
Corollary 3.1.
Every B σ ( D ) is a closed linear subspace of E . Corollary 3.2. If f belongs to B σ ( D ) then for any σ ≥ σ, σ ≥ σ one has (3.9) B ( r ) D ( σ ) f = B ( r ) D ( σ ) f, r ∈ N . Let us introduce the notation B D ( σ ) = B (1) D ( σ ) . One has the following ”power” formula which easily follows from the fact thatoperators B D ( σ ) and D commute on any B σ ( D ). Corollary 3.3.
For any r ∈ N and any f ∈ B σ ( D )(3.10) D r f = B ( r ) D ( σ ) f = B rD ( σ ) f, where B rD ( σ ) f = B D ( σ ) ... B D ( σ ) f. Let us introduce the following notations B (2 m − D ( σ, N ) f = (cid:16) σπ (cid:17) m − X | k |≤ N ( − k +1 A m,k e πσ ( k − / D j f, B (2 m ) D ( σ, N ) f = (cid:16) σπ (cid:17) m X | k |≤ N ( − k +1 B m,k e πkσ D j f. One obviously has the following set of approximate Boas-type formulas.
Theorem 3.5. If f ∈ B σ ( D ) and r ∈ N then (3.11) D ( r ) f = B ( r ) D ( σ, N ) f + O ( N − ) . The next Theorem contains another Boas-type formula.
Theorem 3.6. If f ∈ B σ ( D ) then the following sampling formula holds for t ∈ R and n ∈ N (3.12) e tD D n f = X k e kπσ D f − f kπσ (cid:26) n sinc (n − (cid:18) σ t π − k (cid:19) + σ t π sinc (n) (cid:18) σ t π − k (cid:19)(cid:27) . In particular, for n ∈ N one has (3.13) D n f = Q nD ( σ ) f, where the bounded operator Q nD ( σ ) is given by the formula (3.14) Q nD ( σ ) f = n X k e kπσ D f − f kπσ h sinc (n − ( − k) + sinc (n) ( − k) i . Proof. If f ∈ B σ ( D ) then for any g ∗ ∈ E ∗ the function F ( t ) = (cid:10) e tD f, g ∗ (cid:11) belongsto B ∞ σ ( R ).We consider F ∈ B σ ( R ) , which is defined as follows. If t = 0 then(3.15) F ( t ) = F ( t ) − F (0) t = (cid:28) e tD f − ft , g ∗ (cid:29) , BOAS-TYPE FORMULAS IN BANACH SPACES and if t = 0 then F ( t ) = ddt F ( t ) | t =0 = h Df, g ∗ i . We have F ( t ) = X k F (cid:18) kπσ (cid:19) sinc (cid:18) σ t π − k (cid:19) . From here we obtain the next formula (cid:18) ddt (cid:19) n F ( t ) = X k F (cid:18) kπσ (cid:19) sinc (n) (cid:18) σ t π − k (cid:19) and since (cid:18) ddt (cid:19) n F ( t ) = n (cid:18) ddt (cid:19) n − F ( t ) + t (cid:18) ddt (cid:19) n F ( t )we obtain (cid:18) ddt (cid:19) n F ( t ) = n X k F (cid:18) kπσ (cid:19) sinc (n − (cid:18) σ t π − k (cid:19) + σ t π X k F (cid:18) k πσ (cid:19) sinc (n) (cid:18) σ t π − k (cid:19) Since (cid:0) ddt (cid:1) n F ( t ) = (cid:10) D n e tD f, g ∗ (cid:11) , and F (cid:18) kπσ (cid:19) = * e kπσ D f − f kπσ , g ∗ + we obtain that for t ∈ R , n ∈ N ,D n e tD f = X k e kπσ D f − f kπσ (cid:20) n sinc (n − (cid:18) σ t π − k (cid:19) + σ t π sinc (n) (cid:18) σ t π − k (cid:19)(cid:21) . Theorem is proved. (cid:3)
The next Theorem shows that Boas-type formulas make sense for a dense set ofvectors.
Theorem 3.7.
The set S σ ≥ B σ ( D ) is dense in E .Proof. Note that if φ ∈ L ( R ) , k φ k = 1 , is an entire function of exponential type σ then for any f ∈ E the vector g = Z ∞−∞ φ ( t ) e tD f dt belongs to B σ ( D ) . Indeed, for every real τ we have e τD g = Z ∞−∞ φ ( t ) e ( t + τ ) D f dt = Z ∞−∞ φ ( t − τ ) e tD f dt. Using this formula we can extend the abstract function e τD g to the complex planeas e zD g = Z ∞−∞ φ ( t − z ) e tD f dt. Since by assumption h is an entire function of exponential type σ and k φ k L ( R ) = 1we have k e zD g k ≤ k f k Z ∞−∞ | φ ( t − z ) | dt ≤ k f k e σ | z | . This inequality implies that g belongs to B σ ( D ). OAS-TYPE FORMULAS IN BANACH SPACES 9
Let h ( t ) = a (cid:18) sin( t/ t (cid:19) and a = Z ∞−∞ (cid:18) sin( t/ t (cid:19) dt ! − . Function h will have the following properties:(1) h is an even nonnegative entire function of exponential type one;(2) h belongs to L ( R ) and its L ( R )-norm is 1;(3) the integral(3.16) Z ∞−∞ h ( t ) | t | dt is finite.Consider the following vector(3.17) R σh ( f ) = Z ∞−∞ h ( t ) e tσ D f dt = Z ∞−∞ h ( tσ ) e tD f dt, Since the function h ( t ) has exponential type one the function h ( tσ ) has the type σ .It implies (by the previous) that R σh ( f ) belongs to B σ ( D ).The modulus of continuity is defined as in [4]Ω( f, s ) = sup | τ |≤ s k ∆ τ f k , ∆ τ f = ( I − e τD ) f. Note, that for every f ∈ E the modulus Ω( f, s ) goes to zero when s goes to zero.Below we are using an easy verifiable inequality Ω ( f, as ) ≤ (1 + a ) Ω( f, s ) , a ∈ R + . We obtain k f − R σh ( f ) k ≤ Z ∞−∞ h ( t ) (cid:13)(cid:13) ∆ t/σ f (cid:13)(cid:13) dt ≤ Z ∞−∞ h ( t )Ω ( f, t/σ ) dt ≤ Ω (cid:0) f, σ − (cid:1) Z ∞−∞ h ( t )(1 + | t | ) dt ≤ C h Ω (cid:0) f, σ − (cid:1) , where the integral C h = Z ∞−∞ h ( t )(1 + | t | ) dt is finite by the choice of h . Theorem is proved. (cid:3) Analysis on compact homogeneous manifolds
Let M , dim M = m , be a compact connected C ∞ -manifold. One says that acompact Lie group G effectively acts on M as a group of diffeomorphisms if thefollowing holds true:(1) Every element g ∈ G can be identified with a diffeomorphism g : M → M of M onto itself and g g · x = g · ( g · x ) , g , g ∈ G, x ∈ M , where g g isthe product in G and g · x is the image of x under g .(2) The identity e ∈ G corresponds to the trivial diffeomorphism e · x = x. (3) For every g ∈ G, g = e, there exists a point x ∈ M such that g · x = x . A group G acts on M transitively if in addition to 1)- 3) the following propertyholds: 4) for any two points x, y ∈ M there exists a diffeomorphism g ∈ G suchthat g · x = y. A homogeneous compact manifold M is an C ∞ -compact manifold on which tran-sitively acts a compact Lie group G . In this case M is necessary of the form G/K ,where K is a closed subgroup of G . The notation L p ( M ) , ≤ p ≤ ∞ , is used for theusual Banach spaces L p ( M , dx ) , ≤ p ≤ ∞ , where dx is the normalized invariantmeasure.Every element X of the Lie algebra of G generates a vector field on M which wewill denote by the same letter X . Namely, for a smooth function f on M one has Xf ( x ) = lim τ → f (exp τ X · x ) − f ( x ) τ for every x ∈ M . In the future we will consider on M only such vector fields.Translations along integral curves of such vector field X on M can be identifiedwith a one-parameter group of diffeomorphisms of M which is usually denoted asexp τ X, −∞ < τ < ∞ . At the same time the one-parameter group exp τ X, −∞ <τ < ∞ , can be treated as a strongly continuous one-parameter group of operatorsin a space L p ( M ) , ≤ p ≤ ∞ which acts on functions according to the formula f → f (exp τ X · x ) , τ ∈ R , f ∈ L p ( M ) , x ∈ M . The generator of this one-parametergroup will be denoted as D X,p and the group itself will be denoted as e τD X,p f ( x ) = f (exp τ X · x ) , t ∈ R , f ∈ L p ( M ) , x ∈ M . According to the general theory of one-parameter groups in Banach spaces [4],Ch. I, the operator D X,p is a closed operator in every L p ( M ) , ≤ p ≤ ∞ . In orderto simplify notations we will often use notation D X instead of D X,p .It is known [12], Ch. V, that on every compact homogeneous manifold M = G/K there exist vector fields X , X , ..., X d , d = dim G , such that the second orderdifferential operator X + X + ... + X d , d = dim G , commutes with all vector fields X , ..., X d on M . The corresponding operator in L p ( M ) , ≤ p ≤ ∞ , (4.1) − L = D + D + ... + D d , D j = D X j , d = dim G , commutes with all operators D j = D X j . This operator L which is usually calledthe Laplace operator is involved in most of constructions and results of our paper.The operator L is an elliptic differential operator which is defined on C ∞ ( M )and we will use the same notation L for its closure from C ∞ ( M ) in L p ( M ) , ≤ p ≤ ∞ . In the case p = 2 this closure is a self-adjoint positive definite operatorin the space L ( M ). The spectrum of this operator is discrete and goes to infinity0 = λ < λ ≤ λ ≤ ... , where we count each eigenvalue with its multiplicity. Foreigenvectors corresponding to eigenvalue λ j we will use notation ϕ j , i. e. L ϕ j = λ j ϕ j . The spectrum and the set of eigenfunctions of L are the same in all spaces L p ( S d ).Let ϕ , ϕ , ϕ , ... be a corresponding complete system of orthonormal eigenfunc-tions and E σ ( L ) , σ > , be a span of all eigenfunctions of L whose correspondingeigenvalues are not greater σ . OAS-TYPE FORMULAS IN BANACH SPACES 11
In the rest of the paper the notations D = { D , ..., D d } , d = dim G , will be usedfor differential operators in L p ( M ) , ≤ p ≤ ∞ , which are involved in the formula(4.1). Definition 4.1 ([24], [28]) . . We say that a function f ∈ L p ( M ) , ≤ p ≤ ∞ , belongs to the Bernstein space B pσ ( D ) , D = { D , ..., D d } , d = dim G , if and only iffor every ≤ i , ...i k ≤ d the following Bernstein inequality holds true (4.2) k D i ...D i k f k p ≤ σ k k f k p , k ∈ N . We say that a function f ∈ L p ( M ) , ≤ p ≤ ∞ , belongs to the Bernstein space B pσ ( L ) , if and only if for every k ∈ N the following Bernstein inequality holds true kL k f k p ≤ σ k k f k p , k ∈ N . Since L in the space L ( M ) is self-adjoint and positive-definite there exists aunique positive square root L / . In this case the last inequality is equivalent tothe inequality kL k/ f k ≤ σ k/ k f k , k ∈ N . Note that at this point it is not clear if the Bernstein spaces B pσ ( D ) , B pσ ( L ) arelinear spaces. The facts that these spaces are linear, closed and invariant (withrespect to operators D j ) were established in [28].It was shown in [28] that for 1 ≤ p, q ≤ ∞ the following equality holds true B pσ ( D ) = B qσ ( D ) ≡ B σ ( D ) , D = { D , ..., D d } , which means that if the Bernstein-type inequalities (4.2) are satisfied for a single1 ≤ p ≤ ∞ , then they are satisfied for all 1 ≤ p ≤ ∞ . Definition 4.2. E λ ( L ) , λ > , be a span of all eigenfunctions of L whose corre-sponding eigenvalues are not greater λ . The following embeddings which describe relations between Bernstein spaces B n and eigen spaces E λ ( L ) were proved in [28](4.3) B σ ( D ) ⊂ E σ d ( L ) ⊂ B σ √ d ( D ) . These embeddings obviously imply the equality [ σ> B σ ( D ) = [ λ E λ ( L ) , which means that a function on M satisfies a Bernstein inequality (4.2) in the normof L p ( M ) , ≤ p ≤ ∞ , if and only if it is a linear combination of eigenfunctions of L . As a consequence we obtain [28] the following Bernstein inequalities for k ∈ N , (4.4) kL k ϕ k p ≤ (cid:0) dλ (cid:1) k k ϕ k p , d = dim G , ϕ ∈ E λ ( L ) , ≤ p ≤ ∞ . One also has [28] the Bernstein-Nikolskii inequalities k D i ...D i k ϕ k q ≤ C ( M ) λ k + mp − mq k ϕ k p , k ∈ N , m = dim M , ϕ ∈ E λ ( L ) , and kL k ϕ k q ≤ C ( M ) d k λ k + mp − mq k ϕ k p , k ∈ N , m = dim M , ϕ ∈ E λ ( L ) , where 1 ≤ p ≤ q ≤ ∞ and C ( M ) is constant which depends just on the manifold.It is known [36], Ch. IV, that every compact Lie group G can be identified with asubgroup of orthogonal group O ( N ) of an Euclidean space R N . It implies that everycompact homogeneous manifold M can be identified with a submanifold which is trajectory of a unit vector e ∈ R N . Such identification of M with a submanifold of S N − is known as the equivariant embedding into R N .Having in mind the equivariant embedding of M into R N one can introduce thespace P n ( M ) of polynomials of degree n on M as the set of restrictions to M ofpolynomials in R N of degree n . The following relations were proved in [28]: P n ( M ) ⊂ B n ( D ) ⊂ E n d ( L ) ⊂ B n √ d ( D ) , d = dim G , n ∈ N , and(4.5) [ n ∈ N P n ( M ) = [ σ ≥ B σ ( D ) = [ j ∈ N E λ j ( L ) . The next Theorem was proved in [11], [32].
Theorem 4.3. If M = G/K is a compact homogeneous manifold and L is definedas in (4.1), then for any f and g belonging to E ω ( L ) , their pointwise product f g belongs to E dω ( L ) , where d is the dimension of the group G .Using this Theorem and (4.3) we obtain the following Corollary 4.1. If M = G/K is a compact homogeneous manifold and f, g ∈ B σ ( D ) then their product f g belongs to B dσ ( D ) , where d is the dimension of the group G . An example. Analysis on S d We will specify the general setup in the case of standard unit sphere. Let S d = (cid:8) x ∈ R d +1 : k x k = 1 (cid:9) . Let P n denote the space of spherical harmonics of degree n , which are restrictionsto S d of harmonic homogeneous polynomials of degree n in R d . The Laplace-Beltrami operator ∆ S on S d is a restriction of the regular Laplace operator ∆ in R d . Namely, ∆ S f ( x ) = ∆ e f ( x ) , x ∈ S d , where e f ( x ) is the homogeneous extension of f : e f ( x ) = f ( x/ k x k ). Another way tocompute ∆ S f ( x ) is to express both ∆ S and f in a spherical coordinate system.Each P n is the eigenspace of ∆ S that corresponds to the eigenvalue − n ( n + d − Y n,l , l = 1 , ..., l n be an orthonormal basis in P n .Let e , ..., e d +1 be the standard orthonormal basis in R d +1 . If SO ( d + 1) and SO ( d ) are the groups of rotations of R d +1 and R d respectively then S d = SO ( d +1) /SO ( d ).On S d we consider vector fields X i,j = x j ∂ x i − x i ∂ x j which are generators of one-parameter groups of rotations exp tX i,j ∈ SO ( d + 1) inthe plane ( x i , x j ). These groups are defined by the formulasexp τ X i,j · ( x , ..., x d +1 ) = ( x , ..., x i cos τ − x j sin τ, ..., x i sin τ + x j cos τ, ..., x d +1 )Let e τX i,j be a one-parameter group which is a representation of exp τ X i,j in aspace L p ( S d ). It acts on f ∈ L p ( S d ) by the following formula e τX i,j f ( x , ..., x d +1 ) = f ( x , ..., x i cos τ − x j sin τ, ..., x i sin τ + x j cos τ, ..., x d +1 ) . OAS-TYPE FORMULAS IN BANACH SPACES 13
Let D i,j be a generator of e τX i,j in L p ( S d ). In a standard way the Laplace-Beltramioperator L can be identified with an operator in L p ( S d ) for which we will keep thesame notation. One has ∆ S = L = X ( i,j ) D i,j . Applications
Compact homogeneous manifolds.
We return to setup of subsection 5.1.Since D j , ≤ j ≤ d generates a group e τD j in L p ( M ) the formulas (3.6) give for f ∈ B σ ( D )(5.1) D m − j f ( x ) = B (2 m − j ( σ ) f ( x ) = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e πσ ( k − / D j f ( x ) , m ∈ N , (5.2) D mj f = B (2 m ) j ( σ ) f = (cid:16) σπ (cid:17) m X k ∈ Z ( − k +1 B m,k e πkσ D j f, m ∈ N ∪ . Note, that every vector field X on M is a linear combination P dj =1 a j ( x ) D j , x ∈ M . Thus we can formulate the following fact. Theorem 5.1. If f ∈ B σ ( D ) then for every vector field X = P dj =1 a j ( x ) D j on M (5.3) Xf = d X j =1 a j ( x ) B j ( σ ) f, where B j ( σ ) = B D j ( σ ) . Moreover, every linear combination X = P dj =1 a j X j with constant coefficientscan be identified with a generator D = P dj =1 a j D j of a bounded strongly continuousgroup of operators e tD in L p ( M ) , ≤ p ≤ ∞ .A commutator(5.4) [ D l , D m ] = D l D m − D m D l = d X j =1 c j D j where constant coefficients c j here are known as structural constants of the Liealgebra is another generator of an one-parameter group of translations. Formulas(5.1) imply the following relations. Theorem 5.2. If D = P dj =1 a j D j then for operators B D ( σ ) = B D ( σ ) and all f ∈ B σ ( D )(5.5) B D ( σ ) f = d X j =1 a j B j ( σ ) f. In particular (5.6) [ D l , D m ] f = B l ( σ ) B m ( σ ) f − B m ( σ ) B l ( σ ) f = d X j =1 c j B j ( σ ) f. Moreover, (5.7) D j ...D j k = B j ( σ ) ... B j k ( σ ) f. Clearly, for any two smooth functions f, g on M one has D j ( f g )( x ) = f ( x ) D j g ( x ) + g ( x ) D j f ( x ) . If D = P dj =1 a j ( x ) D j then for f, g ∈ B σ ( D ) the following equality holds D ( f g ) ( x ) = d X j =1 a j ( x ) { f ( x ) B j ( σ ) g ( x ) + g ( x ) B j ( σ ) f ( x ) } . It is the Corollary 4.1 which allows to formulate the following result.
Theorem 5.3. If f, g ∈ B σ ( D ) and D = P dj =1 a j D j , where a j are constants then B D (2 dσ )( f g )( x ) = d X j =1 a j { f ( x ) B j ( σ ) g ( x ) + g ( x ) B j ( σ ) f ( x ) } . The formula −L = D + D + ... + D d implies the following result. Theorem 5.4. If f ∈ B σ ( D ) then (5.8) L f = B L ( σ ) f = d X j =1 B j ( σ ) f. Remark 5.5.
Note that it is not easy to find ”closed” formulas for groups like e it L .Of course, one always has a representation (5.9) e it L f ( x ) = Z M K ( x, y ) f ( y ) dy, with K ( x, y ) = P λ j e itλ j u j ( x ) u j ( y ) , where { u j } is a complete orthonormal systemof eigenfunctions of L in L ( M ) and L u j = λ j . But the formula (5.9) doesn’t tellmuch about e it L . In other words the explicit formulas for operators B L ( σ ) usuallyunknown. At the same time it is easy to understand the right-hand side in (5.8)since it is coming from translations on a manifold in certain basic directions (thinkfor example of a sphere). Let us introduce the notations B (2 m − j ( σ, N ) f = (cid:16) σπ (cid:17) m − X | k |≤ N ( − k +1 A m,k e πσ ( k − / D j f, B (2 m ) j ( σ, N ) f = (cid:16) σπ (cid:17) m X | k |≤ N ( − k +1 B m,k e πkσ D j f. The following approximate Boas-type formulas hold.
Theorem 5.6. If f ∈ B σ ( D ) then (5.10) d X j =1 α j ( x ) D j f = d X j =1 α j ( x ) B j ( σ, N ) f + O ( N − ) , OAS-TYPE FORMULAS IN BANACH SPACES 15 and if a j are constants then (5.11) B D ( σ ) f = d X j =1 a j B j ( σ, N ) f + O ( N − ) , D = d X j =1 a j D j . Moreover, (5.12) [ D l , D m ] f = d X j =1 c j B j ( σ, N ) f + O ( N − ) , (5.13) L f = d X j =1 B j ( σ, N ) f + O ( N − ) . The Heisenberg group.
In the space R n +1 with coordinates ( x , ..., x n , y , ..., y n , t )we consider vector fields X j = ∂ x j − y j ∂ t , Y j = ∂ y j + 12 x j ∂ t , T = ∂ t , ≤ j ≤ n. As operators in the regular space L p ( R n +1 ) , ≤ p ≤ ∞ , they generate one-parameter bounded strongly continuous groups of operators. In fact, these opera-tors form in L p ( R n +1 ) a representation of the Lie algebra of the so-called Heisen-berg group H n . The corresponding one-parameter groups are e τX j f ( x , ..., t ) = f ( x , ..., x j + τ, ....., x n , y , ..., y n , t − y j τ ) e τY j f ( x , ..., t ) = f ( x , ..., x n , y , ..., y j + τ, ..., y n , t + 12 x j τ ) . As we already know for every σ > B σ ( X j ) such thattheir union is dense in L p ( R n +1 ) , ≤ p ≤ ∞ , and for which the following formulashold with m ∈ N X m − j f = (cid:16) σπ (cid:17) m − X k ∈ Z ( − k +1 A m,k e πσ ( k − / X j f,X m − j f = (cid:16) σπ (cid:17) m − X | k | Take 3-dimensional Heisenberg groupand consider what is known as its Schr¨odinger representation in the regular space L ( R ) [10]. The infinitesimal operators of this representation are differentiaton D = πi ddx and multiplication by independent variable x which will be denoted as X . Every linear combination pD + qX with constant coefficients p, q generates aunitary group in L ( R ) according to the formula(5.14) e πi ( pD + qX ) f ( x ) = e πiqx + πipq f ( x + p ) , and in particular(5.15) e πiqX f ( x ) = e πiqx f ( x ) , e πipD f ( x ) = f ( x + p ) . For every σ > B σ ( pD + qX ) , B σ ( D ) , B σ ( X )and corresponding operators B and Q . 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