HARDY TYPE INEQUALITIES AND GAUSSIAN MEASURE

. In this paper we prove some improved Hardy type inequalities with respect to the Gaussian measure. We show that they are strictly related to the well-known Gross Logarithmic Sobolev inequality. Some applications to elliptic P.D.E.’s are also given.

1. Introduction. The classical Hardy inequality states that, for every u ∈ W 1,2 (R N ), N > 2, where the constant N −2 2 2 is sharp and it is not attained in W 1,2 (R N ), even if one replaces W 1,2 (R N ) with W 1,2 0 (Ω), where Ω R N contains the origin. On the other hand, if u ∈ W 1,2 0 (Ω), inequality (1) can be improved by adding remainder terms. For example in [9] it has been proved that for every u in W 1,2 0 (Ω) where ω N and |Ω| denote, respectively, the Lebesgue measure of the unit ball and of Ω, and z 0 is the first zero of the Bessel function J 0 (z). Since the error term in (2) is given in terms of a rearrangement-invariant norm of u, Schwarz symmetrization allows to reduce to the radial case and, then, specify the constant. In [31] inequality (1) is further improved as follows where C is a positive constant depending only on Ω and q. Inequalities (1), (2) and (3) have been extended to the case p = 2, 1 < p < N (see, e.g., [18], [3], [1]), obtaining for instance On the other hand, Hardy type inequalities, where the singularity is given in terms of the function δ(x) = dist(x, ∂Ω), have been extensively studied. The related literature is wide and here we mention the following classical one-dimensional result (see [20], [21], [26], [22]) Inequality (5) has been extended to the case N ≥ 2 when p = 2 in [14], while the case p = 2 is treated, for example, in [25], where the authors prove when Ω is an open, bounded, convex set in R N having C 1 boundary. When the domain is just Lipschitz (not necessarely convex) or when it is unbouned with smooth compact boundary, an inequality of the type (6) still holds true with a constant smaller than p−1 p p (see [24], [12]). Hardy type inequalities have been applied in several contexts as, for instance, the study of the stability of solutions to semilinear elliptic and parabolic equations (see [9]) and the stability of eigenvalues in elliptic problems (see [10]). Finally, classical Hardy inequality (1) provides the embedding of W 1,2 (R N ) into the Lorentz space L 2 * ,2 (R N ). It is rather natural to wonder if there is a relationship between Hardy inequality with respect to the Gaussian measure and the well-known Sobolev Logarithmic embedding theorem proved in [19]. Therefore, in this paper we investigate such a kind of inequalities with respect to the Gaussian measure In this setting it is natural to work with weighted Sobolev spaces, briefly recalled in Section 2. Our starting point is the following inequality and its extension to R N whose simple proofs are contained in Sections 3 and 4. The constants appearing in (7) and (8) are sharp and we will also verify that (7) and (8) cannot be improved by adding any L p norm of u.
As in the classical case one might ask whether, if we substitute R N with a proper subset, some improved Hardy inequalities hold. As we will see, this question has an affirmative answer. Our results rely on some one-dimensional inequalities, since the Gaussian measure is factorized and for it there exists a suitable notion of rearrangement, the so-called Gaussian rearrangement. It transforms the level sets of a positive function u into parallel half-spaces having the same Gaussian measure. In this way one obtains the Gaussian symmetrized of u, u ♯ , which is, therefore, a HARDY TYPE INEQUALITIES AND GAUSSIAN MEASURE   413 one-dimensional, increasing function. This will be a useful tool in passing from R to R N , since a Pólya-Szegö principle with respect to the Gaussian measure holds.
Let 1 ≤ q < p, a ∈ R and let us denote by I a the interval (a, ∞); then the following inequalities hold true for every u in W 1,p 0 (I a , dγ 1 ) and The function ̺ p above is smooth in I a and it has the following asymptotic behavior So "near" a our results are in agreement with the classical one-dimensional Hardy inequality (5), while at infinity we get a singularity of the type x p 1 , in agreement with (7). Moreover there existsx 1 ∈ I a such that ̺ ′ Thus, we generalize (10) to the N -dimensional case as follows where Ω is an open, connected subset of R N , with Ω dγ < 1 (i.e. Ω is essentially different from R N ), and We need to truncate the weight near a so that the left hand side of (11) decreases under Gaussian rearrangement, while the right hand side does not change.
In order to extend (9) to R N we assume that Ω is contained in an half-space and, thanks to factorization arguments, we prove that , ∀u ∈ W 1,p 0 (Ω, dγ).
As we will show, these Hardy type inequalities are, in some sense, equivalent to the Logarithmic Sobolev embedding (see Section 4). This paper is organized as follows. In Section 2 some definitions and notation about weighted spaces, rearrangements and isoperimetric inequality are recalled. Then, Section 3 and Section 4 provide a detailed proof of the Hardy-type inequalities announced before. Finally Section 5 is devoted to investigating some applications of these Hardy inequalities to existence and uniqueness of solutions to Dirichlet problems for non-linear elliptic equations. We also give an example of pathological solution, in the same spirit of Serrin [27], of a one-dimensional problem.

2.
Preliminaries. We begin this section by recalling some definitions about weighted spaces and Gaussian symmetrization that will be useful in the following. From now on Ω will be any connected, open subset of R N such that Ω dγ < 1.
It is known (see, for example [6]) that a measurable function satisfies (12) if and only if where f * (t) is the decreasing of f with respect to the Gaussian measure (defined in the sequel) and is the Gaussian measure of Ω. Obviously |Ω| γ ∈ (0, 1) and R N γ = 1. The last quantity contained in (13) defines a norm with respect to which the space The Gaussian perimeter of a set Ω is defined as where div γ (v(x)) denotes the Gaussian divergence given by When Ω is (N − 1)-rectifiable, then where a is taken such that Ω ♯ γ = |Ω| γ . A straightforward calculation gives where k(t) is the function We recall the isoperimetric inequality for Gaussian measure (see [28], [7], [16] and [11]).
Now we can define the rearrangement, with respect to Gaussian measure, of any measurable function u. To this end let µ be the distribution function of u defined by µ(t) = |{x ∈ Ω : |u(x)| > θ}| γ , θ ≥ 0, and let u * be its decreasing rearrangement defined by We will say that two functions u and v are equimeasurable, or equivalently that v is a rearrangement of u, if they have the same distribution function.
The determination of the cases of equality in the isoperimetric inequality (16) leads to the determination of a particular rearrangement of a measurable function u on Ω: the Gaussian rearrangement u ♯ . More precisely, u ♯ is that rearrangement of u whose level sets are half-spaces. Since the measure of each level set of u is given by the distribution function, then where k is defined in (15).
By definition u ♯ actually depends on one variable only (say x 1 ) and it is an increasing function with respect to it, therefore its level sets are half-spaces. Moreover, since, by definition, u and u ♯ are equimeasurable, by Cavalieri's principle (see [13], p. 30) we have For our purposes we also need the following Hardy-Littlewood inequality Lemma 2.4. Let u be any function in L p (Ω, dγ), with 1 ≤ p ≤ ∞, and v any function in L p ′ (Ω, dγ), with 1/p + 1/p ′ = 1; then It has been shown in [16], [29] and [11] that the following Pólya-Szegö principle holds true.
Moreover equality holds if and only if u = u ♯ , modulo a rotation.
Now we recall the following generalization of the classical Hardy inequality due to Maz'ja (see [26], pages 40ff).
Lemma 2.6. Let µ and ν be nonnegative Borel measures on (a, ∞) and let ν * be the absolutely continuous part of ν. The inequality holds for all Borel functions and 1 ≤ p ≤ q ≤ ∞ if and only if Moreover, if C is the best constant in (20), then In the case q = ∞ the condition (21) means that We end this section by recalling some well-known convexity inequalities that will be used systematically in the paper (for the proofs see [23], [4]).
i) If 1 < p < 2, ii) If p ≥ 2, We will use the simpler notation ̺ p (x 1 ) when no possibility of ambiguity arises. It is elementary to verify that We begin by proving the following Hardy inequality.
Proof. By density arguments we can argue with C ∞ 0 (I a ) functions. A trick, which we learned in [30], consists in integrating over the interval I a the following convexity inequality Observing that ̺ p solves the differential equation we get the claim. Let us show now that the constant p−1 p p in (26) is sharp. Let us introduce the following sequence of functions (see also [3]) Note that u n ∈ W 1,p 0 (I a , dγ 1 ); we want to show that A straightforward calculation gives by using the definition of ̺ p (24), we get the claim (28). Now, let us show that we can improve the above result by adding a remainder term to the right hand side of (26) depending on the L q −norm of u ′ . Theorem 3.2. For any q < p there exists a positive constant C, depending on a, p and q, such that, for every u ∈ W 1,p 0 (I a , dγ 1 ), it holds In order to prove our theorem we need the following Lemma. From now on we will denote by C a positive constant whose value may change from line to line.
Proof. It suffices to observe that the function x1 a e σ 2 /2(q−1) dσ x1 a e σ 2 /2(p−1) dσ is bounded from above and this is an immediate consequence of the following two limits Proof of Theorem 3.2 (The case p ≥ 2). The function solves the differential equation Note that χ / ∈ W 1,p 0 (I a , dγ 1 ) since χ ′ / ∈ L p (I a , dγ 1 ) for any p. Let u ∈ C ∞ 0 (I a ); integrating by parts we can easily obtain since, by (31) and (27), On the other hand, by (23) Moreover, setting u = vχ, by the convexity of the function | · | q and (31), we get We claim that which is an Hardy type inequality with respect to the measure χ q e −x 2 1 /2 dx 1 . In order to prove (33) we need the following auxiliary result.
Proof. The function g is a solution of the following Cauchy problem in the interval (a, A). For any function u in C ∞ 0 (I a ), it holds |u ′ | p ≥ |gu| p + p |gu| p−2 gu(u ′ − gu).

Equation in (35) and inequality (36), together with an integration by parts, yield
On the other hand the function g solves problem (35) also in (A, ∞). Arguing as before we obtain Adding (37) and (38) we get the claim.
Now we come back to the proof of Theorem 3.2 and, in order to treat the term ∞ a χ q ̺ q p |v| q dγ 1 , we use Lemma 3. where (40) Claim (33) is proved once we show By (40) and (25), we easily verify which give (41). Combining (33) and (32), we conclude Then, by Hölder inequality

HARDY TYPE INEQUALITIES AND GAUSSIAN MEASURE 421
Proof of Theorem 3.2 (The case 1 < p < 2). Hölder inequality gives Using again Hölder inequality, by the convexity of the function | · | q and (31), we have At this point, recalling (30), we can use Hardy's inequality (26) with respect to q which gives Collecting inequalities (42), (43) and (44) we have Arguing as in the case p ≥ 2, setting u = χv, we obtain On the other hand inequality (22), with N = 1, ξ 1 = u χ ′ χ and ξ 2 = u ′ , gives In the end we gather Remark 1. From (29) and the embedding of W 1,q 0 (I a , dγ 1 ) in L q (I a , dγ 1 ) (see e.g. [15]) we can deduce that In general we cannot expect a remainder term of the kind ||u|| L p (Ia,dγ1) and, a fortiori, ||u|| W B,p 0 (Ia,dγ1) . Indeed, in the simplest case p = 2 and a = 0, setting as before by Lemma 2.6 the ratio above is not bounded from below by any positive constant.

Remark 2.
Here we consider Sobolev functions defined on the whole real line. By using a standard substitution (see, for instance, [17], [5]) we get that for every u ∈ W 1,2 (R, dγ 1 ) the following inequality holds true Indeed, setting v = ue −x 2 1 /4 , integrating by parts we have The constant 1 4 appearing in the right hand side of (46) is sharp and inequality (46) cannot be improved. Indeed, it is easy to check that the function u ε = e ( 1 4 −ε)x 2 1 belongs to W 1,2 (R, dγ 1 ) and thus, any inequality of the type cannot hold true for any q < 2.
Remark 3. All the arguments used in the case of the Gaussian measure can be adapted for a more general measure. Indeed, let a ∈ R, p > 1, and let φ : [a, ∞) → (0, ∞) be a smooth function such that

4.
The N -dimensional case. The results proved in Section 3 can be generalized to the N -dimensional case, depending on the domain. First of all, in the whole space R N , by factorization arguments, it can been immediately deduced from (46) that and, as in dimension one, (47) cannot be improved.
When Ω is an half-space, or it is contained in an half-space, one can extend (29) using again factorization arguments.
Theorem 4.1. Let a ∈ R and Ω = Ω ♯ = {x = (x 1 , ..., x N ) ∈ R N : x 1 > a}; then there exists a positive constant C such that for every u ∈ W 1,p 0 (Ω, dγ) Proof. Recalling that smooth factorized functions with compact support are dense in W 1,p 0 (Ω, dγ), we just prove the claim for u = Multiplying by N j=2 |u j (x j )| p we get More generally, when Ω ⊂ R N is an open, connected set such that |Ω| γ < 1, one can reduce to the one-dimensional case by means of Gaussian symmetrization. It transforms Ω in the half-space Ω ♯ = {x = (x 1 , ..., x N ) ∈ R N : x 1 > a}, with a = k −1 (|Ω| γ ) and, recalling that u ♯ (x) depends only on x 1 and it is an increasing function, by (19) and (17)  On the other hand, since the function ̺ p (x 1 ) = ̺(a, p; x 1 ) is not increasing with respect to x 1 , the term Ω ̺ p (x 1 ) p |u| p dγ in general, does not increase under Gaussian symmetrization. Therefore we consider the following new weight wherex 1 denotes the point where ̺ p achieves its minimum. So we preserve the singularity at infinity, where the Gaussian function degenerates. Therefore, the following result appears meaningful when the domain is unbounded. These considerations, together with (45), are the main tools in the proof of the following result.