Algebras of singular integral operators with kernels controlled by multiple norms Alexander Nagel Conference in Harmonic Analysis in Honor of Michael Christ
This is a report on joint work with Fulvio Ricci, Elias M. Stein, and Stephen Wainger.
Calderón-Zygmund kernels A Calderón-Zygmund kernel on R n associated with the dilations λ a (x) = λ a (x 1,..., x n ) = (λ a1 x 1,..., λ an x n ) is a tempered distribution K on R n with the following properties. 1. K is given away from the origin by integration against a smooth function. 2. For every multi-index α = (α 1,..., α n ) N n and all x 0 α K(x) ( ) Cα x1 1 a 1 + + x n 1 Qa n j=1 an ajαj where Q a = a 1 + + a n is the homogeneous dimension. 3. K satisfies appropriate cancellation conditions so that in particular the Fourier transform K = m is a bounded function. K will often have compact support or have rapid decay outside the unit ball.
Homogeneous nilpotent Lie groups G denotes a homogeneous nilpotent Lie group with underlying manifold R n and automorphic dilations λ d (x 1,..., x n ) = ( λ d1 x 1,..., λ dn x n ) with d 1 d 2 d n. For x = (x 1,..., x n ), y = (y 1,..., y n ) G x y = ( x 1 + y 1, x 2 + y 2 + M 2 (x, y),..., x n + y n + M n (x, y) ) where M j (x, y) is a polynomial vanishing if x = 0 or y = 0 and M j ( λd (x), λ d (y) ) = λ dj M j (x, y). Convolution on G is given by f g(x) = f(x y 1 )g(y) dy = R n f(y)g(y 1 x) dy. R n
Example: The Heisenberg group The m-dimensional Heisenberg group H m has underlying space R m R m R with automorphic dilations λ 1,1,2 (x, y, t) = ( λx, λy, λ 2 t ) and group multiplication (x 1, y 1, t 1 ) (x 2, y 2, t 2 ) = ( x 1 + x 2, y 1 + y 2, t 1 + t 2 2 x 1, y 2 + 2 x 2, y 1 ) where x, y is the Euclidean inner product. If then X j = xj + 2y j t, Y j = yj 2x j t, T = t, {X 1,..., X n, Y 1,..., Y n, T } is a basis for the left-invariant vector fields on H n.
The problem Consider two different dilation structures on G given by λ a (x 1,..., x n ) = (λ a1d1 x 1,..., λ andn x n ) with a 1 a 2 a n, λ b (x 1,..., x n ) = (λ b1d1 x 1,..., λ bndn x n ) with b 1 b 2 b n. Let K a and K b be Calderón-Zygmund kernels on R n with compact support associated with these dilations. For ϕ S(R n ) set T a [ϕ] = ϕ K a and T b [ϕ] = ϕ K b. What are the mapping properties of the operator T a T b on various function spaces? What can be said about size and cancellation properties of the Schwartz kernel of T a T b? Are there reasonably small algebras of convolution operators containing Calderón-Zygmund kernels associated to both dilations?
Background and Motivation Let Ω C n+1 be a strictly pseudoconvex domain and let = + : L 2 (0,1) (Ω) L 2 (0,1) (Ω). Given f C (0,1) (Ω), the -Neumann problem is to solve u = f on Ω u ρ = 0 on Ω u ρ = 0 on Ω. Greiner and Stein (1977) constructed a parametrix for this boundary value problem which involves the composition of two kinds of operators: operators of elliptic type coming from Poisson integrals and the Green s function for (which is essentially the Laplacian); operators of non-isotropic type coming from inverting the sub-laplacian L = n j=1 (X2 j + Yj 2 ) on the Heisenberg group H n.
Phong and Stein (1982) studied compositions of convolution operators on R d R T E [ϕ] = ϕ K E and T H [ϕ] = ϕ K H where is either Euclidean or Heisenberg convolution, and K E and K H are Calderón-Zygmund kernels on R d R associated with the two different dilations: λ E (x, t) = (λx, λt) and λ H (x, t) = (λx, λ 2 t). The local (near 0) and the global (near infinity) behavior of compositions T E T H or T H T E are different. Phong and Stein established: necessary and sufficient conditions for the compositions of such operators to be of weak-type (1,1); boundedness of the composition on non-isotropic Lipschitz spaces.
On the Heisenberg group H n, T = t and the sub-laplacian is L = n (Xj 2 + Yj 2 ). j=1 Müller, Ricci, and Stein (1995) studied general functions of L and T, and in particular established Marcinkiewicz-type theorems for multiplier operators m(l, T ) where α ξ β τ m(ξ, τ) ξ α τ β. Such multiplier operators are bounded on L p (H n ) for 1 < p <. The corresponding kernels satisfied differential inequalities α z β t K(z, t) z 2n α ( z 2 + t ) 1 β. This last estimate reflects the multi-parameter structure but is stronger than a product estimate.
A flag kernel associated with the dilations λ a (x) = (λ a1 x 1,..., λ an x n ) is a tempered distribution K with appropriate cancellation and singularities on an increasing sequence of subspaces: α K(x) ( ) x1 1 a a1(1+α 1 1)( ) x1 1 a 1 + x 2 1 a a2(1+α 2) 2 n = ( x 1 1 a 1 + x 2 1 1 ) a a 2 + + x j 1 j 1 + xj 1 aj(1+α j) a j. j=1 N., Ricci, Stein (2001) studied such operators when they arise in the study of the Bergman projection in tube domains over polyhedral cones and in solving b on certain quadratic submanifolds having the structure of step-2 nilpotent Lie groups. N., Ricci, Stein, Wainger (2012) and G lowacki (2010), (2013) extended this theory to homogeneous nilpotent Lie groups G of higher step, established boundedness in L p (G) for 1 < p <, and showed that such operators form an algebra under convolution.
A Step 3 Example Let K 1 and K 2 be Calderón-Zygmund kernels with compact support associated with the dilations λ 1 (ξ, η, τ) = (λξ, λη, λτ) and λ 2 (ξ, η, τ) = (λξ, λ 2 η, λ 3 τ). Thus α x β y γz K 1 (x, y, z) ( x + y + z ) α β γ, α x β y γz K 2 (x, y, z) ( x + y 1 2 + z 1 3 ) α 2β 3γ. The Fourier transforms m 1 = K 1 and m 2 = K 2 are smooth bounded functions satisfying the differential inequalities α ξ β η γ τ m 1 (ξ, η, τ) ( 1 + ξ + η + τ ) α β γ, α ξ β η γ τ m 2 (ξ, η, τ) ( 1 + ξ + η 1 2 + τ 1 3 ) α 2β 3γ.
Using Euclidean convolution, let T j [ϕ] = ϕ K j so T 1 T 2 is convolution with a distribution K and K(ξ, η, τ) = m(ξ, η, τ) = m 1 (ξ, η, τ)m 2 (ξ, η, τ). To estimate derivatives of m note that ξ gains (1 + ξ + η + τ ) 1 or (1 + ξ + η 1 2 + τ 1 3 ) 1, η gains (1 + ξ + η + τ ) 1 or (1 + ξ 2 + η + τ 2 3 ) 1, τ gains (1 + ξ + η + τ ) 1 or (1 + ξ 3 + η 3 2 + τ ) 1. Using just the product formula, the best estimates for m are ξ α η β τ γ m(ξ, η, τ) ( 1 + ξ + η 1 1 ) α 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ. 1 + ξ + η + τ How should one think about such estimates?
One possibility is to use the theory of flag kernels and multipliers. If α ξ η β τ γ m(ξ, η, τ) ( 1 1 ) α 1 + ξ + η 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ 1 + ξ + η + τ then α ξ β η γ τ m(ξ, η, τ) ξ α ( ξ + η ) β ( ξ + η + τ ) γ, ( ξ + η 1 2 + τ 1 3 ) α ( η + τ 2 3 ) β τ γ. Thus m is a flag multiplier for two opposite flags (0) {(ξ, 0, 0)} {(ξ, η, 0)} {(ξ, η, τ)} = R 3, (0) {(0, 0, τ)} {(0, η, τ)} {(ξ, η, τ)} = R 3.
The distribution K = K 1 K 2 is thus a flag kernel satisfying α x y β z γ K(x, y, z) x 1 α ( x + y ) 1 β ( x + y + z ) 1 γ, ( x + y 1 2 + z 1 3 ) 1 α ( y + z 2 3 ) 1 β z 1 γ. for the two opposite flags (0) {(x, 0, 0)} {(x, y, 0)} {(x, y, z)} = R 3, (0) {(0, 0, z)} {(0, y, z)} {(x, y, z)} = R 3.
In the 2-step case on R n R, the differential inequalities α x β t K(x, t) ( x + t ) n α ( x 2 + t ) 1 β are equivalent to the pair of flag inequalities α x β t K(x, t) x n α ( x 2 + t ) 1 β ( x + t ) n α t 1 β and However in step 3, the two-flag kernel inequalities from the last slide, α x y β z γ K(x, y, z) x 1 α ( x + y ) 1 β ( x + y + z ) 1 γ, ( x + y 1 2 + z 1 3 ) 1 α ( y + z 2 3 ) 1 β z 1 γ seem to give no information when x = z = 0. These flag estimates, together with the flag cancellation conditions, do imply that K is singular only at the origin.
A second possible way of thinking about the inequalities α ξ η β τ γ m(ξ, η, τ) ( 1 1 ) α 1 + ξ + η 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ 1 + ξ + η + τ is to observe that the ξ, η, and τ derivatives are controlled by different norms and hence different families of dilations: ξ N 1 (ξ, η, τ) = ξ + η 1 2 + τ 1 3 η N 2 (ξ, η, τ) = ξ + η + τ 2 3 η N 3 (ξ, η, τ) = ξ + η + τ
The class P(E) We introduce a class P(E) of distributions on R n singular only at the origin and depending on an n n matrix E. We study: A. Properties of distributions K P(E) a. Significance of the rank of E b. Characterization using the Fourier transform c. Marked partitions d. Characterizations via dyadic decompositions e. Two-flag kernels B. Convolution operators T K ϕ = ϕ K on a homogeneous nilpotent groups a. Continuity on L p (G) b. P(E) is an algebra c. Composition of Calderón-Zygmund kernels with different homogeneities
Let 1 e(1, 2) e(1, 3) e(1, n) e(2, 1) 1 e(2, 3) e(2, n) E = e(3, 1) e(3, 2) 1 e(2, n)....... e(n, 1) e(n, 2) e(n, 3) 1 be an n n matrix satisfying the basic hypotheses: e(j, k) > 0 for all 1 j, k n, e(j, j) = 1 for all 1 j n, e(j, l) e(j, k)e(k, l) for all 1 j, k, l n. Recall that the automorphic dilations on the group G are given by λ d (x 1,..., x n ) = (λ d1 x 1,..., λ dn x n ).
For 1 j n let N j (x 1,..., x n ) = x 1 e(j,1)/d1 + + x n e(j,n)/dn, Each N j is a homogeneous norm for a family of dilations on R n : N j ( λ d 1 e(j,1) x1,..., λ dn e(j,n) xn ) = λ N(x1,..., x n ). P(E) is a class of tempered distributions, depending on the matrix E and smooth away from the origin on R n, defined in terms of differential inequalities and cancellation conditions.
Differential inequalities: If K P(E) then away from the origin K is given by a smooth function and for each multi-index α = (α 1,..., α n ) where Note that α K(x 1,..., x n ) C α n j=1 N j (x 1,..., x n ) dj(1+αj) N j (x 1,..., x n ) = x 1 e(j,1)/d1 + + x n e(j,n)/dn. N j ( λ d 1 e(j,1) x1,..., λ dn e(j,n) xn ) = λ N(x1,..., x n ) so N j is a homogeneous norm for the family of dilations on R n given by λ j (x 1,..., x n ) = ( λ d 1 ) e(j,1) x1,..., λ dn e(j,n) xn. Derivatives of K with respect to the variable x j are controlled by the norm N j.
Cancellation Conditions: These are defined in terms of the action of the distribution K on dilates of normalized bump functions. Let 0 m n 1 and let ψ be any normalized bump function of n m variables. There are two requirements: If R = (R m+1,..., R n ) and each R j > 0 set ψ R (x m+1,..., x n ) = ψ(r m+1 x m+1,..., R n x n ). Define a tempered distribution K # R on Rm by setting K # R, ϕ = K, ϕ ψ R for all ϕ S(R m ). Away from the origin K # R is given by a smooth function and α 1 x 1 x αm m K # R (x 1,..., x m ) C α m j=1 with C α independent of ψ and R. N j (x 1,..., x m, 0,..., 0) aj(1+αj) The same holds for any permutation of the variables x 1,..., x n.
The rank of E Let E be an n n satisfying the basic hypotheses. Lemma (a) Rank (E) = 1 if and only if there is a dilation structure on R n for which P(E) is the space of Calderón-Zygmund kernels. (b) If rank (E) > 1 and K P(E) then K is integrable at infinity. (c) Suppose the rank of E is m. If K P(E) then for λ 1 { x R n : K(x) > λ} λ 1 log(λ) m 1. Moreover there exists K P(E) so that { x R n : K(x) > λ} λ 1 log(λ) m 1.
Characterization via the Fourier transform For rank(e) > 1 only the local behavior is important. Set { } P 0 (E) = K P(E) : K is rapidly decreasing outside the unit ball. Such distributions can be characterized by the behavior of their Fourier transform K. Recall that N j (x) = x 1 e(j,1)/d1 + + x n e(j,n)/dn, 1 j n. The dual norms are then given by Put N j (ξ) = ξ 1 1/e(1,j)d1 + + ξ n 1/e(n,j)dn, 1 j n. M (E) = {m C (R n ) : α m(ξ) n Cα Theorem K P 0 (E) if and only if K = m M (E). j=1 ( 1 + Nj (ξ) ) α jd j }.
Marked Partitions The analysis of distributions K P 0 (E) relies on a partition of the unit ball B(1) R n into regions where one summand is dominant in each norm N j (x). Similarly the analysis of m M (E) depends on a partition of R n \ B(1) into regions where one term in each dual norm N j (ξ) is dominant. For example, on the unit ball, the principal region S 0 is the set where for 1 j n the term x j e(j,j)/dj = x j 1/dj is dominant in N j (x) = n k=1 x k e(j,k)/d k : S 0 = { x B(1) : x k e(j,k)/d k x j 1/dj for all j, k }. Note that if x S 0 then N j (x) x j 1/dj. On this region, the differential inequalities for K P 0 (E) simplify: α K(x 1,..., x n ) n N j (x 1,..., x n ) dj(1+αj) j=1 n x j (1+αj). On the subset S 0 the differential inequalities for K are exactly the same as the differential inequalities for a product kernel. j=1
The following simple observation is a key to the study of other regions. Lemma Suppose x B(1) and suppose the term x k e(j,k)/d k is dominant in the norm N j (x). Then x k e(k,k)/d k = x k 1/d k is dominant in N k (x). Proof. Since x k e(j,k)/d k is dominant in the norm N j (x), x k e(j,k)/d k x l e(l,k)/d l, 1 l n. According to the basic hypothesis, e(l, k) e(l, j)e(j, k) for any 1 j, k, l n. Then since x l 1, x k 1/d k x l e(l,k)/e(j,k)d l x l e(l,j)e(j,k)/e(j,k)d l = x l e(l,j)/d l.
A marked partition S of {1,..., n} is a collection of disjoint subsets I 1,..., I s {1,..., n} whose union is all of {1,..., n}, together with a marked element k r I r, 1 r s. Write S = ( (I 1, k 1 ),..., (I s, k s ) ). The decomposition of B(1) is parameterized by marked partitions. For x B(1) outside a set of measure zero, there is exactly one dominant term in each norm N j (x). Let k 1,..., k s be the distinct integers which arise as subscripts of these dominant terms. Let { } I r = j {1,..., n} : x kr e(j,kr)/d kr is dominant in Nj (x), 1 r s. The subsets I 1,..., I s are disjoint, I 1 I s = {1,..., n}, and by the Lemma, k r I r. Thus S = ( (I 1, k 1 ),..., (I s, k s ) ) is a marked partition.
Characterization of P(E) via dyadic decompositions Let ϕ C0 (R n ). ϕ has strong cancellation if Set 1 k n. R ϕ(x 1,..., x k,..., x n ) dx k = 0 for [ϕ] I (x) = 2 n k=1 i k ϕ(2 i1 x 1,..., 2 in x n ) if I = (i 1,..., i n ) Z n. Γ Z (E) = { } (i 1,..., i n ) Z n : e(j, k)i k i j < 0 for all 1 j, k n. Lemma Let { ϕ I : I Γ(E) } C0 (R n ) be a family of normalized bump functions. Assume that each ϕ I has strong cancellation. Then K(x) = [ϕ I ] I (x) I Γ(E) converges in the sense of distributions to an element of P 0 (E).
The formulation of the converse assertion uses the decomposition of R n based on marked partitions. Let S = ( (I 1, k 1 ),..., (I s, k s ) ) be a marked partition. Write R n = R C1 R Cs and (x 1,..., x n ) = ( x 1,..., x s ) where R Cr is the set of variables x j R n with j I r. Then There is an s s matrix E S satisfying the basic hypotheses. There is a space of distributions P(E S ) P(E). There is a cone Γ Z (E S ) = { } (i 1,..., i n ) Z n : e S (j, k)i k i j < 0. If K P(E) then K = ψ 0 + S K S where ψ 0 S(R n ) and K S P(E S ). Moreover K S (x) = [ϕ I ] I (x). I Γ(E S )
The Basic Hypotheses If the principal region is not empty, then there exists (x 1,..., x n ) with each x j < 1 so that x j 1 d j xk e(j,k) d k ( x l e(k,l) ) d e(j,k) l = xl d(j,k)e(k,l) d l. (1) But since x S 0, x j x l e(j,l). (2) Unless the inequalities defining S 0 are self-improving, inequality (2) should imply (1), and so Since x l 1 it follows that x l e(j,l) x l e(j,k)e(k,l). e(j, l) e(j, k)e(k, l).
The basic hypotheses also arise as follows. Let F = { f(j, k) } be any n n matrix with positive entries, and let Γ(F) = { (t 1,..., t n ) R n : f(j, k)t k t j < 0 for all 1 j, k n }. Lemma If Γ(F) there exists a unique n n matrix E = { e(j, k)} with positive entries such that Γ(E) = Γ(F) and such that the entries of E satisfy Moreover, e(j, k) f(j, k). e(j, j) = 1 1 j n, e(j, l) e(j, k)e(k, l) 1 j, k, l n.
Two-flag kernels Consider a distribution K which is a flag kernel relative to two opposite flags with dilations λ a (x) = (λ a1 x 1,..., λ an x n ), λ b (x) = (λ b1 x 1,..., λ bn x n ). Then K satisfies appropriate cancellation conditions and the following differential inequalities: α K(x) ( ) 1 αj n j=1 x 1 aj/a1 + x 2 aj/a2 + + x j 1 aj/aj 1 + x j ) 1 αj n j=1 ( x j + x j+1 bj/bj+1 + + x n 1 bj/bn 1 + x n bj/bn Note that the differential inequalities give no information if x 1 = x n = 0 and n 3.
Theorem Suppose that a1 b 1 a2 b 2 an b n with at least one strict inequality. (a) The function K is integrable at infinity, and we can write K = K 0 + K where K L 1 (R N ) C (R N ), and K 0 is a two-flag kernel supported in B(1). (b) The kernel K 0 belongs to the class P 0 (E) associated to the matrix 1 b 1 /b 2 b 1 /b 3 b 1 /b n 1 b 1 /b n a 2 /a 1 1 b 2 /b 3 b 2 /b n 1 b 2 /b n a 3 /a 1 a 3 /a 2 1 b 3 /b n 1 b 3 /b n E =......... a n 1 /a 1 a n 1 /a 2 a n 1 /a 3 1 b n 1 /b n a n /a 1 a n /a 2 a n /a 3 a n /a n 1 1
Convolution on groups When studying convolution on general homogeneous nilpotent Lie groups, we need to impose an additional condition on the matrix E = { e(j, k) } called double monotonicity: each row is weakly increasing from left to right and each column is weakly decreasing from top to bottom. Explicitly e(j, k) e(j, k + 1) and e(j, k) e(j + 1, k). Lemma Suppose E is doubly monotone, and let I = (i 1,..., i n ) Γ Z (E), J = (j 1,..., j n ) Γ Z (E). If ϕ, ψ are normalized bump functions, then [ϕ] I [ψ] J = [θ] K where (a) θ is a normalized bump function; (b) K = (k 1,..., k n ) Γ(E) and k m = max{i m, j m }.
Convolution operators are bounded on L p Theorem Let G = R n be a homogeneous nilpotent Lie group and let E be a doubly monotone matrix. If K P 0 (E) then the operator T K ϕ = ϕ K, defined initially on the Schwartz space S(R n ), extends uniquely to a bounded operator on L p (G) for 1 < p <. In fact, every kernel K P(E) is a flag kernel on an appropriate flag.
P(E) is an algebra under convolution Theorem Let G = R n be a homogeneous nilpotent Lie group and let E be a doubly monotone matrix. If K, L P 0 (E) then there exists M P 0 (E) such that T K T L = T M. The proof is quite technical and uses the dyadic decomposition of kernels. If S and T are marked partitions, one studies [ϕ I ] I [ψ J ] J. I Γ(E S ) J Γ(E T
Composition of Calderón-Zygmund kernels Let K a, K b be Calderón-Zygmund kernels associated with the dilations λ a (x) = ( λ a1 x 1,..., λ an x n ) λ b (x) = ( λ b1 x 1,..., λ bn x n ). Assume that a k a k+1 and b k b k+1. Theorem Let E = {e(j, k)} where { aj e(j, k) = max, b } j. a k b k The operator T K1 T K2 is given by convolution with a tempered distribution L with L P(E). Final Remarks: 1. For appropriate a and b the rank of E can be as large as n. 2. However L = n 1 j=1 L j where L j P(E j ) and E j has rank 2.