TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY. To Tadeusz Iwaniec on his 60th birthday

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1 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY GIUSEPPE MINGIONE To Tadeusz Iwaniec on his 60th birthday Contents 1. Road map 1 2. The linear world 2 3. Iwaniec opens the non-linear world 8 4. Review on measure data problems Nonlinear Adams theorems Beyond gradient integrability 36 References Road map Calderón-Zygmund theory is a classical topic in the analysis of partial differential equations, and deals with determining, possibly in a sharp way, the integrability and differentiability properties of solutions to elliptic and parabolic equations, and especially of their highest order derivatives, once an initial, analogous information is known on the given datum involved. Now it happens that while a linear theory has been developed in a quite satisfactory way, a complete theory for gradient estimates for solutions to quasilinear equations of the type (1.1) div a(x, Du) = suitable right hand side is not yet developed, at least up to that complete extent one could wish for. The reason for such a difference lies of course in that linear structures allow, via certain explicit representation formulas, for applying rather abstract tools from Harmonic Analysis, semigroup theory, abstract Functional Analysis, interpolation theory and so forth. The use of all such tools is clearly ruled out in the case of non-linear structures as in (1.1), basically because representation formulas are not available, and more in general because no linear or sub-linear operator can be associated to the non-linear problem in question. The purpose of this paper is now to collect all those pieces - that is the available theorems - that put together should form what we may call a non-linear Calderón-Zygmund theory. In fact we shall review some recent and less recent results concerning the integrability and (weak) differentiability properties of solutions to non-homogeneous equations involving operators of the type in (1.1), with a final emphasis on the content of a couple of recent papers we wrote [105, 106]. We would like to remark that a very deep, fully non-linear Calderón-Zygmund theory for fully non linear problems of the type (1.2) F (x, D 2 u) = f is available, being a fundamental contribution of Caffarelli - see [32, 31]. For obvious reasons the phenomena and the techniques involved for the case (1.2), where 1

2 2 GIUSEPPE MINGIONE solutions are intended in the viscosity sense and equations cannot be differentiated, are quite different from the divergence form/variational case (1.1), where a notion of weak solution in the integral sense is adopted. For this reason we will not touch the theory available for operators of the type (1.2), instead we refer the reader to the excellent monograph [33]. On the other hand, while the theory for (1.2) is essentially scalar, due to the fact that a vectorial analog of viscosity solutions has not been developed yet, we shall see that in the quasilinear case (1.1) systems of PDE can be nevertheless dealt with, at least up to certain extent. A bridge between the viscosity methods and quasilinear structures has been anyway built in [34], a paper who eventually inspired the proof of many results for divergence form operators. General notation. From now on, and for the rest of the paper, we will use CZ as an acronym for Calderón-Zygmund. By Ω we denote a bounded open domain of R n, with n 2. We shall denote by B R (x 0 ) the open ball in R n of radius R and center x 0, that is B R (x 0 ) = {x R n : x x 0 < R}. When the center will be unimportant we shall simply denote B R (x 0 ) B R. With B R R n being a a ball with positive and finite radius, if g : B R R k is an integrable map, the average of g over B R is (g) BR := g(x) dx := 1 g(x) dx. B R B R B R When considering a function space X(Ω, R k ) of possibly vector valued measurable maps defined on an open set Ω R n, with k N, e.g.: L p (Ω, R k ), W β,p (Ω, R k ), we shall define in a canonic way the local variant X loc (Ω, R k ) as that space of maps f : Ω R k such that f X(Ω, R k ), for every Ω Ω. Moreover, also in the case f is vector valued, that is k > 1, we shall also use the short hand notation X(Ω, R k ) X(Ω), or even X(Ω, R k ) X when the domain is not important, and we want to emphasize a qualitative property. Finally, several times, in order to simplify the exposition, we shall not specify the domain of integration when stating certain results; in such cases we shall mean that the domain is not important, or that the result in question holds in a local way, and then also up to the boundary provided suitable boundary conditions are made. Other times, the domain considered will be simply the whole space R n. Acknowledgments. This research is supported by the ERC grant Vectorial Problems. 2. The linear world The material in this section is classical, and we are just giving a short survey of results in order to settle a background of linear results to later present in a more efficient way the forthcoming non-linear ones. We shall prefer here a more informal presentation, not giving all the details but rather aiming to give a general overview. References for the next results and to more details can be for instance [53, 64, 116] Lebesgue spaces. The basic example is at this stage the Poisson equation (2.1) u = f, which for simplicity we shall initially consider in the whole R n, for n 2; here f is the datum. The previous equation, as all the other ones considered in this section is satisfied in the distributional sense, while all solutions are supposed to be at least of class W 1,1.

3 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 3 A classical result, going back to the fundamental work of Calderón & Zygmund asserts the solvability in the right function spaces (2.2) f L γ = D 2 u L γ for every γ > 1. Of course the last result comes along together with an a priori estimate (2.3) D 2 u L γ f L γ for every γ > 1. We remark that well-known counterexamples show that the same implication is false for γ = 1, even locally. As a consequence of (2.2) and of Sobolev embedding theorem we have also (2.4) f L γ = Du L nγ n γ for every γ (1, n). The classical proof of (2.2)-(2.4) goes via a representation formula involving the so called fundamental solution, that is the Green s function, say for n 2 (2.5) u(x) G(x, y)f(y) dy where (2.6) G(x, y) { x y 2 n if n 3 log x y if n = 2. Then, after differentiating twice (2.5) one arrives at a new representation formula (2.7) D 2 u(x) K(x y)f(y) dy where now K(, ) is a so called CZ kernel, that is (2.8) ˆK L B, where ˆK denotes the Fourier transform of K( ), and moreover the following Hörmander cancelation condition holds: (2.9) K(x y) K(x) dx B for every y R n. x 2 y A this point the standard CZ theory of singular integrals comes into the play: the linear operator f I 0 (f) where, in fact (2.10) I 0 (f)(x) := K(x y)f(y) dy, is bounded from L γ to L γ, for every γ (1, ) and therefore (2.2) follows from (2.7). Related a priori estimates for solutions to (2.1) follow from the a priori bounds on I 0 in the various function spaces involved. The crucial point in the Calderón-Zygmund theory of singular integrals is that although the kernel K( ) is singular - i.e. not integrable - in the sense that K(x) 1 x n, condition (2.9) encodes enough cancelations to ensure the convergence of I 0 (f) in L γ for γ > 1. Related to the equation (2.1) is the following one: (2.11) u = div F, and since this can be re-written as div Du = div F,

4 4 GIUSEPPE MINGIONE we expect, for homogeneity reasons - that is Du scales as F - that Du enjoys the same integrability of F. This is the case, indeed it holds that (2.12) Du L γ F L γ for every γ > 1, as desired, and this is again achievable via the use of singular integrals [68, Proporition 1]. As mentioned above, (2.4) follows from (2.2) applying Sobolev embedding theorem, but there is another, more direct way to get it, without appealing to the the theory of singular integrals, but rather relying on a lighter one: that of Riesz potentials, also called fractional integrals. In fact, once again starting from (2.5), but differentiating it once we gain yet another representation formula (2.13) Du(x) K 1 (x, y)f(y) dx, where, accordingly (2.14) K 1 (x, y) 1 x y n 1. This motivates the introduction of so called fractional integrals. Definition 2.1. Let β [0, n); the linear operator, acting on measurable functions and defined by f(y) I β (f)(x) := dy, x y n β is called the β-riesz potential of f. R n Needless to say it is possible to define the action of Riesz potentials over measures with finite total mass as follows as dµ(y) I β (µ)(x) := x y n β. As a matter of fact the following theorem holds: Theorem 2.1 ([66]). Let β [0, n); for every γ > 1 such that βγ < n we have (2.15) I β (f) L nγ n βγ (R n ) c(n, β, γ) f L γ (R n ). R n See also [99, Theorem 1.33] for a proof. At this point the derivation of (2.4) is straightforward from (2.13), (2.14) and (2.15). This is not a surprise since a less general version of Theorem 2.1 was used - and actually re-derived - by Sobolev in order to prove his celebrated embedding theorem for the case t > 1 - the one for the case t = 1 actually necessitates other tools, that is the celebrated Gagliardo- Nirenberg inequalities. Remark 2.1. There is a basic difference between fractional and singular integrals: in the theory of fractional integrals one does not use cancelation properties of the kernel as the one in (2.9). Indeed estimate (2.15) only uses the size of x y β n, and the constant c(n, β, γ) blows-up for β 0, if using the technique leading to Theorem 2.1. In other words the crucial fact is that for β > 0 the function is locally integrable in R n. 1 x n β

5 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY The borderline case γ = 1. A natural question raises now: What happens in the borderline case γ = 1? The results presented up to now fail, but an answer can be nevertheless obtained by considering suitable functions spaces. Let me start by the so called Marcinkiewic spaces M t (A, R k ), also called Lorentz spaces and denoted by L t, (A), or by L t w(a), when they are called weak-l t spaces, or Lorentz spaces - see Definition 5.4 below. Definition 2.2. Let t 1. A measurable map w : Ω R k belongs to M t (Ω, R k ) M t (Ω) iff (2.16) sup λ t {x A : w > λ} =: w t M t (Ω) <. λ 0 It turns out that linear CZ integral operators send L 1 into M 1 ; therefore, in the borderline case γ = 1 estimates (2.3) and (2.12) turn to (2.17) D 2 u M 1 f L 1, and (2.18) Du M 1 F L 1, respectively, while (2.15) turns to (2.19) I β (f) M n n β c(n, β) f L 1, so that for the Poisson equation (2.1) it holds that (2.20) f L 1 = Du M n n 1. For the limiting embedding property of Riesz potential see the classical paper of Adams [6]; this paper contains results we shall examine in greater detail later. Remark 2.2. There is a problem in (2.17)-(2.18); these have to be thought as a priori estimates, since the derivatives involved there are not the distributional ones. To avoid complications, we shall consider (2.17)-(2.18) when f, F are smooth, and therefore u is also smooth, and we shall retain (2.17)-(2.18) only in such a qualitative form. The importance of the space M t clearly lies in the fact that it serves to describe in a sharp way certain limiting integrability situations, very often occurring in modern non-linear analysis, as those given by potential functions. In fact, the prototype of M t functions is given by the potential x n/t ; note that 1 (2.21) M t (B x n 1 ) \ L t (B 1 ), for every t 1. t In general the following inclusions hold: (2.22) L t M t L t ε for every ε > 0. As for the first one, observe that { w > λ} = { w >λ} { w >λ} dx w t λ t dx (2.23) so that w L t λ t w M t w L t holds, and in fact the estimation in (2.23) motivates the definition of Marcinkiewicz spaces. Marcinkiewicz spaces are nowadays of crucial importance in the analysis of

6 6 GIUSEPPE MINGIONE problems with critical non-linearities as those involving harmonic, p-harmonic, and bi-harmonic maps, Euler equations and other pdes from fluid-dynamics. Inclusions (2.22) tell us that Marcinkiewicz spaces interpolate Lebesgue spaces; we shall see later a more refined way of interpolating Lebesgue spaces, when Lorentz spaces will be introduced, extending both Marcinkiewicz and Lebesgue spaces, and introducing finer scales for measuring the size of a function. For more on Marcinkiewicz spaces we refer to Section 5.3 below. The second natural question is now to find a condition for which in estimates (2.17)-(2.18) we can obtain the full Lebesgue integrability scale instead of the Marcinkiewicz one. The question is the same that finding a function space slightly smaller than L 1, which is mapped into L 1 by singular integral operators. For the developments we are concerned with the answer is actually doublefold. In order to replace M 1 by L 1 in the left hand side of (2.17) one has to consider a slightly larger space for the right hand datum f, called Hardy space and denoted by H 1. The elements of such space are functions enjoying enough cancelation properties to match with those by singular integrals and finally yielding convergence in L 1. The story of the Hardy spaces is essentially a complex function theory one until the fifties, when a completely real function theory characterization of such spaces was settle down mainly by authors like Stein and Weiss; see the classical paper of Fefferman & Stein [58]. A central concept is the one of atomic decomposition, initially due to Coifman [38], see also [92], allowing to give a particularly simple definition which we adopt here. We recall that an atom a over a cube Q is a function such that supp a( ) Q and moreover a(x) dx = 0 a L 1 Q, Q hold. The atom a( ) can actually be thought as a bump function exhibiting cancelations. Then a function f belongs to the Hardy H 1 (R n ) iff there exists a sequence of atoms {a k } such that the following atomic decomposition holds: (2.24) w(x) = λ k a k (x) and λ k <. k N The inf of the sums λ k over all possible such decompositions naturally defines the Hardy space norm of w. We won t any longer deal with Hardy spaces here; this in another story, too much unrelated to the non-linear setting we are going to switch to in the next sections. As a matter of fact the atomic decompositions of Hardy functions allows a perfect match with the cancelations properties of the CZ kernel. In fact, using both the cancelation properties of CZ kernels and the zero-average property of a over the cube Q, it is easy to see that I 0 (a) L 1 c, where the constant c ultimately depends on the constant B occurring in (2.8)-(2.9), but is actually independent of the atom considered a. Using this fact, and the very definition of Hardy spaces, that is the decomposition in (2.24), the boundedness in H 1 follows: I 0 (f) L 1 f H 1. Such a proof cannot be apparently extended to the case of non-linear operators, in that such a kind of delicate cancelation phenomena are apparently destroyed by general non-linearities, in other words, non-linear operators seems not to read cancelation properties of the Hardy functions. It remains to try with additional size conditions. The space L log L(Ω), with Ω R n being a bounded domain, is therefore defined as those of the functions f satisfying w log(e + w ) dx <. Ω k N

7 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 7 Such space, a particularly important instance of what are called Orlicz spaces, becomes a Banach space when equipped with the following Luxemburg norm: { (2.25) w L log L(Ω) := inf λ > 0 : w ( log e + w ) } dx 1 <. λ λ Note that for homogeneity reasons that will be clear later we have incorporated in the above definition a dependence on the measure Ω, by considering an averaged integral in (2.25). The following equivalence due to T. Iwaniec [73]: (2.26) w L log L(Ω) w log Ω ( e + Ω A ) w dx =: w L log L(Ω), w(y) dy and the striking fact is that the last quantity actually defines a true norm in L log L(Ω), which is therefore equivalent to the usual Luxemburg one (2.25) via a constant independent of the domain Ω. An obvious consequence of the definitions above is the following inclusion: L log L(Ω) L 1 (Ω) and w dx w L log L(Ω). As a matter of fact the space L log L is sent into L 1 by singular integrals operators Ω I 0 (f) L 1 f L log L. as a consequence we have limiting L 1 -estimates in (2.3) and (2.12): these turn to (2.27) D 2 u L 1 f L log L, and (2.28) Du L 1 F L log L, respectively, while (2.15) turns to (2.29) I β (f) L n n β c f L log L, so that for the Poisson equation (2.1) it holds that (2.30) f L log L = Du L n n 1. For the last two results we again refer to Adams paper [6] Perturbations of the linear theory. The results on the differentiability properties explained in Section 2.1 extend to linear elliptic equations of the type (2.31) AD 2 u = f where A is a constant coefficients matrix such that ν λ 2 Aλ, λ L λ 2 holds for any λ R n. In fact, up to changing coordinates, the equation (2.31) behaves as the usual Poisson equation. The same results hold for equations with variable, continuous coefficients, that is A A(x) C 0 (R n, R n2 ); in fact in this case, due to the continuity of the coefficients, the equation can be considered as a local perturbation of the Laplace operator, and CZ estimates follow by mean of local perturbation and fixed point arguments; see for instance [62, Paragraph 10.4]. It is now obvious that the same results cannot hold when the coefficient matrix A(x) is just measurable, due to well-known counterexamples. Anyway certain types of mild discontinuities for the matrix A( ) can still be allowed. This is the case when the entries of A(x) are assumed to be VMO functions - see Section 3 below. These are functions which are basically continuous up to an averaging process. The first results in this direction have been obtained by Chiarenza & Frasca & Longo [39] by using a few deep theorems from Harmonic Analysis on the boundedness of commutator operators involving BMO coefficients. This strategy, applied

8 8 GIUSEPPE MINGIONE to a variety of problems with different boundary conditions, heavily relies on the linearity of the problem considered, in that it is based on the use of representation formulas via fundamental solutions. This approach, relying on a series of sophisticated tools, can be actually completely bypassed, via the use of suitable maximal function operators, as we shall see more in detail in Section 3.3 below; moreover various type of boundary value problems with very weak regularity assumptions on the boundary are treatable, see for instance [28] and related references Extremals of the linear theory. In the last years the problem of determining a CZ for linear problem not involving treatable kernels - for instance kernels which are not regular enough to satisfy Hörmander condition (2.9) - has been very often dealt with. This are problems in which the linearity of the equations considered still allows to apply - in suitably tailored sophisticated forms - abstract methods from operator theory, interpolation theory, and Harmonic Analysis. Very often, for instance, the linearity of the equations involved suffices to associate to them a suitable sub-linear operator, so that interpolation methods and techniques are applicable. For such developments we refer the reader to the interesting papers of Auscher & Martell [13, 14, 15], and their related references. Plan for the next sections. At the end of this introductory part we would like to conclude with a few remarks, and a plan for the next sections. The very basic survey of results in this section settles the mood for the rest of the paper. The purpose here is to describe some extension of the integrability properties in (2.2) and (2.4) to solutions to non-linear elliptic problems. We would like to emphasize the doublefold character of the issue: basic integrability of the gradient in (2.4), and second order differentiability in (2.2), which is the maximal regularity result since equation (2.1) is a second order one. Of course, in general (2.2) implies (2.4) via Sobolev embedding theorem, but there are anyway cases in which (2.4) holds, but (2.2) does not, unless using additional assumptions. The case of equations with low regularity coefficients is an instance. Section 3 will be dedicated to results concerning the integrability of the gradient for large exponents, and we shall deal with classical weak or energy solutions. Sections 4 and 5 will be still dedicated to integrability issues, this time for small exponents. We shale therefore deal with so-called very weak solutions, i.e. solutions not belonging to the natural energy space. In particular, in Section 4 we shall give a rapid introduction to measure data problems, including basic regularity results. In Section 5 we shall present more delicate integrability results in different scales of spaces as Morrey spaces, and in finer scales as Lorentz spaces. Finally in Section 6 we shall concentrate on the higher differentiability of solutions. 3. Iwaniec opens the non-linear world 3.1. The notion of solution. The general setting we are going to examine concerns non-linear equations and systems which in the most general form look like (3.1) div a(x, u, Du) = H in Ω, where a: Ω R N R Nn R Nn is a measurable vector field, with n 2 and N 1, which is continuous in the last two arguments, and initially satisfies the following p-growth assumption: (3.2) a(x, u, z) L(1 + z 2 ) p 1 2 for p > 1. When N = 1 (3.1) reduces to an equation and we are in the scalar case. On the right hand side of (3.1) we initially assume that H D (Ω, R N ).

9 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 9 The notion of distributional solution prescribes that a map u W 1,1 (Ω, R N ) is a weak solution to (3.1) iff u is such that a(x, u, Du) L 1 (Ω, R N ) and satisfies (3.3) a(x, u, Du)Dϕ dx = H, ϕ for every ϕ Cc (Ω, R N ). Ω As it will be clear later, this definition is too general. following: Therefore we recall the Definition 3.1. An energy solution to (3.1), under the assumption (3.2), is a distributional solution in the sense of (3.3) enjoining the additional property u W 1,p 0 (Ω, R N ). In turn, when dealing with energy solutions, the natural conditions to impose on H is of course H W 1,p (Ω); this allow to test (3.3) with W 1,p 0 (Ω) functions - this follows by simple density arguments - and in particular with multiples of the solution itself. In turn, when having H W 1,p (Ω) and a suitable monotonicity assumption on a( ), the existence of an energy solution follows by classical monotonicity methods. For this we refer to the classical [98]. In the rest of this section, unless otherwise stated, we shall deal with the notion of energy solution; non-energy solutions will appear later, when dealing with measure data problems A starting point. The starting point here is the following natural p-laplacean analog of equation (2.11): (3.4) div ( Du p 2 Du) = div ( F p 2 F ) for p > 1, which indeed reduces to (2.11) for p = 2. Note that the right hand side of (3.4) is written in the peculiar form div ( F p 2 F ) in order to facilitate a more elegant presentation of the results, and also because such form naturally arises in the study of certain projections problems motivated by multi-dimensional quasi-conformal geometry [68]. Anyway, one could immediately consider a right hand side of the type div G by on obvious change of the vector field G F p 1 F F F G 1 G p 1 G. The following fundamental result in essentially due to Tadeusz Iwaniec, who, in the paper [68] established the foundations of the non-linear CZ theory. Theorem 3.1 ([68]). Let u W 1,p (R n ) be a weak solution to the equation (3.4) in R n. Then F L γ (R n, R n ) = Du L γ (R n, R n ) for every γ p. The local version of this result is Theorem 3.2. Let u W 1,p (Ω) be a weak solution to the equation (3.4) in Ω, where Ω is a bounded domain in R n. Then (3.5) F L γ loc (Ω, Rn ) = Du L γ loc (Ω, Rn ) for every γ p. Moreover, there exists a constant c c(n, p, γ) such that for every ball B R Ω it holds that ( ) 1 γ ( ) 1 ( ) 1 (3.6) Du γ dx c Du p p dx + c F γ γ dx. B R/2 B R B R

10 10 GIUSEPPE MINGIONE A proof can be adapted from [3], for instance. From now on, for ease of presentation we shall confine ourselves to treat local regularity results. The non-trivial extension to the case when (3.4) is a system has been obtained by DiBenedetto & Manfredi, who caught a borderline case too. Theorem 3.3 ([49]). Let u W 1,p (Ω, R N ) be a weak solution to the system (3.4), where Ω is a bounded domain in R n, and N 1. Then (3.5) holds. Moreover (3.7) F BMO loc (Ω, R Nn ) = Du BMO loc (Ω, R Nn ) BMO and VMO functions. There appears a new function space in (3.7), the space of functions with bounded mean oscillations, introduced by John & Nirenberg [74]. In order to introduce BMO functions let us introduce the quantity (3.8) [w] R0 [w] R0,Ω := sup w(x) (w) BR dx. B R Ω,R R 0 B R Then a measurable map w belongs to BMO(Ω) iff [w] R0 <, for every R 0 <. It turns out that BMO L γ for every γ <, while a deep and celebrated result of John & Nirenberg tells that every BMO function actually belongs to a suitable weak Orlicz space generated by an N-function with exponential growth [74], and depending on the BMO norm of w. Specifically, we have ( (3.9) {x Q R : w(x) (w) QR > λ} c 1 (n) exp c ) 2λ [w] 2R,QR where Q R is a cube whose sidelength equals R, and c 1, c 2 are absolute constants. Anyway BMO functions can be unbounded, as shown by log(1/ x ). For the proof of (3.9) a good reference is for instance [53, Theorem 6.11]. Related to BMO functions are functions with vanishing means oscillations. These have been originally defined by Sarason [110] as those BMO functions w, such that lim [w] R,Ω = 0. R 0 In this way one prescribes a way to allow only mild discontinuities, since the oscillations of w are measured in an integral, averaged way. As outlined in Section 2.3, the linear CZ theory can be extended to those problems/operators involving VMO coefficients. This happens also in the non-linear case, as proved by Kinnunen & Zhou who considered a class of degenerate equations whose model is given by (3.10) div (c(x) Du p 2 Du) = div ( F p 2 F ) for p > 1, where the coefficient c( ) is a VMO function satisfying (3.11) c( ) VMO(Ω) and 0 < ν c(x) L <. The outcome is now Theorem 3.4 ([84]). Let u W 1,p (Ω) be a weak solution to the equation (3.10) in Ω, where Ω is a bounded domain in R n, and the function c( ) satisfies (3.11). Then assertions (3.5)-(3.6) hold for u, and the constant in estimate (3.6) also depends on the coefficient function c( ). Before going on, we will comment on the strategy adopted in order to prove Theorems ; in turn, up to non-trivial complications due to the increasing level of generality, this goes back to the original Iwaniec s paper [70]. The idea of Iwaniec is very natural; in the linear case the estimates are obtained using exactly two ingredients: global representation formulas as in (2.5), and then the use of CZ

11 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 11 theory of singular integrals (2.10). Then Iwaniec replaces the first ingredient using a local comparison argument with solutions to the homogeneous p-laplace equation (3.12) div ( Dv p 2 Dv) = 0. The pointwise regularity estimates of Uhlenbeck-Ural tseva [121, 122] are then applied in order to provide an analog of the local representation formula in the linear case. Then finally Iwaniec shows that it is possible to pointwise estimate the sharp maximal function of Du with the maximal function of the datum F, and the conclusion follows by applying the well-known maximal theorems of Hardy-Littlewood, and Fefferman-Stein, which, at this stage, play the role of the boundedness of singular integrals More general operators. We will now turn to more general equations of the type (3.13) div a(x, Du) = div ( F p 2 F ) for p > 1, where a: Ω R n R n is a continuous vector field satisfying the following strong p-monotonicity and growth assumptions: a(x, z) + (s 2 + z 2 ) 1 2 Da(x, z) L(s 2 + z 2 ) p 1 2 (3.14) ν(s 2 + z 2 ) p 2 2 λ 2 Da(x, z)λ, λ a(x, z) a(x 0, z) Lω ( x x 0 ) (s 2 + z 2 ) p 1 2, whenever x, x 0 Ω, z, λ R n. Here we take p > 1, s [0, 1], while ω( ) is a modulus of continuity, that is a non-decreasing function defined on [0, ] such that (3.15) lim t 0 + ω(t) = 0. We have seen from the previous section that the regularity of solutions to homogeneous related homogeneous equations (3.16) div a(x, Dv) = 0, is an important ingredient in the proof of the gradient estimates, in that the regularity estimates for solutions to (3.16) are then used in a comparison scheme to get proper size estimates for the gradient of solutions to (3.13). Therefore in order to state a theorem of the type 3.2 one has to consider operators such that solutions v to (3.16) enjoy the maximal regularity, which in our case is Dv L γ for every γ <. On the other hand, an obvious a posteriori arguments is that if an analog of Theorem 3.2 would hold for equation (3.13), then applying it with the choice F 0 would in fact yield Dv L γ for every γ <. This is the case for solutions to (3.16) under assumptions (3.14). Therefore it holds the following: Theorem 3.5. Let u W 1,p (Ω) be a weak solution to the equation (3.13), where Ω is a bounded domain in R n, and such that assumptions (3.14) are satisfied. Then (3.5) holds for u and moreover (3.17) ( Du γ dx B R/2 where c c(n, p, ν, L, γ). ) 1 γ ( c ( Du p + s p ) dx B R ) 1 ( ) 1 p + c F γ γ dx, B R See for instance [3], from which a proof of the previous result can be adapted. There is a number of possible variants of the previous result; we shall outline a couple of them.

12 12 GIUSEPPE MINGIONE The first deals with non-linear operators with VMO coefficients, more precisely with equations of the type (3.18) div [c(x)a(du)] = div ( F p 2 F ) for p > 1, where the vector field a: R n R n satisfies (3.14) - obviously recast for the case where there is no x-dependence, while the coefficient function c( ) satisfies (3.11). We have Theorem 3.6. Let u W 1,p (Ω) be a weak solution to the equation (3.18) in Ω, where Ω is a bounded domain in R n, and such that assumptions (3.11) and (3.14) are satisfied. Then the assertion in (3.5) and (3.17) hold for u, and the constant c appearing in (3.17) depends also on the coefficient c( ). For a proof one could for instance adapt the arguments from [3, 107]. In particular, in the last paper a version of Theorem 3.6 in the so called Heisenberg group has been obtained, while in the first one CZ estimates have been obtained for a class of non-uniformly elliptic operators. The previous result can be also extended to the boundary when considering the Dirichlet problem - as (3.23) below, under very mild assumptions on the regularity of the boundary Ω; we will not deal very much with boundary regularity, and for such issues we for instance refer to [28] and related references. The second generalization goes in another direction; we have seen that the possibility of getting a priori regularity estimates for homogeneous equations as (3.16) is crucial for proving related CZ estimates. As matter of fact the assumptions guaranteeing that solutions to (3.19) div a(dv) = 0, are Lipschitz can be considerably relaxed with respect to those in (3.14). More precisely we may consider a: R n R n to be a continuous vector field satisfying ν(s 2 + z z 2 2 ) p 2 2 z 2 z 1 2 a(z 2 ) a(z 1 ), z 2 z 1 (3.20) a(z) L(s 2 + z 2 ) p 1 2, whenever z 1, z 2 R n, where p > 1 and s [0, 1]. Note that here the vector field a( ) is not even assumed to be differentiable. We then have Theorem 3.7. Let u W 1,p (Ω) be a weak solution to the equation div a(du) = div ( F p 2 F ), where Ω is a bounded domain in R n, and such that assumptions (3.20) are satisfied. Then (3.5) and (3.17) hold for u. The proof can be obtained using the method in [3], but using the a priori regularity estimates for solutions to (3.19) developed for instance [85, 57] - note that the proofs there extend to the general non-variational case of general operators in divergence form. Finally we remark that in Theorem 3.7 we can allow VMO coefficients as in (3.18) The case of systems. Theorem 3.3 tells us that CZ estimates extend to the case of systems when considering the specific p-laplacean system. We now wonder up which extend the results of the previous section extend to general systems. The reason for Theorem 3.3 to hold is that, as first shown by Uhlenbeck, solutions to the homogeneous p-laplacean system (3.12) are actually of class C 1,α for some α > 0. This makes the local comparison argument work, finally leading to Theorem 3.3. In fact a major effort in [49] is to show suitable form of a priori estimates for solutions to (3.12). We also recall that, as pointed out in the previous section, the regularity

13 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 13 of solutions to associated homogeneous problems is crucial to obtain the desired CZ estimates. In the case of general systems as (3.13), and satisfying (3.14), we cannot expect a theorem like 3.3 to hold, and for a very simple reason. It is known that solutions to general homogeneous systems as (3.21) div a(dv) = 0, are not everywhere regular; they are C 1,α -regular only when considered outside a closed negligible subset of Ω, in fact called the singular set of the solution. Moreover, even for p = 2, and in the case of a smooth vector field a( ), Šverák & Yan [113] have shown that solutions to (3.21) may even be unbounded in the interior of Ω; for such issues see for instance the recent survey paper [103]. This rules out the validity of Theorem 3.3 for general systems in that, should it hold, when applied to the case (3.21) it would imply the everywhere Hölder continuity of v in Ω, clearly contradicting the existence of unbounded solutions proved in [113]. On the other hand an intermediate version of Theorem 3.3 which is valid for general systems holds in the following form: Theorem 3.8 ([86]). Let u W 1,p (Ω, R N ) be a weak solution to the system div a(x, Du) = div ( F p 2 F ), for N 1, where Ω is a bounded domain in R n and the continuous vector field a: Ω R Nn R Nn satisfies (3.14) when suitably recast for the vectorial case. Then there exists δ δ(n, N, p, L/ν) > 0 such that whenever F L γ loc (Ω, RNn ) = Du L γ loc (Ω, RNn ), (3.22) p γ < p + 2p n 2 + δ when n > 2, while no upper bound is prescribed on γ in the two-dimensional case n = 2. Moreover, the local estimate (3.17) holds. Note that the previous theorem does not contradict the counterexample in [113], since this does not apply when n = 2. The previous result comes along with a global one. For this we shall consider the Dirichlet problem { div a(x, Du) = 0 in Ω (3.23) u = v on Ω for some boundary datum v W 1,p (Ω, R N ); here we assume for simplicity that Ω C 1,α, but such an assumption can be relaxed. The main result for (3.23) is Theorem 3.9 ([86]). Let u W 1,p (Ω, R N ) be the solution to the Dirichlet problem (3.23) for N 1, where Ω is a bounded domain in R n and the continuous vector field a: Ω R Nn R Nn satisfies (3.14) when suitably recast for the vectorial case. Then there exists δ δ(n, N, p, L/ν) > 0 such that Du γ dx c ( Dv γ + s γ ) dx, Ω Ω holds whenever (3.22) is satisfied for n > 2, while no upper bound is imposed on γ in the two-dimensional case n = 2; the constant c depends only on n, N, p, ν, L, γ, Ω. The previous theorem reveals to be crucial when deriving certain improved bounds for the Hausdorff dimension of the singular set of minima of integral functions - see [86] - and when proving the existence of regular boundary points for solutions to Dirichlet problems involving non-linear elliptic systems - see [54]. Moreover the peculiar upper bound on γ appearing in (3.22) perfectly fits with the parameters

14 14 GIUSEPPE MINGIONE values in order to allow the convergence of certain technical iterations occurring in [86, 54]. The proof of Theorems is based on an argument different from those in [70], but rather relying on some more recent methods used by Caffarelli & Peral [34] in order to prove higher integrability of solutions to some homogenization problems. Although quite different from the previous ones, such method still relies on the use of maximal operators Parabolic problems. The extension to the parabolic case of the results of the previous sections is quite non-trivial, and in fact the validity of Theorem 3.2 for the parabolic p-laplacean system (3.24) u t div ( Du p 2 Du) = div ( F p 2 F ) remained an open problem for a while in the case p 2, even in the case of one scalar equation N = 1; it was settled only recently in [4]. All the parabolic problems in this section, starting by (3.24), will be considered in the cylindrical domain (3.25) Ω T := Ω (0, T ), where, as usual, Ω is a bounded domain in R n, and T > 0. Let us now explain where are the additional difficulties coming from. As we repeatedly pointed out in the previous sections, the proof of the higher integrability results strongly relies on the use of maximal operators. This approach is completely rules out in the case of (3.24). This is deeply linked to the fact that the homogeneous system (3.26) u t div ( Du p 2 Du) = 0 locally follows an intrinsic geometry dictated by the solution itself. This is essentially DiBenedetto s approach to the regularity of parabolic problems [46] we are going to briefly streamline - see also Remark 3.1. The right cylinders on which the problem (3.26) enjoys good a priori estimates when p 2 are of the type (3.27) Q z0 (λ 2 p R 2, R) B R (x 0 ) (t 0 λ 2 p R 2, t 0 + λ 2 p R 2 ), where z 0 (x 0, t 0 ) R n+1 and the main point is that λ must be such that (3.28) Du p λ p. Q z0 (λ 2 p R 2,R) The last line says that Q z0 (λ 2 p R 2, R) is defined in an intrinsic way. It is actually the main core of DiBenedetto s ideas to show that such cylinders can be constructed and used. Now the point is very simple: since the cylinders in (3.27) depend on the size of the solution itself, then it is not possible to associate to them, and therefore to the problem (3.26), a universal family of cylinders - that is independent of the solution considered. In turn this rules out the possibility of using parabolic type maximal operators. In the paper [4] we overcame this point by introducing a completely new technique bypassing the use of maximal operators, and giving the first Harmonic Analysis free, purely pde proof, of non-linear CZ estimates. The result is split in the case p 2 and p < 2. Theorem 3.10 ([4]). Let u C(0, T, L 2 (Ω, R N )) L p (0, T, W 1,p (Ω, R N )) be a weak solution to the parabolic system (3.24), where Ω is a bounded domain in R n, and p 2. Then (3.29) F L γ loc (Ω T, R Nn ) = Du L γ loc (Ω T, R Nn ) for every γ p.

15 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 15 Moreover, there exists a constant c c(n, N, p, ν, L, γ) such that for every parabolic cylinder Q R B R (x 0 ) (t 0 R 2, t 0 + R 2 ) Ω T it holds that ( ) 1 γ Du γ dx dt Q R/2 (3.30) [ ( ) 1 ( c ( Du p p + 1) dx dt + Q R Q R F γ dx dt ) 1 γ ] p 2 We note the peculiar form of the a priori estimate (3.30), which fails to be a reverse Hölder type inequality as (3.17) due to the presence of the exponent p/2, which is the the scaling deficit of the system (3.24). The presence of such exponent is natural, and can be explained as follows: in fact, let us consider the case F 0, that is (3.26). We note that if u is a solution, then, with c R being a fixed constant, the function cu fails to be a solution of a similar system, unless p = 2. Therefore, we cannot expect homogeneous a priori estimate of the type (3.17) to hold for solutions to (3.24), unless p = 2, when (3.30) becomes in fact homogeneous. Instead, the appearance of the scaling deficit exponent p/2 in (3.30) precisely reflects the lack of homogeneity. Another sign of the lack of scaling is the presence of the additive constant in the second integral, this is a purely parabolic fact, linked to the presence of a diffusive term - that is u t - in the system. Remark 3.1 (Intrinsic geometry and self-rebalancing). We will explain here some basic principles of DiBenedetto s intrinsic geometry [46], confining ourselves to the case p > 2; the case p < 2 can be treated by similar means. The reason for considering cylinders as in (3.27) appears natural if we use the following heuristic argument: the relation (3.28) roughly tells us that Du λ in the cylinder Q z0 (λ 2 p R 2, R). Therefore in the same cylinder we may think to system (3.26) as actually (3.31) u t div (λ p 2 Du) = 0. Now, switching from the intrinsic cylinder Q z0 (λ 2 p R 2, R) to Q 0 (1, 1), that is making the change of variables v(x, t) := u(x 0 + Rx, t 0 + λ 2 p R 2 t) (x, t) B 1 ( 1, 1) Q 1, we note that (3.31) gives that (3.32) v t v = 0, holds in the cylinder Q 1. Therefore this argument tells us that on an intrinsic cylinder of the type in (3.27) the solution u approximately behaves as a solution to the standard heat system, and therefore enjoys good estimates. Note that for p = 2 the cylinders considered in (3.27) are actually the standard parabolic ones - that is those equivalent to the balls generated by the parabolic metric in R n+1 d par ((x, t), (y, s)) := x y + t s 1 2, x, y R n, s, t R. These are in turn independent of the solution, and therefore if ones wants to derive CZ estimates for solutions to (3.24) in the case p = 2, then the standard elliptic proof works, provided using the parabolic maximal operator, that is the one defined by considering as defining family the one parabolic cylinders [M par f](x, t) := sup f(y, s) dy ds, Q r B r (x 0 ) (t 0 r 2, t 0 + r 2 ). (x,t) Q r Q r Instead, in the case p > 2 one is lead to consider intrinsic cylinders as in (3.27) which depend on the solution themselves, and therefore do not define a universal.

16 16 GIUSEPPE MINGIONE family. In a sense, we are considering the locally deformed parabolic metric given by (3.33) d par,λ ((x, t), (y, s)) := max x y + λ p 2 2 t s 1 2, where, again the number λ depends on the solution via (3.28). Remark 3.2 (Interpolation nature of Theorem 3.10). A closer look at the proofs in [4, 56] reveals a more explicit structure of estimate (3.30), which actually looks like ( Du γ dx dt Q R/2 ) 1 γ [ ( ) 1 ( c 1 ( Du p p + 1) dx dt + c2 (γ) Q R Q R F γ dx dt ) 1 γ ] p 2 where the constant c 1 depends on n, N, p, ν, L, but is independent of q. Therefore, considering the case F 0 and eventually letting γ the previous estimate yields (3.34) sup Q R/2 Du c 1 ( ( Du p + 1) dx dt Q R ) 1 2, which is the original L -gradient estimate obtained by DiBenedetto-Friedmann [47, 48] for solutions to the homogeneous p-laplacean system (3.12). This phenomenon reflects the interpolation nature of Theorem 3.10, which in some sense provides an estimate which interpolates the trivial L p estimate - that is (3.30) with γ = p, after absorbing the intermediate integral via standard methods - which comes from testing the system with the solution, and the L one (3.34). A similar remark applies to the case p < 2 treated a few lines below. We turn now to the case p < 2. This is the so called singular case since when Du approaches zero, the quantity Du p 2, which roughly speaking represents the lowest eigenvalue of the operator div ( Du p 2 Du), tends to infinity. Anyway, this interpretation is somewhat misleadins here: we are interested in determining the integrability rate of Du, therefore we are interested in the large values of the gradient. Here a new phenomenon appears: we cannot consider values of p which are arbitrarily close to 1, as described in [46]. The right condition turns out to be (3.35) p > 2n n + 2, otherwise, as shown by counterexamples, solutions to (3.26) maybe even unbounded. This can be explained by looking at (3.26) when Du is very large: if p < 2, and it is far from 2, then the regularizing effect of the elliptic part - the diffusion - is too weak as Du p 2 is very small, and the evolutionary part develops singularities like in odes, where no diffusion is involved. For the case p < 2 the result is now Theorem 3.11 ([4]). Let u C(0, T, L 2 (Ω, R N )) L p (0, T, W 1,p (Ω, R N )) be a weak solution to the parabolic system (3.24), where Ω is a bounded domain in R n, and p < 2 satisfies (3.35). Then F L γ loc (Ω T, R Nn ) = Du L γ loc (Ω T, R Nn ) for every γ p.,

17 TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY 17 Moreover, there exists a constant c c(n, N, p, ν, L, q) such that for every parabolic cylinder Q R B R (x 0 ) (t 0 R 2, t 0 + R 2 ) Ω T it holds that ( ) 1 γ (3.36) Du γ dx dt Q R/2 [ ( ) 1 ( c ( Du p p + 1) dx dt + Q R Q R F γ dx dt ) 1 ] 2p p(n+2) 2n γ Note how in the previous estimate the scaling deficit exponent p/2 in (3.30) is replaced by 2p/(p(n+2) 2n), a quantity that stays finite as long as (3.35) is satisfied. Therefore estimate (3.36) exhibits in quantitative way the role of assumption (3.35). Theorems admit of course several possible generalizations; a first one concerns general parabolic equations of the type u t div a(du) = div ( F p 2 F ), where the vector field a( ) satisfies (3.14). In this case Theorems hold in the form described above, for a constant c depending also on ν, L. We shall now outline a couple of non-trivial extensions, recently obtained in [56]. The first concerns the evolutionary p-laplacean system with coefficients (3.37) u t div (c(x)b(t) Du p 2 Du) = div ( F p 2 F ). The point here is that while the function depending on the space variable c(x) is assumed to be VMO regular, that is to satisfy assumptions (3.11), this time dependent measurable function b(t) is assumed to satisfy only 0 < ν b(t) L <, while no pointwise regularity is assumed other than the obvious measurability. Under such assumptions for solutions to (3.37) Theorem 3.10 holds exactly in the form presented above, but for the fact that the constant c also depends on the function c( ). This result, and the related one obtained in [56], is a far reaching extension of recent analogous results due to Krylov [89] and his students, who consider a similar situation in the case of linear parabolic equations (when, in particular, p = 2, and no intrinsic geometry needs to be considered). The second result from [56] we are presenting is the parabolic analog of Theorem 3.8, we are therefore treating general parabolic systems of the type u t a(x, t, Du) = div ( F p 2 F ), while the assumptions on a( ) are a(x, t, z) + (s 2 + z 2 ) 1 2 Da(x, t, z) L(s 2 + z 2 ) p 1 2 (3.38) ν(s 2 + z 2 ) p 2 2 λ 2 Da(x, t, z)λ, λ a(x, t, z) a(x 0, t, z) Lω ( x x 0 ) (s 2 + z 2 ) p 1 2, whenever x, x 0 Ω, t (0, T ), z, λ R Nn, where p 2 and s [0, 1]. Note that, again, we are assuming no continuity of t a(x, t, z), this being just a measurable map. The result is finally Theorem 3.12 ([56]). Let u C(0, T, L 2 (Ω, R N )) L p (0, T, W 1,p (Ω, R N )) be a weak solution to the general parabolic system (3.24), where Ω is a bounded domain in R n, and p 2, N 1, and where the vector field satisfies (3.38). Then there exists a positive number δ δ(n, N, p, ν, L) > 0 such that F L γ loc (Ω T, R Nn ) = Du L γ loc (Ω T, R Nn ),.

18 18 GIUSEPPE MINGIONE whenever (3.39) p γ < p + 4 n + δ. Moreover, there exists a constant c c(n, N, p, ν, L, γ) such that for every parabolic cylinder Q R B R (x 0 ) (t 0 R 2, t 0 + R 2 ) Ω T the local estimate (3.30) holds Open problems. For the sake of brevity we shall restrict here to the model equation (3.4). Let us start from one simple observation. The minimum degree of integrability required to Du and F in order to give meaning to the weak formulation of (3.4), that is Du p 2 DuDϕ dx = F p 2 F Dϕ dx for every ϕ C Ω is clearly given by Ω u W 1,p 1 (Ω, R N ) and F L p 1 (Ω, R Nn ). This leads to consider those distributional solutions to (3.4) which do not belong to the natural space W 1,p, and therefore are not energy solutions; these are very weak solutions. We shall encounter such solutions also later on, when dealing with measure data problems, and we shall see that they can exist beside the usual energy solutions. Now a comparison between the result of Theorem 3.2 and the linear one in (2.12), which regards solutions to (2.11), naturally leads to the following open problem, which is actually a conjecture of Iwaniec: Open problem 1. Prove that the results of Theorems 3.1 and 3.2 hold in the full range p 1 < γ. The only result known up to now in this direction is due independently to Iwaniec & Sbordone [72, 69], and Lewis [90], who were able to prove that the statement of Theorem 3.2 holds in the range p ε < γ < for some ɛ > 0, depending on the exponent p and the dimension n, but independent of all the other entities considered, in particular of the solution. The methods of proof in [72] uses Iwaniec s non-linear Hodge decomposition, a powerful and deep tool of its own interest. The method in [90] relies instead on the truncation of maximal operators, the so-called Lipschitz truncation method. The conjecture above extends to solutions to the parabolic system (3.24). Again, in this direction Kinnunen & Lewis [82, 83] proved the validity of Theorems in the range p ε < γ < p + ε, thereby finding the right extension of so called Gehring s lemma [63, 70] in the case of parabolic problems. Bögelein, in a remarkably deep paper [26], recently extended the results of Kinnunen & Lewis on very weak solutions [83] to the case of parabolic systems depending on higher order spatial derivatives; see also [25], which features the higher order extension of [82]. We close this section noting that the weaker integrability assumption F L γ with γ < p, puts the right hand side of (3.4) outside the natural dual space div ( F p 2 F ) W 1,p (Ω), and therefore leads us to consider those problems, as measure data ones, for which we face the problem of proving gradient estimates below the duality exponent Obstacle problems. We conclude with further integrability results, recently obtained in [27], and concerning gradient estimates for obstacle problems. A point of interest here is that, differently from the usual results available in the literature, the obstacles considered here are just Sobolev functions, and therefore discontinuous, in general.