{'fraction_non_alphanumeric': 0.0697035780631613, 'fraction_numerical': 0.02180760843227526, 'mean_word_length': 3.580836108028117, 'pattern_counts': {'":': 0, '<': 20, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Logarithmic Sobolev Trace Inequalities

We prove a logarithmic Sobolev trace inequality in a gaussian space and we study the trace operator in the weighted Sobolev space W^{1,p}(\Omega,\gamma) for sufficiently regular domain. We exhibit examples to show the sharpness of the results. Applications to PDE are also considered.


Introduction
Sobolev Logarithmic inequality states that R N |u| p log |u| dγ ≤ p 2 R N |∇u| 2 |u| p−2 sign u dγ + u p L p (R N ,γ) log u L p (R N ,γ) , (1.1) where 1 < p < +∞, γ is the Gauss measure and L p R N , γ is the weighted Lebesgue space (see §2 for the definitions). This inequality was first proved in [17] (see also [3] for more general probability measure). It has many applications in quantum field theory and differently from classical Sobolev inequality it is independent of dimension and easily extends to the infinite dimension.
In terms of functional spaces inequality (1.1) implies the imbedding of weighted Sobolev space W 1,p (R N , γ) into the weighted Zygmund space L p (log L) 1 2 R N , γ . The imbedding holds also for p = 1 and it is connected with gaussian isoperimetric inequality and symmetrization (see [19], [14] and [23]). For p = +∞ one obtains (see [21] and [1]) that if u is a Lipschitz continuous function, then u ∈ L ∞ (log L) − 1 2 R N , γ .
This kind of imbeddings are also studied in [8] in the more general case of rearrangement-invariant spaces.
In [10] a set Ω ⊆ R N and the space W 1,p 0 (Ω, γ) are considered using properties of rearrangements of functions; the authors prove that if u ∈ W 1,p 0 (Ω, γ) with 1 ≤ p < +∞, then u ∈ L p (log L) The constants C 1 ,C 2 depend only on p and γ(Ω). Analogue inequalities have been obtained in infinite dimensional case and in the Lorentz-Zygmund spaces (see the appendix of [16]). A first result of our paper is to obtain (1.2) when u ∈ W 1,p (γ, Ω) (see §3); in this case, as one can expect, smoothness assumption on ∂Ω have to be made. Besides the continuity also the compactness of the imbedding of W 1,p (Ω, γ) in a Zygmund space is studied. As a consequence we obtain a Poincaré-Wirtinger type inequality. We analyze also the case p = +∞. These results are sharp and counterexamples in this direction are given. Applications of these results to PDE are also considered.
The results explained above are used to investigate Sobolev trace inequalities. This kind of inequalities play a fundamental role in problems with nonlinear boundary conditions. In the euclidean case the Sobolev trace inequality (cf. e.g. [18]) tell us that if Ω is smooth enough and 1 ≤ p < N, then there exists a constant C (depending only on Ω and on p) such that where T is the trace operator. This kind of inequalities has been developed via different methods and in different settings by various authors including Besov [6], Gagliardo [11], Lions and Magenes [22]. Trace inequality that involves rearrangement-invariant norms are considered in [?].
To investigate about trace operator in the weighted Sobolev space W 1,p (Ω, γ) in §4 we need a Sobolev trace inequality. We prove that if Ω is a smooth domain and u ∈ C ∞ (Ω) then This inequality is sharp and captures the spirit of the Gross inequalities: the logarithmic function replaces the powers in this case too. We analyze also the case p = +∞. Using (1.4), we can define the trace operator and to prove continuity and compactness of the operator W 1,p (Ω, γ) into L p (∂Ω, γ) for sufficiently regular domain Ω ⊆ R N . Moreover we prove a Poincaré trace inequality is obtained in a suitable subspace of W 1,p (Ω, γ). We give also some applications of these results to PDE.
An other Sobolev trace inequality is obtained in [24] as limit case of the classical trace Sobolev inequality.

Preliminaries
In this section we recall some definitions and results which will be useful in the following.

Gauss measure and rearrangements
Let γ be the N -dimensional Gauss measure on R N defined by We will denote by Φ (τ ) the Gauss measure of the half-space x ∈ R N : x N < τ : We define the decreasing rearrangement with respect to Gauss measure (see e.g. [13]) of a measurable function u in Ω as the function where γ u (t) = γ ({x ∈ Ω : |u| > t}) is the distribution function of u.

Sobolev and Zygmund space
The weighted Lebesgue space L p (Ω, γ) is the space of the measurable functions u on Ω such that Ω |u| p dγ < +∞. We recall also that the weighted Sobolev space W 1,p (Ω, γ) for 1 ≤ p < +∞ is defined as the space of the measurable functions u ∈ L p (Ω, γ) such that there exists g 1 ,...,g N ∈ L p (Ω, γ) that verify We stress that u ∈ W 1,p (Ω, γ) is a Banach space with respect to the norm u W 1,p (Ω,γ) = u L p (Ω,γ) + ∇u L p (Ω,γ) . The Zygmund space L p (log L) α (Ω, γ) for 1 ≤ p ≤ +∞ and α ∈ R is the space of the measurable functions on Ω such that the quantity is not trivial if and only if p < +∞ or p = +∞ and α ≤ 0. The Zygmund spaces are the natural spaces in the context of Gauss measure, because of the following property of isoperimetric function is (see [20]): We remind same inclusion relations among Zygmund spaces. If 1 ≤ r < p ≤ +∞ and −∞ < α, β < +∞, then we get It is clear from definition (2.1) that the space L p (log L) α (Ω, γ) decreases as α increases. For more properties we refer to [5].

Smoothness assumptions on the domain
In this paper we deal with integrals involving the values of a W 1,p −function on ∂Ω. To this aim we need to have a suitable local description of the set Ω and ∂Ω is a finite union of graphs. More precisely we will consider smooth domain Ω which verifies the following condition (cfr. Chapter 6 of [18] for bounded domain).
iii) a number β > 0 such that the sets are subsets of ∂Ω, ∂Ω = m ∪ r=1 Λ r and the sets are subset of Ω (after a suitable orthonormal transformation of coordinates).
We observe that the set 3 Sobolev logarithmic inequalities in W 1,p (Ω, γ) In this section we prove continuity and compactness of imbedding of W 1,p (Ω, γ) into L p (log L) 1 2 (Ω, γ). We will deal also with the case p = +∞. The first step is to obtain the analogue of (1.2) and (1.3) when u ∈ W 1,p (Ω, γ) for 1 ≤ p ≤ +∞.
and Ω satisfies condition 2.1, then there exists a positive constant C depending only on p and Ω such that To prove Proposition 3.1 we need an extension operator P from W 1,p (Ω, γ) into W 1,p (R N , γ). When u ∈ W 1,p 0 (Ω, γ) the natural extension by zero outside Ω is continuous without any assumptions on the regularity of the boundary. Working with the space W 1,p (Ω, γ) the situation is more delicate and the regularity of the boundary of Ω plays a crucial role. Using classical tools (see e.g. [4] ) it is possible to prove the existence of an extension operator P from W 1,p (Ω, γ) into W 1,p (R N , γ) which is linear and continuous. The extension operator allows us to prove the density ( for the classical case see e.g. [4] Proof of Proposition 3.1. We consider the extension operator P from W 1,p (Ω, γ) into W 1,p (R N , γ) and using (1.2) we obtain for some constant c (Ω, γ) obtained in the Proposition 3.1 is sharp in the class of the Zygmund spaces as the following example shows.
Let now consider Lipschitz continuous functions.
(Ω, γ) obtained in the Proposition 3.2 is sharp in the class of the Zygmund spaces as the following example shows.
this means that u δ is a Lipschitz continuous function and In order to prove Proposition 3.2 we can argue as in the proof of Proposition 3.1: we need the extension operator P and the inequality (1.3). Let us observe that the boundary conditions lim Proof. It is enough to prove the compactness of the embedding of W 1,p (Ω,γ) into L 1 (Ω,γ).Indeed we have that any bounded set of L p (log L) Using a characterization of precompact sets of Lebesgue spaces (see e.g. Theorem 2.21 of [2]) we have to prove that for any number ε > 0 there exists a number δ > 0 and a subset G ⊂⊂ Ω such that for any u ∈ S and every h ∈ R N with |h| < δ the following conditions hold: where u is the zero extension of u outside Ω. Let ε > 0 and Ω j = x ∈ Ω : dist(x, ∂Ω) > 1 j for j ∈ N. By (3.1) we have for some constant c Since the Gauss measure of Ω is finite, we can choose j big enough to have In the last inequalities we have used (3.1) and the fact that Because of the density of C ∞ (Ω) in W 1,p (Ω, γ), (3.7) holds for every u in W 1,p (Ω, γ) and then for |h| small enough by (3.6) and (3.7) we obtain (3.3) Remark 3.4 Obviously the compactness results holds for W 1,p 0 (Ω, γ) for any domain Ω.
Remark 3.5 The compactness proved in Proposition 3.3 implies the compact embedding of Sobolev space W m,p (Ω,γ) m ≥ 1 into the Zygmund space L p (log L) mβ (Ω,γ) for β < 1 2 . The compactness can be used to obtain a Poincaré-Wirtinger type inequality.
Proof. We precede as in the classical case. We argue by contradiction, then there would exist for any k ∈ N a function u k ∈ W 1,p (Ω, γ) such that

Remark 3.6
The previous proof works in a more general case. Let Ω be a connected domain satisfying condition 2.1 and let V ⊂ W 1,p (Ω, γ) be a linear subspace of W 1,p (Ω, γ) with 1 ≤ p < +∞ which is closed and such that the only constant function belonging to V is the function which is identically zero. Then there exists a positive constant C, depending only on p and Ω, such that  Arguing in a classical way (see e.g. [4] Theorem 8.6.1), using inequality (3.8) and the compactness of the embedding from W 1,2 (Ω, γ) into L 2 (Ω, γ), it follows that there exists an increasing sequence of eigenvalues of the problem (3.13) which tends to infinity and a Hilbertian basis of eigenfunctions in L 2 (Ω, γ). Moreover for λ 1 = 0, the corresponding eigenfunction u 1 = const = 0 and the first nontrivial eigenvalue λ 2 has the following characterization

Sobolev logarithmic trace inequalities
In this section we deal with integrals involving the values of a C ∞ −function on ∂Ω. We prove that a certain integral of the function on ∂Ω is bounded by the W 1,p −norm on Ω. This inequality will be crucial to define trace operator (see §5).  Proof. Following classical tools (see Chapter 6 of [18] ) it is enough to prove the existence of a constant C T > 0 such that for any function u ∈ C ∞ (Ω) whose supports is in Λ r ∪ U + r we have (4.1). After suitable transformation that maps ∆ r × ]0, β[ onto U + r and ∆ r × {0} onto Λ r , we can reduce to consider u such that the support is in ∆ r × [0, β[. Then it is sufficient to prove the existence of a constant C > 0 such that for any We observe that the function . Indeed γ f (t) = 2Φ (−t) and using (2.2) we have Then we obtain .

(4.4)
Moreover using Hölder inequality, we obtain because log(2 + u ⊛ (t)) is dominated by a multiple of (1 − log t). Indeed L p (log L) Remark 4.2 In (4.1)the exponent p−1 2 of the logarithmic is sharp as the following example shows. We consider Ω = x ∈ R N : x N < ω with ω ∈ R and the function u δ (x) = Φ δ (x N ) with − 1 p < δ < 0 as in Remark 3.1. We have that

By (2.2) we have that
Moreover A 2 ≤ cA 1 and If p = +∞ we can prove the following result.
Proof of Proposition 4.2. As in the proof of Proposition 4.1 it is sufficient to prove for any functions u ∈ C ∞ (∆ r × [0, β[) whose supports is in ∆ r × [0, β[ and any λ ∈ (0, 1) the following inequality holds for some positive constant C not depending on u . Now we prove (4.9). For some constant c that can varies from line to line we have (4.12) For any λ ∈ (0, 1) the integrals in (4.11) and (4.12) are finite and  for some constant c depending on λ and ∆ r × [0, β[ . Putting (4.13) and (4.14) in (4.10) and using (3.2) we obtain (4.9).

Remark 4.3
In (4.8) the power 2 in the argument of the exponential is sharp as the following example shows. In order to show that we need to consider Ω = x ∈ R N : x N < ω with ω ∈ R and the function u δ (x) = u δ (x) = (1 − log Φ(x N )) δ with 0 < δ ≤ 1 2 as in Remark 3.3 and argue as in Remark 4.2.

Trace operator
In this section the "boundary values" or trace of functions in Sobolev spaces are studied.
If Ω is a domain satisfying condition 2.1, given a smooth function u ∈ C ∞ (Ω) ⊂ W 1,p (Ω, γ) we can define the restriction to the boundary u| ∂Ω . It turn out that this restriction operator can be extended from smooth functions to W 1,p (Ω, γ) giving a linear continuous operator from W 1,p (Ω, γ) to L p (∂Ω, γ), the space of the measurable functions defined almost everywhere on ∂Ω such that ∂Ω |u| p ϕ dH N −1 < +∞.