FINITE PROPAGATION SPEED AND KERNELS OF STRICTLY ELLIPTIC OPERATORS

We establish estimates of the resolvent and other related kernels i n and discuss LP-theory for a class of strictly elliptic operators on The clas of operators considered in the paper is of the form A 0 + B wlth he leading elliptic part A 0 and a "singular" perturbation B, whose coefficients have LP-type and are modeled after Schrdinger operators.

I. INqRODUCTION.-I In this paper we shall study the resolven= kernel R_(x,y) (-A) and other related "functions of A" for a class of strictly elliptic op=rators AO [ aa (x)D a n on and their perturbations A 0 + B.
Underlying ali those ;s ehe qu=ion of existence and estimates of the reeivent kernel R_.
Typically the estimate ha3 :he form of a convolution-t::pe bound i-r xth an L adial decreasin functlcn H In the uniformly elliptic case kernels of the resolvent R_ (-A) -i and -tA semiroup K e were studied extensively ([!], [2], [3] eta!.).In particular, Eidelman [i] derived :he following radial bound of K D. GURARIE m where m order A, m m-I' and I. Gelfand and G. Shilov [2] applied this estimate to study generalized eigenfunction expansions of A.
In [4], [5] we obtained similar estimates in a different way starting with the resolvent kernel.The latter approach allows us to treat perturbations of A, and consequently operators with "nonregular" coefficients.We applied these results to LP-spectral theory of elliptic operators.
In the present paper we extend the results of [4] to a class of strictly elliptic operators with possibly unbounded coefficients, obtained by linear deformations of uniformly elliptic symbols in the -variable, a a(x,) (X,x()) Here x x means a matrix function (ij(x)) depending smoothly on x E n.
Natural examples of such deformations are second order elliptic operators A aij(x)D2... Indeed any quadratic form a(x,) aij(x)i can be viewed deformation of the simplest one: Il 2 2 (aij " 6 ()I with (x))=. as

a x x
Other examples appear as right (left) invariant operators on nilpotent Lie groups (see [6]).
For the sake of presentation we shall restrict ourselves to the simplest case of deformations when (x)l is scalar ("conformal dilations").Such x transformations correspond to a multiplication of the leading symbol of A with a positive function (x) m, m order of A. The dilating factor (x) is subject to a certain constraint, called "finite propagation speed" condition.
Its precise definition is given in 2. This condition limits the rate of growth of at , which can not exceed O(Ixl) In other words leading coefficients of A can not grow faster than xl m for m-th order operators.
Let us observe that growth restrictions on the leading coefficients are well known in both ordinary (Sturm-Liouville) theory, d 2 A---a(x).---+(see for instance [7], Ch. 9) and also for partial dx differential operators A I aij (x)D'2"lj + ( [8]).A sufficient condition for A to be well defined (essentially selfadjoint or Weyl's "limit circle case") is the divergence of the integral

Ixl<r aij
The latter condition was called in [8] "finite propagation speed."Physically, a(x)ll can be interpreted as a "local propagation speed," then integrals (1.2) measure the "amount of time required to get from a finite point to ".
It is not clear to what extent(l.2)isnecessary to get a well defined operator A.
The form of "finite propagation speed" used in our paper is close but somewhat stronger than (1.2).It is expressed in terms of derivatives of (x) rather than integrals.Also the way we apply it in our setting differs from the standard method of "hyperbolic wave propagators" (cf.[8], [14]).
-th Under assumptions of 2 we show that m order elliptic operators are well defined in different LP-spaces (I < p ) and their kernels (resolvent, semigroup, etc.) share many properties of "constant coefficient" and "uniformly elliptic" kernels (cf.[15], [16], [5], [4]).There is one notable exception however: the radial bound (I) central for the argument of [5], [4]  Maximal functions appear in this context when "deformations" of symbols (of resolvent and related operators) are translated into "deformations" of kernels.
As a result one gets kernels of the type K(x,y) -nxIKrx-Yl ,(x)J where K is Ll-radial, (x) > 0 the dilating factor.If ,(x) < c is bounded (uniformly elliptic case) K is estimated by the usual "Ll-dilation K(x,y) < K() In our case (unbounded (x)) K has no longer a single convolution type 0 bound (I.A).But the maximal function (1.3) applies to give (Kf)(x) < KII f*(x) The maximal operator f f* is well known to be L p (I < p < ) (see for nst. [17]), which yields a bound for the LP-operator norm of K Such bounds are used throughout the paper as a substitute for the radial bound (I.I).
Otherwise our argument is similar to [4].
First we construct the "free" resolvent R 0 (-Ao)-I A 0 is the leading (homogeneous) part of A, via tle "parametrix series" R 0 Here K is a pseudodifferential operator (DO) with symbol -a(x,) a(x,$) being the leading svmbol of A, and L a @DO of order -I whose symbol is computed explicitly ( 4).
The "perturbed" (-A) -I is constructed via the perturbation resolvent R series (1.6) The coefficients of perturbation B b (x)D a are taken in LP-ciasses.
In our case b (x) can be allowed certain growth at {} depending on (x) Such conditions are naturally formulated in terms of weighted LP-spaces, L p with weights w, that are fractional powers of Also the L-domin of A 0 and A ca be characterized as a weighted LP-Sobolev space (Theorem 2).
The central result of the paper, Lemma i, analogous to Le.ma of [5], gives conditions (in terms of LP-classes of coefficients) for relative boundedness of B with respect to A 0 and estimates the norm of BK This result applies in 4 to sum series(l.5)and(l.6),andto derive different corollaries.Among them we ge a) LP-closedness of A A 0 + B and bounds on its LP-spectrum; 2 b) a priori estimates and essential selfadjointness of A in L c) resolvent summability, i.e., convergence Rf(x) f(x) in L p and a.e.d) existence of a strongly continuous holomorphic semigroup -tA} {e Re t > 0" Let us remark that due to the maximal function techniques adopted in the paper the Ll-space is excluded from consideration.The Ll-theory of integral kernels Kd(x,y) (x)-nK((x)-l(x-y)) with K E L poses an interesting harmonic analysis question.At the end of 4 we give two examples which have bearing on Ll-theory.
In conclusion let us mention that the basic L2-theory of elliptic operators ([19], [0], [3]) was extended later in a fairly general setting of pseudodifferential calculus ( [21], [9], [22], [6]).But much less i known about L p, p # 2. Two recent works that discuss specifically LP-theory are those by Nagel and Stein [6] and Beals [23].Both methods allow to treat a variety of "nonelliptic problems" (subelliptic Lap!acians, degenerate elliptic operators, etc.), but remain entirely within the pseudodifferential framework in the sense that symbols (coefficients) are assumed sufficiently smooth.
Another trend ( [15], [16], [5]) was to explore what "amount of nonregularity" of coefficients yields "well defined" LP-operators.The present paper continues the latter trend, but our approach seems to be extendable to various "nonelliptic" situations as well.This extension will be discussed elsewhere. 2. THE CLASS OF OPERATORS.The following assumptions will be made on A 0 and B.
I. The leading part A 0 is assumed to be homogeneous of degree m and have a real symbol a(x,) obtained by the "deformation" of a uniformly elliptic symbol a(x,$) a(x,) (x,(x)) m(x)(x,) Uniform ellipticity as usual means that The dilating factor (x) and the coefficients of (x,) (x).are subject to the following constraints Here and elsewhere u () denote a partial derivative Du.Condition (2.1) is close but somewhat stronger than the "finite propagation speed" condition [8].
-th The latter adjusted for m order operators of the above type requires dr (r) ) ngl+ +" Both conditions do not allow (x) to grow faster than O(Ix I) as x .This implies in particular that the class of perturbations A 0 + B includes all symmetric operators with the leading part A 0 The latter is important both for the argument below and in various applications. 3. THE MAIN LEM.
In this section we shall prove the main Lemma, which gives condition for relative boundedness of the operator B b(x)D a with respect to A 0 of I, and provides an estimate of the operator norm of the product B "parametrix o ('-A O) This result will enable us to prove the convergence of series (1.5) and (1.6) and to derive all consequences in 4.
Let A 0 a(x,D) be an elliptic operator of 2 with symbol a( Define a DO K K (x,z),z x y, with symbol I -a(x,) a parametrix of (-Ao) where C depends on p the L -norm of b(x) and the leading symbol W a(x,) PROOF.The operator BK consists of a DO DK followed by a multiplication with b(x) We start with DK.By the usual product formula of DO's its symbol is equal (x,) [ ()*-88--a) "x We apply the "iterated chain rule" to partial derivatives of (_-) including a stronger version of Young's inequality the Hardy-Littlewood-Sobolev inequality, where K 6 Lq-weak (see for instance [17], Ch. 3).The relation between p, p' and q is the same but p i is excluded in the latter case.
Rephrasing (3.7) in the reciprocal scale 6 [0,i] one can say that a P convolution with K 6 L q (or Lq-weak) shifts the LP-class to the left by the amount (i ) while a multiplication with b E L r shifts it to the right by q !.The product of two operations f bK*f is bounded if the LP-classes of b r i i and K are "Young-dual": I r q This relation explains the limitation on the scale of LP-spaces in Lemma I, p (we do not want a convolution-type term K to "push" the LP-class of P f out of the scale [0,i]).It also explains the definition of the "fractional order" (2.3).Indeed, a multiplication with b 6 L p is "equivalent" by (3.7) to a convolution with the "fractional Laplacian" A s (_)s/2 ;s Now we return to the operator b(x)D=K and want to make the above heuristic argument accurate.It is convenient to multiply and divide BK by the fractional Laplacian As, s --n (s O, of pa =) Then we get from (3.We "pull out" the absolute value of complex re in the denominator of (3.8), and by the homogenuity of a write o. in terms of a "uniformly elliptic" symbol oi --i e -a(x,) r -i/m 6 i(x') 7 i(x; r (x)E) -t The weight 6 (x) will joint the left (multiplication) factor of (bi)MiA-s Remembering that #i is a product of derivatives k (J), S j and the "convolution" (right) factor A -s of the factor i t -s product iMi A "complete" each other in the sense of interpolation formulas (3.7).It remains to study the middle term.We shall show that a DO M. is LP-bounded (l<p<) and estimate its norm.
We first observe that the dilation of symbols .o. (x,pK) with -i/m6 M.l(x,z) p-n i(x; 0-1z) which represents M. in terms of the "uniformly elliptic" kernel ..
The order of a DO o. is nonpositive by (3.9).So two cases are   where c depends on a finite number (N--n + I) symbol class semlnorms of $ (see for inst.[5]).
A straightforward evaluation of seminorms yields sup llNl(aoi)( )I < in C x,( e/2]N, N IBI whence the constant in estimate (3.2) of Lemma i.From (3.11), (3.12) it follows that the kernel M4(x,z) is bounded by the -i/m 6 so-called p-dilation of the radial Ll-function H with p p(x) r (x) .IMi(x,z) < co-n[x)H[io" "' (x)z) H z x Y.
If 0 were bounded o(x) < e (uniformly elliptic case), then H < H and the usual Ll-convolution would immediately imply the result: LP-boundedness of Mo for all i < p < (cf.[4], [5]).In case of unbounded 6(x) the maximal 1 function estimate applies, (Hof)(x) < Hlllf*(x) This yields LP-boundedness of M in all spaces I < p < and proves 1 Lemma I in the first case.We split $i E SI,0 into the sum of 3 terms: o'l c(x)l + 0 +a multiple of the "identity" (in the -variable) function" a homogeneous of order zero (in ) sbol Oo(X,E) with the zero mean-value over the unit sphere i and a negative order sbol  /a c(x), while the second is obviously in SI, 0 sufficiently regular at {0} which is important for the existence of "global" radial bound H of the kernel $ (x,D) (cf.[5]).
Correspondingly kernel splits into the sum of three kernels: c(x)l + M 0 + M_, a multiplication with "nice" (bounded) function c(x) a Calderon-Zygmund kernel o(X'Z) (homogeneous of degree -n in z) and an

&. RESOLVENT KERNELS OF A 0
AND A AND THEIR APPLICATIONS.
After Lemma I we can study the convergence of series (1.5) and (1.6), that give resolvent kernels R 0 (-A0)-1 and R (-A) -1 We start with R 0 THEOREM i. Series (1.5) converges absolutely in the complement of a parabolic region about positive real axis in , in all LP-spaces 1 < p < , and defines the resolvent R 0 of A.
PROOF.We recall that the operator L (-A0)K I.By the product formula (a e 6 L_ m plays the role of b(x)) So the whole argument of Lemma I can be repeated for L" Notice that all DO's o.(x,D) have strictly negative order, -i. 1 Hence by Lemma i each term M of L and consequently the whole operator L is estimated as [sin 8/21 n+l r X i [8[/m ) i m with C depending on L-norms of i and p (I < p < ) Returning to series expansion (1.5), we see that i Lk converges absolutely if the right hand side of ( We want to show that R 0 is the resolvent of A 0 that is RO(-A0)f f for all f in the LP-domain of A 0 The first formula immediately follows from the relation (-Ao)K I L. It We can not apply the above result directly to A 0 to show the existence of the right resolvent, since A 0 is not of homogeneous order.But A 0 A 0 + B, whose lower order terms {beD e }lel<m have coefficients bounded "relative to A 0 Therefore the "perturbation" Theorem (Theorem 4 below) applies to prove the existence of the right resolvent for A 0 outside of another parabolic region ' Once both resolvents, right and left, are shown to exist they must be equal.This completes the proof of Theorem i.
Next we apply Lemma i to characterize the LP-domain of A 0, Pp(Ao), i < p < . -th For constant coefficient elliptic operators P (A O) is known to be the m P Sobolev space i v'm" (I-A)m/2LP and the same is true for uniformly elliptic operators (see [4]).For strictly elliptic operators of the above type we have -th THEOREM  is the sum of terms that appear in Lemma i.
To prove invertibility it is convenient to use (-A) m/2 for sufficiently large > 0 instead of (l-&) m/2 in the definition of ip,m norm and also to w reverse two terms in the product w(%-A) m/2 We observe that where order B < m i and the coefficients b [ c w v'/w are all bounded.Then for sufficiently large X the operator norm Hence the inverse [(l A) m/2 + B] -I exists and is bounded.But A0w-i 0 + with uniformly elliptic 0 w-iA0 and of the same type as B. (0 + )(k &)-m/2 for uniformly elliptic 0 and "relatively small" , which is well known (cf.[4]) Indeed the operator S is a combination of zero-and negative order ,DO's { (x)D( g)m/2: II < m}.The former are given by Calderon-Zygmund kernels, hence L p for all 1 < p < (see [!7], ch.2).The latter have Ll-radially bounded kernels, hence L p for all i-<.p .
In many applications, including the perturbation series (1. is the "fractional order of B, constant C depends on p and the symbol a(x,) From (4.5) immediately follows COROLLARY 2 (cf.[5] [4], [16]).The operator B is Ao-bounded and the LP-domains of A and A 0 are equal (4.5) w i < p < rain Ip p(A) p(A0) 6p,m,w -m, Let us notice that the relative bound in the right hand side of (4.5) can be made as small as one likes, taking sufficiently large r, provided the "fractional" order d < m.In this case the perturbation series (1.6 converges absolutely in the complement of a parabolic region ' {: relative bond < I} The limit case, d m, is important, as it gives the "optimal amount" of LP-type singularity of lower order coefficients and also allows "top order" perturbations.In this case one can claim somewhat less.
PROPOSITION I. Operator BR?II < i in any regioD : II > R; lard I > } R, !ib"'l < e.Consequently, te whole perturbation B splits into the sum B' + B".The first term has order d < m-i hence B'RO!l becomes small outside of a parabolic region .' while the second B"R?I < 1 in some region fR,O" Here depends on b"!l and can be made as small as one likes.This proves the Proposition From Proposition i easily follows THEOREM 5.An operator A of Theorem 3 generates a holomorphic semigroup -tA e in the right half plane Re t > 0 in all L p I < p min {p} -tAf Moreover, the family of functions e (x) f(x) as t 0 in LP-norm REMARK i.As in [4] ( ), all the above results extend to strictly elliptic operators on certain Riemannian manifolds diffeomorphic to Rn and strictly elliptic systems.+ lal m, (see [5]).Another extension of L p Pe classes is discussed in [15], [16].It consists of all functions b (x) whose convolution with H (z) Izl -s (Izl i, and s n m + a) (a "local singularity" of Da(-Ao)-I) is bounded.This condition is close to so called "Rollnik condition" in the theory of Schrdinger operators (see [25]).
We shall conclude this section with two examples, which have bearing on Ll-theory of the above class of operators.This theory poses the following interesting harmonic analysis problem: given an L I function K(x) and a x-y L I I K( (x)) is bounded.
dilating factor (x) > O, when the kernel K n(x) Two examples below indicate that the situation in Ll-case becomes more subtle compared to LP-theory (p > i) Namely, an Ll-radially bounded K and a "finitely propagating" are no longer sufficient for LI: the growth of 6 must be "slower" than O(Ixl) and some additional relation between K and appears.EXAMPLE I. Take (x) Ixi on .Then K(x,y) K(l-is homogeneous of degree -I. It is well known (see, for instance, [17], Appendix) that K is L p if and only if J y-PK(l-y) dy < =.The latter is obviously true for p > i, but fails for p I. XAMPLE 2. Now we take (x) 2vrx on R and write the operator K6f as (K6f)(x) I K(y)f(x (x)y) dy.
We introduce a new variable u x (x)y (x) The range of u is the Y 2 whole real line R but the function u y(X) has a critical point x 0 y and on the interval _y2 u 0 u is "three-fold" (see figure) We divide the line Rx into three intervals I I (-;0]; 12 (0;2y2]; 13 (2y2;+) On I I and 13 the Jacobian (d) is bounded from both sides, 0 < c I (=d__u_) c2 < I f(x (x)y) dx 2 f(u) v------du + 2 I f(u) du.A similar condition on K can be derived for other polynomial functions Ixl has the maximal allowed growth both kernels fail to be L I

Throughout this paper we shall consider operators of the
require d m.

5 )
is needed in order to have B bounded relative to A 0We also need some control of b at {}, depending on 6A convenient way to express it is in terms of weighted LP-spaces, L p {f: IfwlPdx < } Let us notice that the above class of operators is closed under the adjunction A A* provided the coefficients of g are sufficiently smooth (b P the LP-weighted Sobolev space of order I' a(a-)D Remembering that a m with and g subject to (2.1) (2.2), we estimate the g-order coefficient of the adjoint (a-g) a . (m) () () () y '+y=a-as So the lower order terms of A* coming from the top order coefficients: b a (-) are L (w -II) i.e. belong to the above class of perturbations.Similar argument applies to derivatives of the coefficients b of B. Namely, interpolating in 3 parameters: p (for LP-class of b Sobole t (fractional) order s and the exponent t of the weight w g one can show that b 6 LPs;tb (Y) 6 [qs_iyl; t-IYl where eq ep + IYI Let a(x ) (x (x)) and b(x) 6 L e w' satisfy assumptions (2.1) (2.5) of 2, then for all 6 range a(x,)+, the operator BK is LP-bounded, I < p -<. p, and its norm is estimated as follows k 8xS(a) C81-...8 k (-a)-k-l'['[(83Ja) 'i x (3.4) k summation taken over all partitions B8J into the sum of multi-indices {BJ} i with certain combinatorial coefficients CBI "8 k.As a result o treat a possible LP-type singularity of the coefficient E(x we shall use two interpolation inequalities for convolution and multiplication. order of o. is II + s m(k+l) + s mk + d Isl.
is well known to have an Ll-radial A negative order DO SI, 0 convolution-type bound [Bi(x,z)[ < cH(lzl)

L ! -adially
bounded kernel The first two of them are invariant under the 0-dilation (z) -(-z) o while the third is dominated by the maximal function as in case i This completes the proof.
resolvent of A 0 To show the existence of the left resolvent (II) we shall use duality between LP-spaces.Namely: the adjoint of the right resolvent of A 0 in L p becomes the left resolvent of the adjoint * L p i operator A 0 in ( + --I) and vice versa.
2. The LP-domain of A 0 is equal to the m weighted Sobolev spce with )fll in P (A O) and !lwDfll.p p w PROOF We have to show that the operator w(l-) m/2-0 is bounded and invertible in L v But R 0 K(I-L) -I It suffices to show boundedness and invertibility of the operator T w(l-&)m/2K.Boundedness follows directly from Lemma I. Indeed w(l-&)m/2K [ c wDK.

Finally "pulling
out" (\-A) m/2 from the right factor in (4.4), (l A) -m/2 [I + ( A)-m/2B -I the problem reduces to LP-boundedness of the operator S L) is easily shown to converge to I as uniformly in any sector {iarg I > > 0} The problem is thus reduced to estimating BKII The latter is provided by Lemma I.

REMARK 2 .
More general classes of coefficients {b of the perturbation B can be allowed in 3.One such class consists of b (x) defined on quotient a neighborhood of the critical point x 0 ythat causes to "blow up".The Jacobian () is computed explicitely.
s < I) Namely the convolution of yl/2-s K(yl_ s and i must y be L These results can be used to study L I properties of the resolvent kernel R ----d is bounded near {0} and decays exponentially at _m {} K(z) e -Ylzl If 6(x) Ixi with s < m, the condition Y-1/2 -SK(yl-S) , l__y L is obviously satisfied.Hence K s and consequently R i are L On the other hand if6(x)