The maximal regularity operator on tent spaces

Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^{2}$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^{p}$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p,2}$ for $p$ in a certain open range. We also study the case $p=\infty$.


Introduction
Let −L be a densely defined closed linear operator acting on L 2 (R n ) and generating a bounded analytic semigroup (e −tL ) t≥0 . We consider the maximal regularity operator defined by for functions f ∈ C c (R + × R n ). The boundedness of this operator on L 2 (R + × R n ) was established by de Simon in [16]. The L p (R + × R n ) case, for 1 < p < ∞, turned out, however, to be much more difficult. In [10], Kalton and Lancien proved that M L could fail to be bounded on L p as soon as p = 2. The necessary and sufficient assumption for L p boundedness was then found by Weis [17] to be a vector-valued strengthening of analyticity, called R-analyticity. As many differential operators L turn out to generate R-analytic semigroups, the L p boundedness of M L has subsequently been successfully used in a variety of PDE situations (see [14] for a survey).
Recently, maximal regularity was used in a different manner as an important tool in [2], where a new approach to boundary value problems with L 2 data for divergence form elliptic systems on Lipschitz domains, is developed. More precisely, in [2], the authors establish and use the boundedness of M L on weighted spaces L 2 (R + × R n ; t β dtdx), for certain values of β ∈ R, under the additional assumption that L has bounded holomorphic functional calculus on L 2 (R n ). This additional assumption was removed in [3,Theorem 1.3]. Here is the version when specializing the Hilbert space to be L 2 (R n ). Theorem 1.1. With L as above, M L extends to a bounded operator on L 2 (R + × R n ; t β dtdx) for all β ∈ (−∞, 1).
The use of these weighted spaces is common in the study of boundary value problems, where they are seen as variants of the tent space T 2,2 which occurs for β = −1, introduced by Coifman, Meyer and Stein in [6]. For p = 2, the corresponding spaces are weighted versions of the tent spaces T p,2 , which are defined, for parameters β ∈ R and m ∈ N, as the completion of C c (R + × R n ) with respect to , the classical case corresponding to β = −1, m = 1, and being denoted simply by T p,2 . The parameter m is used to allow various homogeneities, and thus to make these spaces relevant in the study of differential operators L of order m. To develop an analogue of [2] for L p data, we need, among many other estimates yet to be proved, boundedness results for the maximal operator M L on these tent spaces. This is the purpose of this note. Another motivation is well-posedness of non-autonomous Cauchy problems for operators with varying domains, which will be presented elsewhere. In the latter case, M L can be seen as a model of the evolution operators involved. However, as M L is an important operator on its own, we thought interesting to present this special case alone. In Section 3 we state and prove the adequate boundedness results. The proof is based on recent results and methods developed in [9], building on ideas from [5] and [8]. In Section 2 we recall the relevant material from [9].

Tools
When dealing with tent spaces, the key estimate needed is a change of aperture formula, i.e., a comparison between the T p,2 norm and the norm for some parameter α > 0. Such a result was first established in [6], building on similar estimates in [7], and analogues have since been developed in various contexts. Here we use the following version given in [9,Theorem 4.3].
where τ = min(p, 2) and C depends only on n and p. 1 Theorem 2.1 is actually a special case of the Banach space valued result obtained in [9]. Note, however, that it improves the power of α appearing in the inequality from the n given in [6] to n τ . This is crucial in what follows, and has been shown to be optimal in [9].
Applying this to (t, y) → t m(β+1) 2 f (t m , y) instead of f , we also have the weighted result, where , where τ = min(p, 2) and C depends only on n and p.
To take advantage of this result, one needs to deal with families of operators, that behave nicely with respect to tent norms. As pointed out in [9], this does not mean considering R-bounded families (which means R-analytic semigroups when one considers (tLe −tL ) t≥0 ) as in the L p (R + × R n ) case, but tent bounded ones, i.e. families of operators with the following L 2 off-diagonal decay, also known as Gaffney-Davies estimates. Definition 2.3. A family of bounded linear operators (T t ) t≥0 ⊂ B(L 2 (R n )) is said to satisfy offdiagonal estimates of order M , with homogeneity m, if, for all Borel sets E, F ⊂ R n , all t > 0, and all f ∈ L 2 (R n ): In what follows · 2 denotes the norm in L 2 (R n ).
1 only on pouréviter les confusions As proven, for instance, in [4], many differential operators of order m, such as (for m = 2) divergence form elliptic operators with bounded measurable complex coefficients, are such that (tLe −tL ) t≥0 satisfies off-diagonal estimates of any order, with homogeneity m. This condition can, in fact, be seen as a replacement for the classical gaussian kernel estimates satisfied in the case of more regular coefficients. ∞), and τ = min(p, 2). If (tLe −tL ) t≥0 satisfies off-diagonal estimates of order M > n mτ , with homogeneity m, then M L extends to a bounded operator on T p,2,m (t β dtdy).

Results
Proof. The proof is very much inspired by similar estimates in [5] and [9]. Let f ∈ C c (R + × R n ). Given (t, x) ∈ R + × R n , and j ∈ Z + , we consider Fixing j ≥ 0, k ≥ 1 we first estimate I k,j as follows. For fixed x ∈ R n , In the second inequality, we use Cauchy-Schwarz inequality for the integral with respect to t, the fact that t − s ∼ t for s ∈ ∪ k≥1 [2 −k−1 t, 2 −k t] ⊂ [0, t 2 ] and Fubini's theorem to exchange the integral in t and the integral in y. The next inequality follows from the off-diagonal estimate verified by (t − s)Le −(t−s)L and again the fact that t − s ∼ t. By Corollary 2.2 this gives where τ = min(p, 2). It follows that ∞ k=1 ∞ j=0 I k,j f T p,2,m (t β dtdy) since M > n mτ and n m + 1 − β > 2n mτ (Note that for p ≥ 2, this requires β < 1). We now turn to J 0 and remark that J 0 ≤ R n J 0 (x) Le −(t−s)L (g(s, ·)(y)ds (y)f (s, y). The inside integral can be rewritten as As M L is bounded on L 2 (R + × R n ; t β− n m dydt) by Theorem 1.1 and (e −tL ) t≥0 is uniformly bounded on L 2 (R n ), we get We finally turn to J j , for j ≥ 1. For fixed x ∈ R n , where we have used Cauchy-Schwarz inequality in the second inequality, the off-diagonal estimates and the fact that s ≤ t in the third, Fubini's theorem and the fact that s ≥ t 2 in the fourth, and the change of variable σ = t t−s in the last. An application of Corollary 2.2, then gives and the proof is concluded by summing the estimates.
An end-point result holds for p = ∞. In this context the appropriate tent space consists of functions such that |g(t, x)| 2 dxdt t is a Carleson measure, and is defined as the completion of the space C c (R + × R n ) with respect to We also consider the weighted version defined by Proof. Pick a ball B(z, r 1 m ). Let We want to show that I 2 r n m f 2 T ∞,2 (t β dtdy) . We set For I 0 we use again Theorem 1.1 which implies that M L is bounded on L 2 (R + × R n , t β dxdt). Thus Next, for j = 0, we proceed as in the proof of Theorem 3.1 to obtain Exchanging the order of integration, and using the fact that t ∼ t − s in the first part and that t ∼ s in the second, we have the following.
where we used β < 1. We thus have and the condition M > n 2m allows us to sum these estimates. Remark 3.3. Assuming off-diagonal estimates, instead of kernel estimates, allows to deal with differential operators L with rough coefficients. The harmonic analytic objects associated with L then fall outside the Calderón-Zygmund class, and it is common (see for instance [1]) for their boundedness range to be a proper subset of (1, ∞). Here, our range ( 2n n+m(1−β) , ∞] includes [2, ∞] as β < 1, which is consistent with [2]. In the case of classical tent spaces, i.e., m = 1 and β = −1, it is the range (2 * , ∞], where 2 * denotes the Sobolev exponent 2n n+2 . We do not know, however, if this range is optimal. Remark 3.4. Theorem 3.2 is a maximal regularity result for parabolic Carleson measure norms. This is quite natural from the point of view of non-linear parabolic PDE (where maximal regularity is often used), and such norm have, actually, already been used in the context of Navier-Stokes equations in [11], and, subsequently, for some geometric non-linear PDE in [12]. Theorem 3.1 is also reminiscent of Krylov's Littlewood-Paley estimates [13], and of their recent far-reaching generalization in [15]. In fact, the methods and results from [9], on which this paper relies, use the same circle of ideas (R-boundedness, Kalton-Weis γ multiplier theorem...) as [15]. The combination of these ideas into a "conical square function" approach to stochastic maximal regularity will be the subject of a forthcoming paper.