Orlicz-space Hardy and Landau-Kolmogorov inequalities for Gaussian measures

We prove Orlicz-space versions of Hardy and Landau-Kolmogorov inequalities for Gaussian measures on the n-dimensional Euclidean space.


Introduction
The classical Hardy inequality on R n states that for u ∈ W 1,2 (R n ) which can be written as It is a natural question to ask for its generalisations: the 'measure' 1 |x| 2 dx on the left hand side of (1.1) can be replaced by dµ, second norm by p−th or q−th, the measure dx on the right hand side by dν.
For n = 1 and functions u vanishing on the boundary, the Hardy inequality (for general measures on [a, ∞)) in L p norms has been thoroughly studied and there is a complete description of measures that allow for such an inequality. We have the following characterization, which can be found in ( [9], Section 1.3.1, Th. 1): Theorem 1.1 ([9]). Suppose that µ, ν are nonnegative measures on (a, ∞), let ν * be the absolutely continuous part of ν. Then the inequality We are concerned with generalisations of (1.1), when the Lebesgue measure is replaced by the standard Gaussian measure on R n , γ n (dx) = (2π) −n/2 exp(−|x| 2 /2) dx, the 'inner' weight w(x) = |x| −1 is replaced by w(x) = |x|, and the L p −norms are replaced by Orlicz norms or Orlicz modular expressions. Inequalities for the Gaussian measure on R n can be reduced to inequalities on [0, ∞), with respect to the measure dµ n (r) = r n−1 e −r 2 /2 dr. Applying Theorem 1.1 with dµ(r) = r q dµ n (r) and dν(r) = dµ n (r), we see that in that case inequality (1.2) with p = q (for Hardy transforms) can hold only if p > n.
Another reason why inequalities (1.2) need to be extended is that we need inequalities for measures µ n holding true not only for Hardy transforms, but also for functions not necessarily vanishing at zero.
More precisely, in this paper we aim at obtaining inequalities of the form: (1.5) with C 1 , C 2 independent of v from a sufficiently large class of functions, but depending on the dimension n. This dependence cannot be suppressed, see the discussion at the end of Section 3. We still call inequality (1.5) the Hardy inequality for the Gaussian measure. The Hardy inequality for Gaussian measures are of separate interest, both in the probability theory and the PDE theory. For other inequalities for the Gaussian measure (Poincaré, log-Sobolev) the reader can consult e.g. [8], while in [2,3,4] one can find the results concerning the importance of Gaussian measures in the PDE theory.
We obtain (1.4) and (1.5) for general N −functions M satisfying the ∆ 2 −condition (doubling), see Proposition 3.2. With some additional condition (close to the property that M (r)/r 2 is non-decreasing) we were able to provide a more detailed analysis of the resulting constants.
Inequalities (1.5) are an example of the so-called U −bounds (see [5]), i.e. inequalities of the form |v| q U dµ ≤ C |∇f | q dµ + D |f | q dµ analyzed in the context of general metric spaces and metric gradients. Examples furnished in that paper indicate that the most interesting U −bounds for a measure dµ(x) = e −ϕ(x) are those with U (x) = |∇ϕ|. Such inequalities can be related e.g. to Poincaré, log-Sobolev and other inequalities for µ. Since for the Gaussian measure one has ϕ(x) = |x| 2 /2, and |∇ϕ(x)| = |x|, the weight w(x) = |x| in (1.5) is the most desirable one. In a somewhat different context, such inequalities were also investigated in [4].
As an application we show, using a general theorem from [7], that inequality (1.5) implies the Orlicz version of the Landau-Kolmogorov inequality for the Gaussian measure: together with its modular counterpart.
In [6], one proves additive Gagliardo-Nirenberg inequalities in weighted Orlicz spaces. In particular, the following inequality for Gaussian measures was obtained: where M was an N −function satisfying the ∆ 2 −condition and increasing faster that r 2 , and Φ 1 , Φ 2 were other N −functions. The functions M, Φ 1 , Φ 2 were tied by certain consistency conditions, which in particular excluded the case M = Φ 1 = Φ 2 , i.e. the results of [6] did not yield the Landau-Kolmogorov inequality (1.6) in Orlicz norms. This is rectified in present paper, see Corollary 4.1.

Notation
Throughout the paper, the symbol ∇ (2) u denotes the Hessian of a function u ∈ C 2 (R n ), i.e. the matrix [ ∂ 2 u ∂x i ∂x j ] n i,j=1 . For a square n × n matrix A, by |A| we denote its Hilbert-Schmidt norm: C ∞ 0 (R n ) stands for smooth compactly supported functions on R n . r , for r > 0. Additional conditions on M will be added as needed.

Weighted Orlicz spaces
Suppose that µ is a positive Radon measure on R n and let M : [0, ∞) → [0, ∞) be an N −function. The weighted space L M (µ) with respect to the measure µ is, by definition, the function space

This norm is complete and turns
We recall the following two properties of Young functionals: When M satisfies the ∆ 2 −condition, then (2.3) becomes an equality.
For more information on Orlicz spaces the reader may consult e.g. [10].
3 The Hardy inequality for the Gaussian measure

Inequalities on the real line
We start with inequalities for measures µ n (dr) = r n−1 e −r 2 /2 dr, r > 0, where n = 1, 2, ... In our approach, we will make the following assumption concerning the function M : Then, obviously, D M ≥ d M and M is an increasing continuous function with M (0) = 0, lim r→∞ M (r) = ∞, and moreover r → r −d M M (r) is non-decreasing.
When we additionally assume that D M > 2 and d M ≥ 2, then in particular lim r→0 + r −2 M (r) exists and is finite. Hence by a natural convention we treat r → r −2 M (r) and r → r −1 M (r) as continuous functions on the whole [0, ∞), the latter taking value 0 at 0.
Then for any λ ≥ 1/d M and r, s ≥ 0 we have and Proof. Because of the continuity we may and will assume that r and s are strictly by setting u = s/r we rewrite both asserted inequalities in the case s ≤ r as For s ≥ r we have M (r) = M (rs −1 · s) ≤ (r/s) d M M (s), so by setting u = s/r we reduce our task to proving The case u = 1 of the above estimate follows by the previous argument, and the proof is finished by observing that d Let n ≥ 1 and dµ n (r) = r n−1 e −r 2 /2 dr. For a continuous and piecewise If additionally D M + n ≥ e + 2 (which holds true whenever n ≥ 3), then 2 −1 , which proves the second inequality of (3.5). Therefore, if D M + n − 2 ≥ e then the right-hand side of (3.6) dominates the right-hand side of (3.5), which proves the last assertion. Hence it suffices to prove that (3.5) or (3.6) holds true. Additionally, let us assume at first that u is compactly supported. By a standard integration by parts argument we obtain We x . Now, for any λ, ρ ≥ 1/d M we can apply (3.3) to estimate the first summand, and (3.4) to bound the second summand, arriving at which ends the proof in the case of compactly supported u.
In the general case let 2N ], and u N (r) = 0 if r ≥ 2N . Let Since u N is compactly supported we have By the Monotone Convergence Theorem we obviously have K N → K and L N → L as N → ∞ (note that |u N | ր |u| and recall that M is non-decreasing). Since there whereas for almost all r ∈ B N we have −→ M, and the proof is finished. The argument fails only if L = ∞, but then the main assertion is trivial.
We may slightly weaken the assertion of Proposition 3.1 by turning it into a more convenient linear estimate: with positive C 1 and C 2 depending only on n, D M and d M . Elementary calculations permit us to obtain e.g.
valid when D M + n ≥ e + 2. Also, when we consider M (r) = r p , no restrictions other that p > 2 are required, which follows from a straightforward calculation which uses integration by parts and Hölder's inequality only. See also Corollary 3.1 below.
Remark 3.1. It is known (see [4]) that for p = 2 one has (i.e. C 1 = 2n, C 2 = 4) and that the constant 1 4 cannot be improved. Additionally, if C 2 = 4 then (3.8) holds true with C 1 = 2n but it fails for C 1 < 2n. In this case (p = 2), our method permits to lower C 1 as close to n as we wish, again at the expense of getting C 2 large. Getting C 1 = n is not possible.
Proposition 3.1 provides reasonable bounds but its assumptions are a bit restrictive in that they require the function r → r −2 M (r) to be non-decreasing. However, we may also prove (3.8)-type inequality if we replace (3.2) by convexity. This time we do not push for the best possible constants. Since M is convex and increasing there must be D M ≥ 1. Observe that when M is an N − function satisfying the ∆ 2 −condition, then the assumptions of Proposition 3.2 are satisfied.
We need a simple lemma. Proof. If b ≥ 1 then and the inequality obviously holds. For b ∈ [0, ε] the inequality is trival. For b ∈ (ε, 1) we have Proof of Proposition 3.2. Again, we first assume additionally that u is compactly supported. Let  By setting ε = (4D M ) −1 and κ = 2(D M + n) 1/2 , upon obvious cancellations we obtain the asserted estimate. Finally, we may remove the compact support assumption in the same way as in the proof of Proposition 3.1.
When M is an N −function satisfying the ∆ 2 −condition, then using standard Orliczspace methods one can obtain the Hardy inequality for norms.

The n-dimensional case
Using the one-dimensional inequality as a tool, now we derive the Hardy inequality for the n−dimensional Gaussian measure. We start with the statement under general assumptions on the function M involved, which however does not give a good control on the resulting constants.

The Landau-Kolmogorov inequality for the Gaussian measure
The Hardy inequalities from Section 3.2 can be used for deriving Landau-Kolmogorov inequalities for Gaussian measures in R n .
To this end, we will use the following theorem (Theorem 3.3 of [7]), applied with P = Q = M. We apply this theorem to Ω = R n and dµ(x) = e −|x| 2 /2 dx. In this case |∇ϕ(x)| = |x|, and the validity of (4.1) is assured by Proposition 3.2 (or Proposition 3.1, provided we assume (M)). Choosing θ = 1 we obtain the following: Corollary 4.1. Suppose M is a differentiable N −function satisfying the ∆ 2 −condition and such that M (r)/r 2 is non-decreasing. Let dγ n (x) = e −|x| 2 /2 dx. Then there exist positive constants C 1 , C 2 such that for any u ∈ C ∞ 0 (R n ) one has and positive constantsC 1 ,C 2 such that for any u ∈ C ∞ 0 (Ω) ∇u L M (R n ,γn) ≤C 1 ∇ (2) u L M (R n ,γn) u L M (R n ,γn) +C 2 u L M (R n ,γn) . (4.5) By usual density arguments, smoothness conditions on u can be relaxed.