BOUNDEDNESS AND APPLICATIONS OF SINGULAR INTEGRALS AND SQUARE FUNCTIONS: A SURVEY STEVE HOFMANN AND ALAN MCINTOSH Abstract. We present a survey of certain aspects of the theory of singular inte- grals and square functions, with emphasis on L2 boundedness criteria and recent applications in partial differential equations. Contents 1. Introduction1 1.1. Singular Integrals2 1.2. Square Functions5 1.3. Notation7 2. Lp and Endpoint Theory of SIOs and Square Functions8 3. L2 Boundedness Criteria 13 3.1. The Convolution Case 13 3.2. The Non-Convolution Case 14 4. Local Tb Theorems and Applications 22 4.1. Local Tb Theorems for Square Functions 26 4.2. Application to the Kato Square Root Problem 30 5. Further Results and Recent Progress 35 5.1. Tb Theory for SIOs 35 5.2. Local Tb Theory for Square Functions and Applications 36 References 37 1. Introduction We survey those aspects of the theory of singular integral operators which have been obtained since the pioneering work of Zygmund, Calderon´ and Mikhlin, con- cerning the Calderon´ program as developed by Coifman and Meyer. Key results in this development include the Calderon´ commutator theorem, L2 bounds on the higher commutators and on the Cauchy integral on Lipchitz curves, the solutions of the Painleve´ problem on analytic capacity and the Kato square root problem for 2000 Mathematics Subject Classification. Primary 42B20, 42B25; Secondary 35J25, 35J55, 47F05, 47B44. Steve Hofmann was supported by the National Science Foundation; Alan McIntosh was supported by the Australian Government through the Australian Research Council. 1 2 S. HOFMANN AND A. MCINTOSH elliptic operators, along with further applications to analytic capacity and partial differential equations. Our emphasis is on L2 boundedness criteria for singular integrals, commonly known as T1 , Tb and local Tb theorems, which have arisen from and contributed to the above-mentioned program. We conclude the survey with a discussion of some recent progress and applications. For the classical theory of singular integrals and square functions, we refer the reader to the excellent monographs of Stein [St3, St4], and of Christ [Ch2]. 1.1. Singular Integrals. A singular integral operator (SIO) in Rn (in the gener- alized sense of Coifman and Meyer), is a linear mapping T from test functions n 1 n 0 n D(R ):= C0 (R ) into distributions D (R ); which is associated to a Calderon-´ Zygmund kernel K(x; y); in the sense that (1.1) hT'; i = (x) K(x; y) '(y) dydx 1 n whenever '; 2 C0 (R ) with disjoint" supports. A Calder´on-Zygmundkernel is one which satisfies the standard size and Holder¨ bounds (1.2) jK(x; y)j ≤ C jx − yj−n and jhjα (1.3) jK(x; y + h) − K(x; y)j + jK(x + h; y) − K(x; y)j ≤ C jx − yjn+α for some α 2 (0; 1]; whenever 2jhj ≤ jx − yj . For now, let us take the point of view that K : Rn × Rn n fx = yg ! C , although in the sequel we shall also mention the case that the range of K is, more generally, a Hilbert space. We remark that, given a closed cube Q ⊂ Rn , T extends to a bounded linear mapping from L2(Q) into L2(Rn n Q) , with the representation Z (1.4) T f (x) = K(x; y) f (y) dy Q for all f 2 L2(Q) and all x 2 Rn n Q . Indeed, this follows readily from the kernel estimate (1.2), and the Hardy inequality Z Z 1 2 Z j j ≤ j j2 n f (y) dy dx Cn f (x) dx ; RnnQ Q jx − yj Rn along with (1.1) and a density argument. The theory can be extended to settings other than Euclidean space, and there are worthwhile reasons for doing so, but for most of this survey we shall just consider functions defined on Rn , for the sake of simplicity of exposition. Let us now mention several examples. The Hilbert transform 1 Z 1 1 Z 1 (1.5) H f (x):= p:v: f (y) dy := lim f (y) dy ; π R x − y "!0 π jx−yj>" x − y relates the real and imaginary parts of a holomorphic function F in the half-space 2 C+ := R+ := f(x; t) 2 R × (0; 1)g , by the formula H(<e F(·; t)) = −=m F(·; t), assuming adequate integrability of F on horizontal slices (say F(·; t) is uniformly SINGULAR INTEGRALS AND SQUARE FUNCTIONS 3 in Lp(R) for some p 2 (1; 1) ). Here, the convergence of the principal value limit holds pointwise a.e. and in Lp , for f 2 Lp; 1 < p < 1 . We shall not explore pointwise convergence further in the present survey, but see, e.g., [St3], Chapters II-III, and [St4], Chapter I, Section 7. In higher dimensions, the operators analogous to H are the Riesz transforms Z 2 x j − y j (1.6) R f (x):= p:v: f (y) dy ; j = 1; 2; :::; n ; j n+1 σn Rn jx − yj n+1 where σn is the volume of the unit n -sphere in R . The Riesz transforms relate the tangential and normal derivatives of a harmonic function u in the half-space n+1 n R+ := f(x; t) 2 R × (0; 1)g , via the formula R j(@tu(·; t)) = @x j u(·; t) , assuming, say, u(·; t) 2 Lp(Rn): They also arise naturally in the study of W2;p regularity of solutions of Poisson’s equation ∆u = f in Rn (see [St3], Chapter III). We observe that the two examples (1.5) and (1.6) are both of convolution type, i.e., K(x; y) = K(x − y) . We shall discuss convolution operators further in Section 3.1. We now mention some examples that are not of convolution type. The Calder´on Commutators are the operators Z !k k i A(x) − A(y) 1 (1.7) CA f (x):= p:v: f (y) dy 2π R x − y x − y where A is a Lipschitz function. Observe that, up to normalization, the case k = 0 1 is the Hilbert transform, and that at least formally, CA is a commutator: i h i i (1.8) C1 f = d H; A f := d (HA f ) − A d (H f ) ; A 2 dx 2 dx dx k while CA is a higher commutator ( k = 2; 3;::: ): " "" # # # i dk Ck f = ::: H; A A ::: A f : A k!2 dxk 1 The operator CA (and its higher dimensional analogues) arose in Calderon’s´ construction of an algebra of SIOs suitable for the treatment of partial differential operators with merely Lipschitz coefficients, thus, a sort of pseudo-differential cal- culus which, in contrast to the classical pseudo-differential calculus, was applicable to operators with rather minimally smooth coefficients [Ca2]. k Moreover, the family of operators CA arise in the power series expansion of the operator i Z 1 (1.9) CA f (x):= p:v: f (y) dy ; 2π R x − y + i(A(x) − A(y)) namely 1 X k k CA = (−i) CA ; k=0 0 at least when kA k1 < 1 . In turn, the operator CA arises when writing the paramet- ric representation of the Cauchy singular integral operator on a Lipschitz graph. 4 S. HOFMANN AND A. MCINTOSH More precisely, set i Z 1 Cγg(z):= p:v: g(v) dv : 2π γ z − v If γ is a Lipschitz curve in the complex plane C parametrized by z = x + iA(x), then (1.10) CA f (x) = Cγg(x + iA(x)) ; where f (y):= (1 + iA0(y))g(y + iA(y)) : Of course, the role of the Cauchy integral in complex function theory is well known. We observe that, for A Lipschitz, the kernels K(x; y) = (A(x)−A(y))k=(x− k+1 −1 k y) and K(x; y) = (x − y + i(A(x) − A(y))) , corresponding to the operators CA and CA respectively, satisfy the Calderon-Zygmund´ kernel conditions (1.2) and (1.3), as the reader may readily verify. Calderon’s´ lecture at the International Congress of Mathematicians in Helsinki in 1978 contains a clear account of the state of the art at that time concerning commutators, Cauchy integrals on Lipschitz curves and applications [Ca4]. Let us note that for all of the operators (1.5), (1.6), (1.7) and (1.9), the kernel K(x; y) is anti-symmetric, i.e., (1.11) K(x; y) = −K(y; x): For all anti-symmetric kernels which satisfy the pointwise kernel bound (1.2) jK(x; y)j ≤ Cjx−yj−n , the associated principal value operator is always well defined, at least in the sense of distributions. Indeed, in that case, one may extend the 1 n representation (1.1) as follows. For all '; 2 C0 (R ) (with supports that are not necessarily disjoint), the principal value (1.12) hT'; i := lim (x) K(x; y) '(y) dydx "!0 jx−yj>" exists. Moreover it satisfies the"Weak Boundedness Property (WBP), i.e. there exists C = C(K; n) such that n (1.13) jhT'; ij ≤ CR fk'k1 + Rkr'k1gfk k1 + Rkr k1g for all R > 0 and x 2 Rn , and all test functions '; supported in the ball B(x; R):= fy 2 Rn : jx − yj < Rg: Indeed, to verify (1.12) and (1.13), we use (1.11) and then a re-labelling of the variables to write hT" '; i := (x) K(x; y) '(y) dydx jx−yj>" " = − (x) K(y; x) '(y) dydx = − (y) K(x; y) '(x) dxdy ; jx−yj>" jx−yj>" and thus " " 1 hT" '; i = K(x; y) ( (x)'(y) − (y)'(x)) dydx : 2 jx−yj>" " SINGULAR INTEGRALS AND SQUARE FUNCTIONS 5 Written this way, the integrand is only weakly singular, in the sense that the kernel bound (1.2) jK(x; y)j ≤ Cjx − yj−n has been improved to jK(x; y)( (x)'(y) − (y)'(x))j ≤ jK(x; y) (x)('(y) − '(x)) + '(x)( (x) − (y))j 0 −n+1 ≤ C jx − yj fk k1kr'k1 + k'k1kr k1g from which it is easy to deduce convergence of the limit in (1.12), along with the bound (1.13).