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Mean Value Properties of Fractional Second Order Operators COMMUNICATIONS ON doi:10.3934/cpaa.2015.14.83 PURE AND APPLIED ANALYSIS Volume 14, Number 1, January 2015 pp. 83{106 MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS Fausto Ferrari Dipartimento di Matematica dell' Universit`adi Bologna Piazza di Porta S. Donato, 5, 40126, Bologna, Italy Abstract. In this paper we introduce a method to define fractional operators using mean value operators. In particular we discuss a geometric approach in order to construct fractional operators. As a byproduct we define fractional linear operators in Carnot groups, moreover we adapt our technique to define some nonlinear fractional operators associated with the p−Laplace operators in Carnot groups. 1. Introduction. In this paper we introduce and discuss a geometric method to define nonlocal operators arising from (local) mean value formulas. The idea we are going to discuss may be considered as indirectly and implicitly suggested in the book [25] by Landkof. We have to add, for completeness, that the afore mentioned book is largely based on the fundamental work by Marcel Riesz, see [34]. The idea that we want to discuss is quite natural and it is worth describing it in the simplest case. Let us consider the mean value operator of a continuous function φ defined for simplicity in n; R Z M[φ](x; r) = φ(σ)dHn−1(σ); @B(x;r) where, as usual, B(x; r) is the open Euclidean ball centered in x with radius r; R n−1 n−1 @B(x; r) is its boundary, @B(x;r) φ(σ)dH (σ) denotes the spherical mean, H is the n − 1-Hausdorff measure and Z Z n−1 1 n−1 φ(σ)dH (σ) := n−1 φ(σ)dH (σ): @B(x;r) H (@B(x; r)) @B(x;r) Let us fix x 2 Rn and for every r > 0 consider the function r !M[φ](x; r) − φ(x): Of course this function gives information about how far the mean value of φ at x is from the value of φ evaluated in x: It is also clear that, if M[φ](x; r) − φ(x) ≥ 0 for every r; then we are describing a function whose values are, averagely, larger than the value of φ(x): For example the function φ(x) = jxj2 in any ball centered at 0 satisfies this condition. It is well 2000 Mathematics Subject Classification. Primary: 35H20, 35J60; Secondary: 35E05, 35J92. Key words and phrases. Mean operators, Carnot groups, fundamental solutions, nonlinear operators. The author is supported by the ERC starting grant project 2011 EPSILON (Elliptic PDEs and Symmetry of Interfaces and Layers for Odd Nonlinearities) n. 277749 and by RFO of the University of Bologna, Italy. 83 84 FAUSTO FERRARI known that the point wise behaviour for small r of r !M[φ](x; r) − φ(x); for each x in a fixed open set Ω; determines necessary and sufficient conditions to say that φ is harmonic in Ω; see e.g. [32] for a long list of authors who studied the relationships among mean properties and harmonic functions. However, in general, we would have to expect positive and negative values of M[φ](x; r) − φ(x) associated with different value of r; especially when r becomes larger and larger. So, it is, in a sense, natural to study the global behaviour of M[φ](x; r) − φ(x) by considering the integral of our function possibly with respect to a fixed measure µ. Namely we are interested in: Z 1 (M[φ](x; r) − φ(x)) dµ(x; r): (1) 0 In this way we are able to evaluate the weight of the positive values of M[φ](x; r) − φ(x) with respect to its negative value and vice versa. On the other hand, we recall that whenever M[φ](x; r) − φ(x) vanishes as r ! 0; we may also suppose that m X k m M[φ](x; r) − φ(x) = ak(x)r + o(r ); (2) k=1 as r ! 0; for some given exponent m: For example, see one more time [32] for the historical details about the authorship of the following result, we know that continuous functions φ such that for every x 2 Ω satisfy M[φ](x; r) − φ(x) = o(r2); (3) as r ! 0; are harmonic functions indeed. Thus, in order to keep in account to the contribution of M[φ](x; r) − φ(x) for large r in a global framework, it is natural to multiply M[φ](x; r) − φ(x) by a singular weight as r ! 0; and then considering the integral of that product in R+: For example we multiply M[φ](x; r)−φ(x) by r−s; where s > 0 is a positive number. As a consequence, we need to ask that previous integral (1) converges. Indeed, we would like that Z +1 (M[φ](x; r) − φ(x))r−sdr < 1: (4) 0 The approximation given in (2) depends on the definition of M; however, we can suppose to normalise this definition assuming that the size r is defined in such a way that, possibly changing the definition of the operator M; we get M[φ](x; r) − φ(x) lim = a2(x) 2 R: r!0+ r2 More precisely, if u is sufficiently smooth, then R n−1 @B(x;r) φ(σ)dH (σ) − φ(x) lim = Cn∆φ(x); r!0+ r2 where Cn is a positive constant independent with x: Hence, φ satisfies Cn∆φ(x) = f(x) if and only if M[φ](x; r) − φ(x) = f(x)r2 + o(r2); as r ! 0: As a consequence, MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 85 we restrict ourselves to consider mean value operators such that 2 2 M[φ](x; r) − φ(x) = a2(x)r + o(r ): (5) Indeed, if M[φ](x; r) − φ(x) = arβ + o(rβ); for some β > 0; then 2 2 2 M[φ](x; r β ) − φ(x) = ar + o(r ); 2 and defining M~ [φ](x; r) := M[φ](x; r β ) our approach applies to M~ on balls of different sizes. R 1 2−s Thus, recalling (4) and (5), on one side we need that 0 r dr converges. That is, we have to ask that s < 3: Moreover, requiring that the integral (4) converges, even for s ! 3−; we may choose a normalising constant c(s; n) such that Z 1 c(s) c(s) c(s) lim c(s) r2−sdr = lim ( − lim r3−s) = lim = 1: s!3− 0 s!3− 3 − s r!0+ 3 − s s!3− 3 − s In particular this means that c(s) = 3 − s: On the other hand, assuming that the behaviour of M[φ](x; r) − φ(x); for large r, is uniformly bounded by a constant, we need to impose that s > 1 in order to have the convergence of the integral Z +1 −s Mr[φ(y) − φ(x)]r dr: 1 Hence, it is natural to put in evidence this remark by writing s = 1 + 2α; where here α 2]0; 1[ and to define, up to a multiplicative constant, the operator Z +1 Lαφ(x) = (M[φ](x; r) − φ(x)) r−1−2αdr (6) 0 that is well defined, for example, in S(Rn): Applying the coarea formula we get immediately Z α 1 φ(y) − φ(x) L φ(x) = n+2α dy; n! n jx − yj n R n where !n denotes the volume of the unit ball in R : In this approach we just used a mean value operator defined on the boundaries of a family of sets. Indeed, in our example, we considered the Euclidean balls. Nevertheless, a similar construction can be made, in principle, using other sets. The operator that we obtained is, up to some possible different normalisation constants, related to the Euclidean fractional Laplace operator. Indeed multiplying the integral in (6) with the right constant c = cα,n; see e.g. [9], we get that Lα := −(−∆)α; where (−∆)α is the Euclidean fractional Laplace operator for α 2]0; 1[: Moreover it α is also true that, see one more time [9], limα!1− L φ(x) = ∆φ(x): After this remark it is quite natural to generalise such idea to a metric context in order to construct fractional operators naturally arising from assigned mean value formulas. Hence, one of the aims of this paper consists in revisiting the classical methods based on the mean value properties that, in the case of the nonlocal operators have been apparently considered less flexible with respect to the extension method proposed in [8]. An application of this approach is given by the proof of the Harnack inequality that we show in Section2. Indeed, by using the Caffarelli-Silvestre approach based on the extension method, see [8], the Harnack inequality boils down thanks to a sophisticated theory based 86 FAUSTO FERRARI on Muckenhoupt weights, see [10]. With our approach, the proof of the Harnack inequality follows from a sort of global mean equation, by mimiking the classical idea, see e.g. in [18], based on the application of the mean value property to positive harmonic functions on concentric balls. Since the definition of these nonlocal operators depends on the mean value operators kept in account, the proof of Harnack inequality may be, in principle, straightforwardly extended to all those situations where the mean value operators are defined on balls that satisfy the doubling property with respect to the assigned measure.
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