Proceedings of Symposia in Pure Mathematics
A local T b Theorem for square functions Steve Hofmann Dedicated to Prof. V. Maz’ya on the occasion of his 70th birthday. Abstract. We prove a “local” T b Theorem for square functions, in which we assume only Lq control of the pseudo-accretive system, with q > 1. We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.
1. Introduction, statement of results, history The T b Theorems of McIntosh and Meyer [McM], and of David, Journ´e and Semmes [DJS], are boundedness criteria for singular integrals, by which the L2 boundedness of a singular integral operator T may be deduced from sufficiently good behavior of T on some suitable non-degenerate test function b. A “local T b theorem” is a variant of the standard T b theorem, in which control of the action of the operator T on a single, globally defined accretive test function b, is replaced by local control, on each dyadic cube Q, of the action of T on a test function bQ , which satisfies some uniform, scale invariant Lp bound along with the non-degeneracy condition ! !" ! ! 1 −1 ! (1.1) ≤ |Q| ! bQ !! , C0 Q for some uniform constant C0 . A collection of such local test functions, ranging over all dyadic cubes Q (or over all cubes or balls) is called a “pseudo-accretive system”. The first local T b theorem, in which the local test functions are assumed to belong uniformly to L∞ , is due to M. Christ [Ch], and was motivated in part by applications to the theory of analytic capacity; an extension of Christ’s result to the non-doubling setting is due to Nazarov, Treil and Volberg [NTV]. A more recent version, in which Christ’s L∞ control of the test functions is relaxed to Lq control, appears in [AHMTT], and this sharpened version (see also [AY], and the unpublished manuscript [H]) has found application to the theory of layer potentials associated to divergence form, variable coefficient elliptic PDE (see [AAAHK]). It is also of interest to consider local T b theorems for square functions (as opposed to singular integrals). These have found application to the solution of the 2000 Mathematics Subject Classification. Primary 42B25; Secondary 35J25 . S. Hofmann was supported by the National Science Foundation.
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c !0000 (copyright holder)
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STEVE HOFMANN
Kato problem [HMc], [HLMc], [AHLMcT] (see also [AT] and [S] for related results), and to variable coefficient layer potentials [AAAHK]. In this note, we consider the square function estimate "" dx dt |θt f (x)|2 ≤ C"f "2L2 (Rn ) , (1.2) n+1 t R+ where θt f (x) :=
"
Rn
ψt (x, y)f (y)dy
and {ψt (x, y)}t∈(0,∞) , satisfies, for some exponent α > 0, |ψt (x, y)| ≤ C and (a) (b)
tα (t + |x − y|)n+α
|h|α (t + |x − y|)n+α |h|α |ψt (x + h, y) − ψt (x, y)| ≤ C (t + |x − y|)n+α
|ψt (x, y + h) − ψt (x, y)| ≤ C
(1.3)
(1.4)
whenever |h| ≤ t/2. Our main result in this paper is the following: # Theorem 1.1. Let θt f (x) := ψt (x, y)f (y)dy, where ψt (x, y) satisfies (1.3) and (1.4). Suppose also that there exists a constant C0 < ∞, an exponent q > 1 and a system {bQ } of functions indexed by dyadic cubes Q ⊆ Rn , such that for each dyadic cube Q # (i) Rn |bQ |q !≤ C0 |Q| ! !# ! (ii) C10 |Q| ≤ ! Q bQ ! %q/2 # $# "(Q) (iii) Q 0 |θt bQ (x)|2 dt dx ≤ C0 |Q|. t Then we have the square function bound (1.2).
Here, and in the sequel, we use the notation $(Q) to denote the side length of a cube Q. The case q = 2 of this theorem was already known, and requires only the first inequality in (1.4) (smoothness in the y variable). See [H2] and [A] for explicit formulations in that case, although in fact the result and its proof were already implicit in [HMc], [HLMc] and [AHLMcT]. As mentioned above, analogous results for singular integrals (as opposed to square functions) were obtained for q = ∞ in [Ch] and [NTV], for q > 1 in the “perfect dyadic” case treated in [AHMTT], and, based on [AHMTT], for q = 2 (or q = 2 + %) in the case of standard singular integrals in [AY] and [H]. It remains an open problem to treat the case q < 2 for singular integrals that are not of perfect dyadic type, but rather satisfy standard Calder´on-Zygmund conditions. The paper is organized as follows: in the next section we prove Theorem 1.1, and in Section 3 we present an application to the theory of layer potentials for variable coefficient divergence form operators with bounded measurable non-symmetric coefficients.
A LOCAL T b THEOREM
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2. Proof of Theorem 1.1 We begin by recalling the following well known fact, due explicitly to Christ and Journ´e [CJ], but also implicit in the work of Coifman and Meyer [CM]. # Proposition 2.1. [CJ] Let θt f (x) ≡ Rn ψt (x, y)f (y)dy, where ψt (x, y) satisfies (1.3) and (1.4) (a). Suppose that we have the Carleson measure estimate " "(Q)" 1 dxdt sup |θt 1(x)|2 ≤ C. (2.1) t Q |Q| 0 Q
Then we have the square function bound (1.2).
Remark. The converse direction (i.e. that (1.2) implies (2.1)) is essentially due to Fefferman and Stein [FS]. Thus, to prove Theorem 1.1, it is enough to establish (2.1). In fact, by covering an arbitrary cube by finitely many dyadic cubes of comparable side length, it is enough to establish a version of (2.1) in which the supremum runs over dyadic cubes only. To this end, we shall use the following lemma of “John-Nirenberg” type. Lemma 2.2. Suppose that there exist η ∈ (0, 1) and C1 < ∞, such that for every dyadic cube Q ∈ Rn , there is a family {Qj } of non-overlapping dyadic sub-cubes of Q, with & |Qj | ≤ (1 − η)|Q| (2.2) and
where τQ (x) :=
)
" '" Q
"(Q)
τQ (x)
2 dt
|θt 1(x)|
t
(q/2
(2.3)
dx ≤ C1 |Q|,
1Qj (x) $(Qj ). Then (2.1) holds.
We shall defer momentarily the proof of Lemma 2.2, and proceed to the proof of Theorem 1.1. We may suppose without loss of generality that 1 < q < 2, as the case q > 2 may be reduced to the known case q = 2 by H¨older’s inequality. We claim that, in the spirit of [S] and [AT] (but using also Lemma 2.2), it is enough to prove that for each dyadic cube Q, there is a family {Qj } of non-overlapping dyadic sub-cubes of Q satisfying (2.2) for which (q/2 (q/2 " '" "(Q) " '" "(Q) 2 dt 2 dt |θt 1(x)| dx ≤ C |θt 1(x) At bQ (x)| dx, (2.4) t t τQ (x) Q 0 Q where At denotes the usual dyadic averaging operator, i.e., " −1 f, At f (x) := |Q(x, t)| Q(x,t)
and Q(x, t) denotes the minimal dyadic cube containing x with side length at least t. Indeed, given (2.4), we may follow [CM] and write (1)
θt 1At = (θt 1) (At − Pt ) + (θt 1Pt − θt ) + θt := Rt
(2)
+ Rt
+ θt ,
where Pt is a nice approximate identity, of convolution type, with a smooth, compactly supported kernel. By hypothesis (iii) of Theorem 1.1, the contribution of θt bQ , to the right hand side of (2.4), is controlled by C|Q|, as desired. Moreover,
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STEVE HOFMANN (2)
Rt 1 = 0, and its kernel satisfies (1.3) and (1.4). Thus, by standard LittlewoodPaley/vector-valued Calder´on-Zygmund theory, we have that (q/2 " '" "(Q) (2) 2 dt dx ≤ Cq "bQ "qq ≤ C|Q|, (2.5) |R bQ (x)| t Q 0 where in the last inequality we have used hypothesis (i) of Theorem 1.1. Further(1) more, the same Lq bound holds for Rt (even though (1.4) fails for this term), as may be seen by following the interpolation arguments of [DRdeF]. We omit the details. Thus, the right hand side of (2.4) is bounded by C|Q|, so that the conclusion of Theorem 1.1 then follows by Lemma 2.2 and Proposition 2.1. Therefore, it is enough to establish (2.4), for a family of dyadic sub-cubes of Q satisfying (2.2). To this end, we follow the stopping time arguments in [HMc], [HLMc] and [AHLMcT] (but see also [Ch], where a similar idea had previously appeared). Our starting point is hypothesis (ii) of Theorem 1.1. Dividing by an appropriate complex constant, we may suppose that " 1 bQ = 1. (2.6) |Q| Q
We then sub-divide Q dyadically, to select a family of non-overlapping cubes {Qj } which are maximal with respect to the property that " 1 (e bQ ≤ 1/2. (2.7) |Qj | Qj By the maximality of the cubes in the family {Qj }, it follows that
1 ≤ (e At bQ (x), if t > τQ (x), 2 so that (2.4) holds with C = 2q . It remains only to verify that there exists η > 0 such that |E| > η|Q|, (2.8) where E ≡ Q\(∪Qj ). By (2.6) we have that " " " &" |Q| = bQ = (e bQ = (e bQ + (e Q
≤ |E|1/q
Q
!
*"
Q
|bQ |q
E
+1/q
+
j
bQ
Qj
1& |Qj |, 2
when in the last step we have used (2.7). From hypothesis (i) of Theorem 1.1, we then obtain that ! 1 |Q| ≤ C|E|1/q |Q|1/q + |Q|, 2 and (2.8) now follows readily. This concludes the proof of Theorem 1.1, modulo Lemma 2.2, whose proof we now give. Proof of Lemma 2.2. We begin by stating Lemma 2.3. Suppose that there exist N < ∞ and β ∈ (0, 1) such that for every dyadic cube Q, |{x ∈ Q : gQ (x) > N }| ≤ (1 − β) |Q|, (2.9)
A LOCAL T b THEOREM
where gQ (x) :=
'"
"(Q)
0
Then (2.1) holds.
dt |θt 1(x)|2 t
5
(1/2
.
We take this lemma for granted momentarily, and prove Lemma 2.2. Fix a dyadic cube Q. For a large, but fixed N to be chosen momentarily, let ΩN := {x ∈ Q : gQ (x) > N }.
Under the hypotheses of Lemma 2.2, with E := Q \ (∪Qj ), we have & |ΩN | ≤ |Qj | + |{x ∈ E : gQ (x) > N }| '" (1/2 "(Q) 2 dt ≤ (1 − η) |Q| + |{x ∈ Q : |θt 1(x)| > N }| t τQ (x)
C1 |Q|, Nq where in the last step we have used Tchebychev’s inequality and (2.3). Choosing N so large that C1 /N q ≤ η/2, we obtain (2.9) with β = η/2. Thus, Lemma 2.3 implies Lemma 2.2. In turn, to prove Lemma 2.3, we proceed as follows. We momentarily fix % ∈ (0, 1), and let N, β be as in the hypotheses of Lemma 2.3. For a dyadic cube Q, set '" (1/2 min("(Q),1/$) 2 dt gQ,$ (x) := |θt 1(x)| , t $ ≤ (1 − η) |Q| +
where we take this term to be 0 if $(Q) ≤ %. Define " 1 2 , gQ,$ K(%) := sup Q |Q| Q
where the supremum runs over all dyadic cubes Q. By the truncation, K(%) is finite for each fixed %, and our goal is to show that sup K(%) < ∞.
0 0, Λ < ∞, and for all ξ ∈ Rn+1 , x ∈ Rn . The divergence form equation is interpreted in the weak sense, i.e., we say that Lu = 0 in a domain Ω if 1,2 u ∈ Wloc (Ω) and " A∇u · ∇Ψ = 0
for all Ψ ∈ C0∞ (Ω). Although the case of real symmetric “radially independent” (i.e., in our context, t-independent) coefficients is now rather well understood (see [JK], [KP], [K] and also [AAAHK]), in general it remains an open problem to establish solvability results, with Lp data, for boundary value problems associated to non-symmetric equations in Rn+1 ± . However, in the case n = 1, i.e., in the domains R2± , solvability of the Lp Dirichlet problem (“Dp ”) has been established in [KKPT] for p sufficiently large (but finite), while solvability of the Lp Neumann (“Np ”) and Regularity (“Rp ”) problems with p near 1 (in fact, dual to the Dirichlet exponent) was obtained in [KR]. We refer to those papers for detailed statements of the boundary value problems (this is not our main emphasis here), but we note that solvability of Dp is equivalent to a scale invariant Lq bound, p−1 + q −1 = 1, for the Poisson kernel. To be precise, fix a boundary cube Q ⊂ Rn , and let A± Q denote the upper and lower “corkscrew points” associated to Q, i.e., if xQ denotes the center of Q, then n+1 A± Q := (xQ , ±$(Q)) ∈ R± . For a point X ∈ Rn+1 (we shall adopt the notational convention that capital letters ± X denote the X := (x, t), Y := (y, s) may be used to denote points in Rn+1 ), let kL,± n+1 Poisson kernel for L with pole at X in R± . It turns out that Dp is solvable for L in the domain Rn+1 if and only if there is a constant B such that the following ± scale invariant bound holds for every cube Q ⊂ Rn : " * ± +q AQ kL,± ≤ B |Q|1−q , (3.2) Rn
with p + q = 1 (see, e.g., [KKPT] or [K]). It is shown in [KKPT] that for every L as above in R2± , there is a q := q(L) > 1 such that (3.2) holds. The proof in [KR] of the solvability of Np and Rp , in R2± , with p near 1, uses in a crucial way the L2 boundedness (but not invertibility) of the layer potentials associated to L. In this paper, we present an alternative (and rather short) proof −1
−1
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STEVE HOFMANN
of this boundedness, based on the local T b Theorem proved in Section 1. The proof in [KR] also uses T b theory (to be precise, the result of [DJS]), but is tied very closely to the 2-dimensionality of the domain. Our proof is in principle not dimension dependent, but rather relies only on the Poisson kernel estimate (3.2). Of course, at present, (3.2) is known to hold for non-symmetric operators only when n + 1 = 2. We conjecture that (3.2) remains true for non-symmetric t-independent operators in all dimensions. The idea to use estimates like (3.2) to prove layer potential bounds (in the setting of symmetric coefficients) has appeared previously in [AAAHK], but the argument there was limited to the case q = 2, as it depended on the local T b theorem for singular integrals [AHMTT], as extended to standard Calder´on-Zygmund operators in [H] or [AY]. In the present paper, we will use the square function/non-tangential maximal function estimates of [DJK] to reduce matters to square functions treatable via Theorem 1.1. We now recall the method of layer potentials. For L as above, let L∗ := − div A∗ ∇ denote the transpose operator (which is also the adjoint, since we are dealing with real coefficients here), and let Γ(X, Y ) and Γ∗ (X, Y ) := Γ(Y, X) denote the corresponding fundamental solutions in Rn+1 . Thus, LX Γ(X, Y ) = δY , L∗Y Γ∗ (Y, X) := L∗Y Γ(X, Y ) = δX , where δX denotes the Dirac mass at the point X. By the t-independence of our coefficients, we have that Γ(x, t, y, s) = Γ(x, t − s, y, 0).
(3.3)
We define the single layer potential operators for L and L by " Γ(x, t, y, 0) f (y) dy, t ∈ R St f (x) ≡ Rn " Γ∗ (x, t, y, 0) f (y) dy, t ∈ R, St∗ f (x) ≡ ∗
(3.4)
Rn St∗
(we apologize for this notation: denotes the single layer potential for L∗ , and is not, in general, equal to the adjoint of St ) and our goal is to show that (3.5)
sup "∇x,t St f "L2 (Rn ) + sup "∇x,t St∗ f "L2 (Rn ) ≤ C"f "L2 (Rn ) t
t
(the latter estimate implies L2 bounds for the corresponding double layer potentials via duality). To be precise, we have the following Theorem 3.1. Suppose that L is an operator of the type described above and A±
Q that there are exponents q(L), q(L∗ ) > 1 and a constant B such that kL,± and
A±
kL∗Q,± (the Poisson kernels for L and L∗ , respectively) satisfy (3.2) for every cube Q ⊂ Rn . Then the layer potential bound (3.5) holds, with a constant depending only on dimension, λ, Λ, B and min (q(L), q(L∗ )). Remarks. In particular, since (3.2) always holds for such operators when n = 1 [KKPT], we recover the boundedness result of [KR]. We also observe that our proof will require that (3.2) hold for both L and L∗ , even if we restrict our attention to the bound for St . Proof. We begin with some preliminary reductions. We treat only St in the case t > 0, as the same argument carries over mutatis mutandi to the case t < 0
A LOCAL T b THEOREM
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and to St∗ . By Lemma 5.2 of [AAAHK], it suffices to prove sup "∂t St f "L2 (Rn ) ≤ C"f "L2 (Rn ) .
(3.6)
t
and
""
Rn+1 ±
|t ∇x,t ∂t St f (x)|2
dxdt ≤ C"f "2L2 (Rn ) . |t|
Moreover, the same lemma shows that (3.7) follows from "" dxdt |t (∂t )2 St f (x)|2 ≤ C"f "2L2 (Rn ) . n+1 |t| R±
(3.7)
(3.8)
A+
Q In addition, from (3.2) for kL,+ , along with the results of [DJK] applied to the solution u(x, t) := ∂t St f (x), we have that (3.7) implies (3.6) (we use here that the arguments of [DJK] carry over to the non-symmetric case - see the comments in the introduction to [KKPT]). Thus, it is enough to prove (3.8). To this end, we first note that by [GW] (if n + 1 ≥ 3), or (if n + 1 = 2) as a consequence of the Gaussian bounds and local H¨ older continuity of the kernel of the heat semigroup e−τ L (see, for example [AMT]), we have that
ψt (x, y) := t (∂t )2 Γ(x, t, y, 0), the kernel of θt := t (∂t )2 St , satisfies (1.3) and (1.4). Thus, it is enough to construct a pseudo-accretive system {bQ } satisfying the hypotheses of Theorem 1.1. We now set A−
bQ ≡ |Q| kL∗Q,− .
Observe that condition (i) of Theorem 1.1 follows immediately from (3.2). Moreover (ii) is an immediate consequence of the following well known estimate of Caffarelli, Fabes, Mortola and Salsa [CFMS], extended to the case of non-symmetric coefficients (as may be done: see the comments in [KKPT] concerning the validity of the results of [CFMS] in the non-symmetric setting): " A− 1 kL∗Q,− (y) dy ≥ . C Q + It remains to establish condition (iii) of Theorem 1.1. Let (x, t) ∈ RQ ≡ n+1 Q × (0, $(Q)). Then, since for fixed (x, t) ∈ R+ , we have that ∂t2 Γ(x, t, ·, ·) is a solution of L∗ u = 0 in Rn+1 − , we obtain
|θt bQ (x)| = |Q| t|
"
A−
(∂t )2 Γ(x, t, y, 0) kL∗Q,− (y) dy| = |Q| t |(∂t )2 Γ(x, t, A− Q )| t t |ψt+"(Q) (x, xQ )| ≤ C . = |Q| t + $(Q) $(Q)
where in the last two steps we have used (3.3) and then (1.3). Hypothesis (iii) now follows readily. This concludes the proof of Theorem 3.1. !
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STEVE HOFMANN
References [AAAHK] M. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann and S. Kim, Analyticity of layer potentials and L2 solvability of boundary value problems for divergence form elliptic equations with complex L∞ coefficients, preprint. [A] P. Auscher, Lectures on the Kato square root problem, Surveys in analysis and operator theory (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ. 40, Austral. Nat. Univ., Canberra, 2002, pp. 1–18. [AHLMcT] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian, The solution of the Kato Square Root Problem for Second Order Elliptic operators on Rn , Annals of Math., 156 (2002), 633–654. [AHMTT] P. Auscher, S. Hofmann, C. Muscalu, T. Tao, C. Thiele, Carleson measures, trees, extrapolation, and T (b) theorems, Publ. Mat., 46 (2002), no. 2, 257–325. [AMT] P. Auscher, A. McIntosh and P. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Functional Analysis, 152 (1998), 22-73. [AT] P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Ast´ erisque Vol. 249 (1998), Soci´ et´ e Math´ ematique de France. [AY] P. Auscher and Q. X. Yang, On local T (b) Theorems, preprint. [CFMS] L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), no. 4, 621–640. [Ch] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloquium Mathematicum, LX/LXI (1990) 601-628. [CJ] M. Christ and J.-L. Journ´ e, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), no. 1-2, 51–80. [CM] R. Coifman and Y. Meyer, Non-linear harmonic analysis and PDE, E. M. Stein, editor, Beijing Lectures in Harmonic Analysis, vol. 112, Annals of Math. Studies, Princeton Univ. Press, 1986. [DJK] B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat., 22 (1984), no. 1, 97–108. [DJS] G. David, J.-L. Journ´ e, and S. Semmes, Op´ erateurs de Calder´ on -Zygmund, fonctions para-accr´ etives et interpolation, Rev. Mat. Iberoamericana, 1 1–56, 1985. [DRdeF] J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561. [FS] C. Fefferman, and E. M. Stein, H p spaces of several variables, Acta Math., 129 (1972), no. 3-4, 137–193. [GW] M. Gr¨ uter and K. O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342. [H] S. Hofmann, A proof of the local T b Theorem for standard Calder´ on-Zygmund operators, unpublished manuscript, http://www.math.missouri.edu/ e hofmann/ [H2] S. Hofmann, Local T b Theorems and applications in PDE, Proceedings of the ICM Madrid, Vol. II, pp. 1375-1392, European Math. Soc., 2006. [HLMc] S. Hofmann, M. Lacey and A. Mc Intosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds, Annals of Math., 156 (2002), 623631. [HMc] S. Hofmann and A. McIntosh, The solution of the Kato problem in two dimensions, Proceedings of the Conference on Harmonic Analysis and PDE held in El Escorial, Spain in July 2000, Publ. Mat., Vol. extra, 2002, pp. 143-160. [JK] D. Jerison and C. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2), 113 (1981), no. 2, 367–382. [K] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, 83. Published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1994 [KKPT] C. Kenig, H. Koch, H. J. Pipher and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math., 153 (2000), no. 2, 231–298.
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[KP] C. Kenig and J. Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math., 113 (1993), no. 3, 447–509. [KR] C. Kenig and D. Rule, The regularity and Neumann problems for non-symmetric elliptic operators, preprint. [McM] A. Mc Intosh and Y. Meyer, Alg` ebres d’op´ erateurs d´ efinis par des int´ egrales singuli` eres, C. R. Acad. Sci. Paris, 301 S´ erie 1 395–397, 1985. [NTV] F. Nazarov, S. Treil and A. Volberg, Accretive system T b-theorems on nonhomogeneous spaces, Duke Math. J., 113 (2002), no. 2, 259–312. [S] S. Semmes, Square function estimates and the T (b) Theorem, Proc. Amer. Math. Soc., 110 (1990), no. 3, 721–726. Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA E-mail address:
[email protected]
