Piotr Graczyk
email: piotr.graczyk@univ-angers.fr

Todd Kemp
email: tkemp@math.ucsd.edu

Jean-Jacques Loeb
email: jean-jacques.loeb@univ-angers.fr

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

We prove an intrinsic equivalence between strong hypercontractivity (sHC) and a strong logarithmic Sobolev inequality (sLSI) for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions, and use it to prove the equivalence of (sHC) and (sLSI) for such log-subharmonic functions.

Introduction

In this paper we study strong versions of logarithmic Sobolev inequalities (sLSI) and strong hypercontractivity (sHC) in the real spaces R n and for logarithmically subharmonic (LSH) functions, continuing our research published in [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF] and solving the conjecture on the equivalence between (sHC) and (sLSI) formulated in [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF]Remark 5.11]. The main difficulty to overcome, as already noticed by Gross and Grothaus in [START_REF] Gross | Reverse hypercontractivity for subharmonic functions[END_REF], was efficient approximating of (logarithmically) subharmonic functions.

If µ is a probability measure, the entropy functional Ent µ relative to µ, defined on all sufficiently integrable positive test functions g, is Ent µ (g) = ˆg ln g g 1 dµ

where g 1 = g L 1 (µ) . (When g 1 = 1, so g is a probability density, this gives the classical Shannon entropy.)

The logarithmic Sobolev inequality is an energy-entropy functional inequality: a measure µ on R n (or more generally on a Riemannian manifold) satisfies a log Sobolev inequality if, for some constant c > 0 and for all sufficiently smooth positive test functions f ,

Ent µ (f 2 ) ≤ c ˆ|∇f | 2 dµ. (LSI)
Making the substitution g = f 2 gives the equivalent form Ent µ (g) ≤ c 4 ´|∇g| 2 /g dµ, the integral on the right defining the Fisher information of g relative to µ. In this form, the inequality was first discovered for the standard normal law µ on R by Stam in [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF]. It was rediscovered and named by Gross in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF], proved for standard Gaussian measures on R n with sharp constant c = 2. Over the past four decades, it has become an enormously powerful tool making fundamental contributions to geometry and global analysis [START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF][START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities true for Markov semigroups[END_REF][START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Bobkov | Christian: Some connections between isoperimetric and Sobolev-type inequalities[END_REF][START_REF] Bobkov | From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities[END_REF][START_REF] Davies | Explicit constants for Gaussian upper bounds on heat kernels[END_REF][START_REF] Davies | Heat kernels and spectral theory[END_REF][START_REF] Davies | Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians[END_REF][START_REF] Ledoux | Isoperimetry and Gaussian analysis[END_REF][START_REF] Ledoux | The geometry of Markov diffusion generators[END_REF][START_REF] Ledoux | Spectral gap, logarithmic Sobolev constant, and geometric bounds[END_REF], statistical physics [START_REF] Holley | Logarithmic Sobolev inequalities and stochastic Ising models[END_REF][START_REF] Yau | Logarithmic Sobolev inequality for the lattice gases with mixing conditions[END_REF][START_REF] Yau | Log-Sobolev inequality for generalized simple exlcusion processes[END_REF][START_REF] Zegarlinski | Dobrushin uniqueness theorem and logarithmic Sobolev inequalities[END_REF], mixing times of Markov chains [START_REF] Bobkov | Modified logarithmic Sobolev inequalities in discrete settings[END_REF][START_REF] Diaconis | Logarithmic Sobolev inequalities for finite Markov chains[END_REF][START_REF] Guionnet | Lectures on logarithmic Sobolev inequalities[END_REF], concentration of measure and optimal transport [START_REF] Ledoux | Concentration of measure and logarithmic Sobolev inequalities[END_REF][START_REF] Ledoux | The concentration of measure phenomenon[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF], random matrix theory [START_REF] Anderson | An introduction to random matrices[END_REF][START_REF] Ledoux | A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices[END_REF][START_REF] Zimmermann | Logarithmic Sobolev inequalities for mollified compactly supported measures[END_REF], and many others.

Gross discovered the log Sobolev inequality through his work in constructive quantum field theory, particularly relating to Nelson's hypercontractivity estimates [START_REF] Nelson | The free Markov field[END_REF]. In fact, Gross showed in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] that the log Sobolev inequality (LSI) is equivalent to hypercontractivity. Later, in [START_REF] Janson | On hypercontractivity for multipliers of orthogonal polynomials[END_REF][START_REF] Janson | On complex hypercontractivity[END_REF], Janson discovered a stronger form of hypercontractivity that holds for holomorphic test functions.

Theorem 1.1 (Janson [START_REF] Janson | On hypercontractivity for multipliers of orthogonal polynomials[END_REF]). If µ is the standard Gaussian measure on C n , and 0 < p ≤ q < ∞, then for all holomorphic functions f ∈ L p (C n , µ), f (e -t • ) q ≤ f p for t ≥ 1 2 ln q p ; for t < 1 2 ln q p , the dilated function f (e -t • ) is not in L q (C n , µ). Remark 1.2. Nelson's hypercontractivity estimates [START_REF] Nelson | The free Markov field[END_REF] involve the semigroup e -tAµ , where A µ is the Dirichlet form operator for the measure µ: ´|∇f | 2 dµ = ´f A µ f dµ. If dµ = ρ dx has a smooth density ρ, integration by parts shows that A µ = -∆-(∇ρ/ρ)•∇, and so when applied to holomorphic (hence harmonic) functions, e -tAµ is the flow of the vector field ∇ρ/ρ. For the standard Gaussian measure, this is just the coordinate vector field x, the infinitesimal generator of dilations Ef (x) = x • ∇f (x), also known as the Euler operator. The perspective of this paper, like its predecessor [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF], is that the strong hypercontractivity theorem is essentially about the dilation semigroup f → f (e -t • ), independent of the underlying measure.

Janson's strong hypercontractivity differs from Nelson's hypercontractivity in two important ways: first, the time-to-contraction is smaller, 1 2 ln q p as opposed to the larger Nelson time 1 2 ln q-1 p-1 , and second, the theorem applies even in the regime 0 < p, q < 1 where the L p "norms" are badly-behaved. Nevertheless, in [START_REF] Gross | Hypercontractivity over complex manifolds[END_REF], Gross showed that Janson's theorem is also a consequence of the same log Sobolev inequality (LSI); moreover, he generalized this implication considerably to complex manifolds (equipped with sufficiently nice measures). The reverse implication, however, was not established: the proof requires (LSI) to hold for non-holomorphic functions (in particular of the form |f | p/2 ). We refer the reader to [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF] for an extensive list of recent literature on strong hypercontractivity in the holomorphic category, and related ideas (notably reverse hypercontractivity) in the subharmonic category.

The aim of the present paper is to prove an intrinsic equivalence of strong hypercontractivity and a log Sobolev inequality. The starting point is a generalization of Theorem 1.1 beyond the holomorphic category. A function on R n is log-subharmonic (LSH for short) if ln |f | is subharmonic; holomorphic functions are prime examples. In [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF], we proved that Theorem 1.1 holds in the larger class LSH, for the Gaussian measure and several others. We also established a weak connection to a strong log Sobolev inequality. Definition 1.3. A measure µ on R n satisfies a strong logarithmic Sobolev inequality if there is a constant c > 0 so that, for non-negative g ∈ LSH sufficiently smooth and integrable,

Ent µ (g) ≤ c 2 ˆEg dµ. (sLSI)
Inequality (sLSI) could be written equivalently in the form Ent µ (f 2 ) ≤ c ´f Ef dµ; we will use it in L 1form throughout. In [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF], we showed the strong log Sobolev inequality holds for the standard Gaussian measure on R n , with constant c = 1 (half the constant from (LSI)), and conjectured that (sLSI) is equivalent in greater generality to the following form of Janson's strong hypercontractivity. Definition 1.4. A measure µ on R n satisfies the property of strong hypercontractivity if there is a constant c > 0 so that, for 0 < p ≤ q < ∞ and for every f ∈ L p (µ) ∩ LSH, we have

f (r • ) L q (µ) ≤ f L p (µ) if 0 < r ≤ (p/q) c/2 .
(sHC)

Remark 1.5. The statement in Definition 1.4 is given in multiplicative notation rather than additive, with r = e -t scaling the variable. It would appear more convenient to use the constant c instead of c 2 in (sLSI) and (sHC). We choose to normalize with c 2 for historical reasons: Gross's equivalence of the log Sobolev inequality and Nelson's hypercontractivity equates c in (LSI) to c 2 scaling the time to contraction.

Notation 1.6. For a function f on R n and r ∈ [0, 1], f r denotes the function f r (x) = f (rx).

Main Results

In [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF], we showed that (sHC) implies (sLSI) in the special case that the measure µ is compactly supported. Our first result is the converse.

Theorem 1.7. Let µ be a compactly supported measure on R n . Suppose that µ satisfies (sLSI) for all sufficiently smooth functions g ∈ LSH(R n ). Then µ satisfies (sHC) for all functions f ∈ LSH(R n ).

Remark 1.8. We emphasize here that the domains in the equivalence consist of log-subharmonic functions a priori defined on all of R n , not just on the support of µ. Indeed, the dilation semigroup is not well-defined if this is not satisfied. In fact, it is not hard to see that this result extends to log-subharmonic functions defined on any star-shaped open region containing the support of µ.

Theorem 1.7 and its converse have non-trivial applications: for example, Proposition [12, Proposition 4.2] implies that (sLSI) holds true for any compactly supported symmetric measure on R, with constant c ≤ 2. Nevertheless, it excludes the standard players in log Sobolev inequalities, most notably Gaussian measures. In [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF]Theorem 5.8], we proved directly that (sLSI) holds true for the standard Gaussian measure on R n , with best constant c = 1. This was proved directly from (LSI), and relied heavily on the precise form of the Gaussian measure; a direct connection to strong hypercontractivity (also proved for the Gaussian measure in [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF]Theorem 3.2]) was not provided. That connection, for a wide class of measures, is the present goal.

The technicalities involved in establishing the equivalence of (sLSI) and (sHC) are challenging because of the rigidity of the class LSH: standard cut-off approximations needed to use integrability arguments in the proof are unavailable for subharmonic functions. To amend this, we use a fundamentally different approximation technique: the dilated convolution introduced in [START_REF] Galaz-Fontes | Reverse hypercontractivity over manifolds[END_REF][START_REF] Gross | Reverse hypercontractivity for subharmonic functions[END_REF], and developed in Section 2.2 below. In [START_REF] Galaz-Fontes | Reverse hypercontractivity over manifolds[END_REF], the authors provided a local condition on the density of µ under which this operation is bounded on L p (µ) (amounting to a bound on the Jacobian derivative of the translation and dilation). Here we present alternative conditions, which require little in terms of the local behaviour of the measure (they are essentially growth conditions near infinity) and achieve the same effect. Definition 1.9. Let p > 0 and let µ be a positive measure on R n with density ρ. Say that µ (or ρ) is Euclidean exponential type p if ρ(x) > 0 for all x and if the following two conditions hold:

sup x sup |y|≤s |x| p ρ(ax + y) ρ(x) < ∞ for any a > 1, s ≥ 0 (1.1) sup x sup 1<a<1+ǫ ρ(ax) ρ(x) < ∞ for some ǫ > 0. (1.2)
If µ is Euclidean exponential type 0, we say it is Euclidean regular.

The terminology derives from the fact that conditions (1.1) and (1.2) insist that the Euclidean group acts on ρ in a controlled manner; exponential type refers to the growth condition involving |x| p (indeed, for p > 0 the measure must have tails that decay faster than any polynomial to be Euclidean exponential type p). For any probability measure µ with strictly positive density ρ, denote for a ≥ 1 and p, s ≥ 0

C p µ (a, s) = C p ρ (a, s) ≡ sup x sup |y|≤s |x| p ρ(ax + y) ρ(x) . (1.3)
Then the condition that µ is Euclidean exponential type p is precisely that C p µ (a, s) < ∞ for each a > 1 and s ≥ 0, and C 0 µ (a, 0) is uniformly bounded for a close to 1. It is clear from the definition that C p µ (a, s) is an increasing function of s. Moreover, if µ is Euclidean exponential type q then it is Euclidean exponential type p for any p < q. For convenience, we will often write C µ for C 0 µ .

Example 1.10. On R, the densities (1 + x 2 ) -α for α > 1 2 are Euclidean regular. On R n the densities e -c|x| a with a, c > 0 are Euclidean exponential type p for all p > 0.

More examples and properties that prove the Euclidean regular measures form a rich class are given in Section 2.1. The purpose of introducing this class at present is its utility in proving a density theorem for an appropriate class of Sobolev-type spaces. Definition 1.11. Let µ be a measure on R n , and let p > 0. Define the Sobolev space L p E (µ) to consist of those weakly-differentiable functions f ∈ L p (µ) for which Ef ∈ L p (µ). To be clear: Ef (x) = n j=1 x j u j (x), where u j is the function (posited to exist) satisfying

-ˆ∂j ϕ f dx = ˆϕ u j dx for any ϕ ∈ C ∞ c (R n ),
where dx denotes Lebesgue measure.

Standard techniques, involving approximation by C ∞ c functions, show that L p E is dense in L p for reasonable measures. However, our goals here involve approximation of log-subharmonic functions, and the usual cut-off approximations fail to preserve subharmonicity. An alternative approach is to use a convolution approximate identity procedure, as is readily available for Lebesgue measure. The problem is that, for a given bump function ϕ, the operation f → f * ϕ is typically unbounded on L p (µ) when µ is not Lebesgue measure. Indeed, for L p of Gaussian measure, even the translation f → f ( • + y) is unbounded. The problem is that the convolution can shift mass in from near infinity. One might hope to dilate this extra mass back out near infinity, to preserve p-integrability; thus the dilated convolution f → (f * ϕ) r . Section 2.2 shows that this operation behaves well in L p spaces of Euclidean regular measures; it also preserves the cone LSH.

The main technical theorem of this paper is the following Sobolev density theorem.

Theorem 1.12. Let p ∈ [0, ∞), and let µ be a Euclidean exponential type p probability measure on R n . Then the cone LSH ∩ L p E (µ) is dense in the cone LSH ∩ L p (µ). Using Theorem 1.12, we will prove the equivalence of (sLSI) and (sHC), the latter in a nominally weaker form. Definition 1.13. Let µ be a probability measure on R n , and let 0 < p < q < ∞. Denote by LSH p<q (µ) the closure of L q (µ) ∩ LSH in L p (µ) ∩ LSH.

For any probability measure, there is a common dense subspace (L ∞ ) for all the full L q -spaces, q > 0, and so the closure of L q in L p is all of L p for p < q; the proof uses cut-offs that do not respect subharmonicity, and indeed, there are no non-constant bounded subharmonic functions. In [START_REF] Gross | Hypercontractivity over complex manifolds[END_REF], Gross showed that, under certain conditions on a measure µ on a complex manifold (in terms of its Dirichlet form operator d * d), in the presence of a full log Sobolev inequality (LSI), there is a common dense subspace for all holomorphic L q spaces of µ. In the present context of logarithmically-subharmonic functions, no such technology is known, and we will content ourselves with the spaces LSH p<q . We will consider the nature of the spaces in a future publication. This brings us to our main theorem: the equivalence of (sLSI) and (sHC) for logarithmically subharmonic functions.

Theorem 1.14. Let µ be an O(n)-invariant probability measure on R n .

(1) If µ is Euclidean exponential type p for all p > 0, and if µ satisfies the strong log Sobolev inequality (sLSI), then µ satisfies strong hypercontractivity (sHC) in the spaces LSH p<q (µ): for 0 < p ≤ q < ∞ and f ∈ LSH p<q , f r q ≤ f p for 0 < r ≤ (p/q) c/2 .

(2) If µ is Euclidean exponential type p for some p > 1, and if µ satisfies (sHC) in the above sense, then µ satisfies the strong log Sobolev inequality (sLSI):

Ent µ (g) ≤ c 2 ˆEg dµ for all g ∈ LSH ∩ L 1 E (µ) ∩ C 1 (R n ). Remark 1.15.
The global assumption of rotational-invariance in Theorem 1.14 is actually quite natural in this situation. The functional g → ´Eg dµ on the right-hand-side of our strong log Sobolev inequality is not generally positive, since the operator E is not generally self-adjoint in L 2 (µ); however, when µ is rotationally-invariant, this functional is positive on the cone LSH, as pointed out in [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF]Proposition 5.1].

We emphasize that Theorem 1.14 is intrinsic. While the two directions of the theorem require slightly different assumptions on the applicable measures, the implications between (sLSI) and (sHC) both stay within the cone LSH of log-subharmonic functions. This is the main benefit of extending Janson's strong hypercontractivity theorem from holomorphic functions to this larger class, and restricting the log-Sobolev inequality to it: here, the two are precisely equivalent.

Alternative Formulation of sHC

The following equivalent characterization of strong hypercontractivity will be useful in what follows.

Proposition 1.16. Fix c > 0 and let q(r) denote the function q(r) = r -2/c . A measure µ satisfies strong hypercontractivity (sHC) if and only if for each function f ∈ L 1 (µ) ∩ LSH,

f r q(r) ≤ f 1 and f r 1 ≤ f 1 , for r ∈ (0, 1].
Remark 1.17. Similarly, the form of strong hypercontractivity given in Theorem 1.14 is equivalent to the same inequalities above holding for all f in the nominally smaller space LSH 1<q(r) .

For the proof, it is useful to note that the class LSH is closed under f → f p for any p > 0.

Proof. First, suppose (sHC) holds with constant c. The case p = q = 1 yields f r 1 ≤ f 1 for 0 < r ≤ (p/q) 2/c = 1. More generally, by (sHC), f r q ≤ f 1 whenever 0 < r ≤ (1/q) c/2 ; i.e. whenever q ≥ r -c/2 = q(r). In particular, it follows that f r q(r) ≤ f 1 as claimed.

Conversely, suppose the above conditions hold true. Fix q ≥ p > 0 and let f ∈ L p (µ) ∩ LSH. Then f p ∈ L 1 (µ)∩LSH, and so by assumption we have (f p ) r q(r) ≤ f p 1 for 0 < r ≤ 1. Since (f p ) r = (f r ) p , it follows immediately that f r p p•q(r) ≤ f p p . Setting q = p • q(r) and solving for r, we have r = r(p, q) ≡ (p/q) c/2 , and so we have proved the equality case of (sHC). Finally, suppose that r ′ ≤ r(p, q) = (p/q) c/2 ; then there is s ∈ (0, 1] so that r ′ = s • r(p, q). Dilations form a multiplicative semigroup, so f r ′ = (f r(p,q) ) s . We have just proved that f r(p,q) ∈ L q , and hence (f r(p,q) ) q is in L 1 (µ). Therefore, by assumption, [(f r(p,q) ) q ] s 1 ≤ (f r(p,q) ) q 1 ; unwinding this yields

f r ′ q q = (f r(p,q) ) s 1 q = [(f r(p,q) ) s ] q 1 = [(f r(p,q) ) q ] s 1 ≤ (f r(p,q) ) q 1 = f r(p,q) q q ≤ f q p
by the equality case, thus proving (sHC).

Remark 1.18. In fact, (sHC) implies the putatively stronger statement that r → f r q(r) is non-decreasing on [0, 1]; however, the weaker form presented above is generally easier to work with.

Convolution property

We will use the convolution operation to prove the Sobolev density theorem at the heart of this paper, as well as Theorem 1.7. We begin by showing that this operation preserves the cone LSH.

Lemma 1.19. Let f ∈ LSH. Let ϕ ≥ 0 be a C ∞ c test function. Then f * ϕ ∈ LSH ∩ C ∞ .
Proof. Since f ∈ LSH, f ≥ 0 and ln f is subharmonic. In particular, ln f is upper semi-continuous and locally bounded above, and so the same holds for f . Thus f is locally bounded and measurable; thus f * ϕ defines an

L 1 loc ∩ C ∞ function. We must show it is LSH.
Any subharmonic function is the decreasing limit of a sequence of C ∞ subharmonic functions, cf. [27, Appendix 1, Proposition 1.15]. Applying this to ln f , there is a sequence

f n ∈ LSH ∩ C ∞ such that f n ↓ f . Let g n = f n + 1
n ; so g n is strictly positive, and g n ↓ f . Since ϕ is ≥ 0, it follows from the Monotone Convergence Theorem that g n * ϕ ↓ f * ϕ pointwise. Now, (g n * ϕ)(x) = ´Rn g n (x -ω)ϕ(ω) dω. Since translation and positive dilation preserve the cone LSH, the function x → g n (x-ω)ϕ(ω) is continuous and LSH for each ω. Moreover, the function ω → g n (x-ω)ϕ(ω) is continuous and bounded. Finally, for small r, sup |t-x|≤r g n (t-ω)ϕ(ω) ≤ ϕ ∞ sup |t|≤|x|+r g n (t) is bounded uniformly in ω. It follows from [START_REF] Graczyk | Hypercontractivity for log-subharmonic functions[END_REF]Lemma 2.4] that g n * ϕ is LSH. (The statement of that lemma apparently requires the supremum to be uniform in x as well, but this is an overstatement; as the proof of the lemma clearly shows, only uniformity in ω is required). Thus, f * ϕ is the decreasing limit of strictly positive LSH functions g n * ϕ. Applying the Monotone Convergence Theorem to integrals of ln(g n * ϕ) about spheres now shows that ln(f * ϕ) is subharmonic, so f * ϕ ∈ LSH as claimed.

Compactly Supported Measures

This section is devoted to the proof of Theorem 1.7. It follows the now-standard Gross proof of such equivalence: differentiating hypercontractivity at the critical time yields the log Sobolev inequality, and vice versa. The technical issues related to differentiating under the integral can be dealt with fairly easily in the case of a compactly supported measure; the remainder of this paper develops techniques for handling measures with non-compact support. The forward direction of the theorem, that (sHC) implies (sLSI) for compactly supported measures, is [12, Theorem 5.2], so we will only include the proof of the reverse direction here.

Proof of Theorem 1.7. By assumption, (sLSI) holds for sufficiently smooth and integrable functions; here we interpret that precisely to mean Ent µ (g) ≤ c

2 ´Eg dµ for all g ∈ C 1 (R n ) for which both sides are finite. Fix f ∈ L 1 (µ)∩ LSH∩ C 1 . Utilizing Proposition 1.16, we must consider the function α(r) = f r q(r) where q(r) = r -2/c . Let β(r) = α(r) q(r) = ´f (rx) q(r) µ(dx) and set β x (r) = f (rx) q(r) so that β(r) = ´βx (r) µ(dx). Then,

∂ ∂r ln β x (r) = q ′ (r) ln f (rx) + q(r) f (rx) x • ∇f (rx).
Since q ′ (r) = -2 rc q(r), and since

x • ∇f (rx) = 1 r (Ef ) r (x) = 1 r E(f r )(x), we have ∂ ∂r β x (r) = - 2 rc f r (x) q(r) ln f r (x) q(r) + 1 r q(r)f r (x) q(r)-1 (Ef r )(x). (1.4)
Fix 0 < ǫ < 1. As f is C 1 , the function (of x) on the right-hand-side of (1.4) is uniformly bounded for r ∈ (ǫ, 1] and x ∈ suppµ (due to compactness). The Dominated Convergence Theorem thus allows differentiation under the integral, and so

β ′ (r) = ˆ∂ ∂r β x (r) µ(dx). (1.5)
Thus, since α(r) = β(r) 1/q(r) and β(r) > 0, it follows that α is C 1 on (ǫ, 1] and the chain rule yields

α ′ (r) = α(r) q(r)β(r) 2 rc β(r) ln β(r) + rc 2 β ′ (r) . (1.6)
From (1.4) and (1.5), the quantity in brackets is

ˆf q(r) r dµ • ln ˆf q(r) r dµ + rc 2 ˆ - 2 rc f q(r) r ln f q(r) r + 1 r q(r)f q(r)-1 r Ef r dµ = ˆf q(r) r dµ • ln ˆf q(r) r dµ -ˆf q(r) r ln f q(r) r dµ + q(r) c 2 ˆf q(r)-1 r Ef r dµ = -Ent µ (f q(r) r ) + c 2 ˆE(f q(r) r ) dµ, (1.7)
where the equality in the last term follows from the chain rule. Since f ∈ C 1 , it is bounded on the compact set suppµ, and so are all of its dilations f r . Hence, both terms in (1.7) are finite, and so by the assumption of the theorem, this term is ≥ 0. From (1.6), we therefore have α ′ (r) ≥ 0 for all r > ǫ. Since this is true for each ǫ > 0, it holds true for r ∈ (0, 1]. This verifies the first inequality in Proposition 1.16. For the second, we use precisely the same argument to justify differentiating under the integral to find

∂ ∂r f r 1 = ˆ∂ ∂r f r (x) µ(dx) = 1 r ˆEf r (x) µ(dx) ≥ 2 cr Ent µ (f r ) ≥ 0
by the assumption of (sLSI). This concludes the proof for f ∈ C 1 . Now, if f ∈ L 1 (µ) ∩ LSH, we consider a smooth approximate identity sequence ϕ k . The inequalities in Proposition 1.16 hold for f * ϕ k by the first part of the proof and Lemma 1.19. Note by simple change of variables that (f * ϕ k ) r = f r * (r n ϕ k ) r , and that (r n ϕ k ) r is also an approximate identity sequence. The function f r is LSH, so it is upper semi-continuous and consequently locally bounded. Thus f r ∈ L q(r) and (f * ϕ k ) r converges to f r in L q(r) . This concludes the proof.

2 Density results through LSH functions

Properties of Euclidean regular measures

In this section, we show several closure properties of the class of Euclidean regular measures (of any given exponential type p ∈ [0, ∞)): it is closed under bounded perturbations, convex combinations, product, and convolution. Throughout, we use µ i (i = 1, 2) to stand for such measures, and ρ i to stand for their densities.

Proposition 2.1. Let µ 1 and µ 2 be positive measures on R n , and suppose

µ 1 is Euclidean exponential type p ∈ [0, ∞). If there are constants C, D > 0 such that Cµ 1 ≤ µ 2 ≤ Dµ 1 , then µ 2 is also Euclidean exponential type p. Proof. The assumption is that Cρ 1 ≤ ρ 2 ≤ Dρ 1 . Let ǫ > 0 be such that sup 1<a<1+ǫ C 0 ρ 1 (a, 0) < ∞.
Then for any such a, ρ 2 (ax)

ρ 2 (x) ≤ Dρ 1 (ax) Cρ 1 (x) ≤ D C C 0 ρ 1 (a, 0)
for all x; thus C 0 ρ 2 (a, 0) ≤ D C C 0 ρ 1 (a, 0), and so sup 1<a<1+ǫ C 0 ρ 2 (a, 0) < ∞. Similarly, for x, y ∈ R n and a > 1,

|x| p ρ 2 (ax + y) ρ 2 (x) ≤ |x| p Dρ 1 (ax + y) Cρ 1 (x) ≤ D C C p ρ 1 (a, |y|) and so C p ρ 2 (a, s) ≤ D C C p ρ 1 (a, s) < ∞.
Proposition 2.2. Let µ 1 and µ 2 be Euclidean regular measures of exponential type p ∈ [0, ∞).

For any t ∈ [0, 1], µ = (1 -t)µ 1 + tµ 2 is Euclidean exponential type p. Proof. Let ǫ > 0 be such that sup 1<a<1+ǫ C 0 ρ i (a, 0) < ∞ for i = 1, 2.
Let ρ be the density of µ. Then for any

x ∈ R n , ρ(ax) = (1 -t)ρ 1 (ax) + tρ 2 (ax) ≤ (1 -t)C 0 ρ 1 (a, 0)ρ 1 (x) + tC 0 ρ 2 (a, 0)ρ 2 (x) ≤ max{C 0 ρ 1 (a, 0), C 0 ρ 2 (a, 0)}ρ(x)
and so C 0 ρ (a, 0) ≤ max{C 0 ρ 1 (a, 0), C 0 ρ 2 (a, 0)} is uniformly bounded for 1 < a < 1 + ǫ, as required. Similarly, for x, y ∈ R n and a > 1,

|x| p ρ(ax + y) ≤ (1 -t)|x| p ρ 1 (ax + y) + t|x| p ρ 2 (ax + y) ≤ (1 -t)C p ρ 1 (a, |y|)ρ 1 (x) + tC p ρ 2 (a, |y|)ρ 2 (x) ≤ max{C p ρ 1 (a, |y|), C p ρ 2 (a, |y|)}ρ(x)
which shows that C p ρ (a, s) ≤ max{C p ρ 1 (a, s), C p ρ 2 (a, s)} < ∞ for a ≥ 1 and s ≥ 0. Proof. For i = 1, 2 let ρ i be the density of µ i ; then

µ 1 ⊗ µ 2 has density ρ 1 ⊗ ρ 2 (x 1 , x 2 ) = ρ 1 (x 1 )ρ 2 (x 2 ). Fix ǫ > 0 so that sup 1<a<1+ǫ C p ρ i (0, a) < ∞ for i = 1, 2. Then, letting x = (x 1 , x 2 ), ρ 1 ⊗ ρ 2 (ax) = ρ 1 (ax 1 )ρ 2 (ax 2 ) ≤ C 0 ρ 1 (a, 0) ρ 1 (x 1 ) • C 0 ρ 2 (a, 0) ρ 2 (x 2 )
and so C 0

ρ 1 ⊗ρ 2 (a, 0) ≤ C 0 ρ 1 (a, 0) • C 0 ρ 2 (a, 0), meaning sup 1<a<1+ǫ C 0 ρ 1 ⊗ρ 2 (a, 0) < ∞.
Similarly, for fixed x, y ∈ R n 1 +n 2 and a > 1,

|x| p ρ 1 ⊗ ρ 2 (ax + y) = (|x 1 | + |x 2 |) p ρ 1 (ax 1 + y 1 )ρ 2 (ax 2 + y 2 ).

By elementary calculus, (|x

1 | + |x 2 |) p ≤ 2 p-1 (|x 1 | p + |x 2 | p )
, and so we have

|x| p ρ 1 ⊗ ρ 2 (ax + y) ≤ 2 p-1 |x 1 | p ρ 1 (ax 1 + y 1 ) • ρ 2 (ax 2 + y 2 ) + 2 p-1 ρ 1 (ax 1 + y 1 ) • |x 2 | p ρ 2 (ax 2 + y 2 ).
For the first term, we have

|x 1 | p ρ 1 (ax 1 + y 1 ) ≤ C p ρ 1 (a, |y 1 |)ρ 1 (x 1 ) while ρ 2 (ax 2 + y 2 ) ≤ C 0 ρ 2 (a, |y 2 |); for the second term, we have ρ 1 (ax 1 + y 1 ) ≤ C 0 ρ 1 (a, |y 1 |)ρ 1 (x 1 ) while |x 2 | p ρ 2 (ax 2 + y 2 ) ≤ C p ρ 2 (a, |y 2 |). If |y| ≤ s then |y i | ≤ s for i = 1, 2. All together, this shows that C p ρ 1 ⊗ρ 2 (a, s) ≤ 2 p-1 C p ρ 1 (a, s)C 0 ρ 2 (a, s) + C 0 ρ 1 (a, s)C p ρ 2 (a, s)
which is finite since both ρ 1 , ρ 2 are Euclidean exponential type p (and hence also Euclidean regular). This proves the proposition.

Proposition 2.4. Let µ 1 and µ 2 be positive measures on R n , each of Euclidean exponential type p ∈ [0, ∞).

Then µ 1 * µ 2 is Euclidean exponential type p.

Proof. Let ρ j be the density of µ j . By assumption, for i = 1, 2 C p i ρ i (a, s) < ∞ for a > 1 and s ≥ 0, and there is ǫ > 0 such that sup 1<a<1+ǫ C 0 ρ i (a, 0) < ∞, cf. (1.3). Then for a ≥ 1 and

x ∈ R n ρ 1 * ρ 2 (ax) = ˆρ1 (ax -u)ρ 2 (u) du = a n ˆρ1 (ax -av)ρ 2 (av) dv. By definition, ρ 1 (a(x -v)) ≤ C 0 ρ 1 (a, 0)ρ 1 (x -v) and ρ 2 (av) ≤ C 0 ρ 2 (a, 0)ρ 2 (v) for all x, v. Thus ρ 1 * ρ 2 (ax) ≤ a n C 0 ρ 1 (a, 0) • C 0 ρ 2 (a, 0) ˆρ1 (x -v)ρ 2 (v) dv = a n C 0 ρ 1 (a, 0) • C 0 ρ 2 (a, 0)ρ 1 * ρ 2 (x). It follows that C 0 ρ 1 * ρ 2 (a, 0) ≤ a n C 0 ρ 1 (a, 0) • C 0 ρ 2 (a, 0
), and hence

sup 1<a<1+ǫ C 0 ρ 1 * ρ 2 (a, 0) ≤ (1 + ǫ) n sup 1<a<1+ǫ C 0 ρ 1 (a, 0) • sup 1<a<1+ǫ C 0 ρ 2 (a, 0) < ∞ (2.1)
as required. Similarly, for x, y ∈ R n and a > 1,

|x| p ρ 1 * ρ 2 (ax + y) = |x| p ˆρ1 (ax + y -u)ρ 2 (u) du = a n ˆ|x| p ρ 1 (a(x -v) + y)ρ 2 (av) dv.
Note (by elementary calculus) that |x| p ≤ 2 p-1 (|x -v| p + |v| p ), and so

|x| p ρ 1 * ρ 2 (ax + y) ≤ 2 p-1 a n ˆ|x -v| p ρ 1 (a(x -v) + y)ρ 2 (av) dv + ˆρ1 (a(x -v) + y)|v| p ρ(av) dv .
In the first term, we have |x -v| p ρ 1 (a(x -v) + y) ≤ C p ρ 1 (a, |y|)ρ 1 (x -v) and ρ 2 (av) ≤ C 0 ρ 2 (a, 0), and so

ˆ|x -v| p ρ 1 (a(x -v) + y)ρ 2 (av) dv ≤ C p ρ 1 (a, |y|) • C 0 ρ 2 (a, 0) ρ 1 * ρ 2 (x).
In the second term, we have ρ 1 (a(x -v) + y) ≤ C 0 ρ 1 (a, |y|)ρ 1 (x -v) and |v| p ρ(av) ≤ C p ρ 2 (a, 0), and so

ˆρ1 (a(x -v) + y)|v| p ρ(av) dv ≤ C 0 ρ 1 (a, |y|) • C p ρ 2 (a, 0) ρ 1 * ρ 2 (x).
All together, for any s ≥ |y|, this gives

C p ρ 1 * ρ 2 (a, s) ≤ 2 p-1 a n C p ρ 1 (a, s) • C 0 ρ 2 (a, 0) + C 0 ρ 1 (a, s) • C p ρ 2 (a, 0) (2.2)
which is finite since both ρ 1 and ρ 2 are Euclidean exponential type p (and thus also Euclidean regular). Equations (2.1) and (2.2) prove the proposition.

Continuity of the Dilated Convolution

One easy consequence of Definition 1.9 is that the operation f → f r is bounded on L p .

Lemma 2.5. Let µ be a Euclidean regular probability measure, let p > 0, and let r ∈ (0, 1). Then

f r L p (µ) ≤ r -n/p C µ 1 r , 0 1/p f L p (µ) .
Proof. We simply change variables u = rx and use Definition 1.9:

ˆ|f r (x)| p µ(dx) = ˆ|f (rx)| p ρ(x) dx = r -n ˆ|f (u)| p ρ(x/r) dx ≤ r -n C µ 1 r , 0 ˆ|f (u)| p ρ(x) dx.
Remark 2.6. By condition (1.2) of Definition 1.9, the constant in Lemma 2.5 is uniformly bounded for r ∈ (ǫ, 1] for any ǫ > 0; that is, there is a uniform (independent of r) constant C ǫ so that, for r ∈ (ǫ, 1],

f r L p (µ) ≤ C ǫ f L p (µ) .
The next proposition shows that, under the assumptions of Definition 1.9, the dilated convolution operation is indeed bounded on L p . As usual, the conjugate exponent

p ′ to p ∈ [1, ∞) is defined by 1 p + 1 p ′ = 1.
Proposition 2.7. Let µ be a Euclidean regular probability measure on R n . Let p ∈ [1, ∞), and let ϕ ∈ C ∞ c be a test function. Then the dilated convolution operation f → (f * ϕ) r is bounded on L p (µ) for each r ∈ (0, 1). Precisely, if K = suppϕ and s = sup{|w| ; w ∈ K}, then

(f * ϕ) r L p (µ) ≤ r -n/p C µ ( 1 r , s r ) 1/p Vol(K) 1/p ϕ L p ′ (K) f L p (µ) ,
where C µ is the constant defined in (1.3).

Proof. Denote by K the support of ϕ. By definition,

(f * ϕ) r p L p (µ) = ˆRn ˆK f (rx -y)ϕ(y) dy p ρ(x) dx.
We immediately estimate the internal integral using Hölder's inequality:

ˆK f (rx -y)ϕ(y) dy p ≤ ˆK |f (rx -y)| p dy • ϕ p L p ′ (K) ,
which is finite since the first integral is the p-th power of the L p -norm of f restricted to the compact set rx -K. Hence,

(f * ϕ) r p L p (µ) ≤ ϕ p L p ′ (K) ˆRn ˆK |f (rx -y)| p dy ρ(x) dx. (2.3)
We apply Fubini's theorem to the double integral, which is therefore equal to

ˆK ˆRn |f (rx -y)| p ρ(x) dx dy = ˆK r -n ˆRn |f (u)| p ρ u + y r du dy (2.4)
where we have made the change of variables u = rx -y in the internal integral. By assumption, ρ is Euclidean regular, and so we have

ρ( 1 r u + 1 r y) ≤ C µ ( 1 r , s r ) ρ(u), y ∈ K. (2.5)
where s = sup{|w| ; w ∈ K}. Substituting (2.5) into (2.4), we see that (2.3) yields

(f * ϕ) r p L p (µ) ≤ r -n C µ ( 1 r , s r ) Vol(K) ϕ p L p ′ (K) ˆ|f (u)| p ρ(u) du.
This completes the proof.

Remark 2.8. The explicit constant in Proposition 2.7 appears to depend strongly on the support set of ϕ, but it does not. Indeed, it is easy to check that the standard rescaling of a test function, ϕ s (x) = s -n ϕ(x/s), which preserves total mass, also preserves the ϕ-dependent quantity above; to be precise, Vol(suppϕ s ) ϕ s p L p ′ (R n ) does not vary with s. In addition, the constant C µ (1/r, s/r) is well-behaved as s shrinks (indeed, it only decreases). It is for this reason that the proposition allows us to use the dilated convolution operation with an approximate identity sequence in what follows.

The use of Proposition 2.7 is that it allows us to approximate an L p function by smoother L p functions, along a path through LSH functions. To prove this, we first require the following continuity lemma. Lemma 2.9. Let µ be a Euclidean regular probability measure, and let r ∈ (0, 1). Then for any f ∈ L p (µ), the map T f : R n → L p (µ) given by [T f (y)](x) = f r (x -y) is continuous.

Proof. First note that, by the change of variables u = rx -ry,

T f (y) p L p (µ) = ˆ|f (rx -ry)| p ρ(x) dx = r -n ˆ|f (u)| p ρ 1 r u + y du,
and the latter is bounded above by

r -n C µ ( 1 r , |y|) f p L p (µ) , showing that the range of T f is truly in L p (µ) for y ∈ R n . Now, fix ǫ > 0 and let ψ ∈ C c (R n ) be such that f -ψ L p (µ) < ǫ. Let (y k ) ∞
k=1 be a sequence in R n with limit y 0 . Then

T f (y k ) -T f (y 0 ) L p (µ) ≤ T f (y k ) -T ψ (y k ) L p (µ) + T ψ (y k ) -T ψ (y 0 ) L p (µ) + T ψ (y 0 ) -T f (y 0 ) L p (µ) .
The first and last terms are simply T ψ-f (y k ) (with k = 0 for the last term), and so we have just proved that

T ψ-f (y k ) L p (µ) ≤ r -n/p C µ 1 r , |y k | 1/p ψ -f L p (µ) < r -n/p C µ 1 r , |y k | 1/p ǫ.
Moreover, there is a constant s so that |y k | ≤ s for all k, and since C µ (a, s) is an increasing function of s, it follows that

T f (y k ) -T f (y 0 ) L p (µ) ≤ T ψ (y k ) -T ψ (y 0 ) L p (µ) + 2r -n/p C µ 1 r , s 1/p ǫ.
For each x, (T ψ (y k )(x) -T ψ (y 0 )(x) = ψ(rx -ry k ) -ψ(rx -ry 0 ) converges to 0 since ry k → ry 0 and ψ is continuous. In addition, ψ r is compactly supported and continuous, so it is uniformly bounded. Since µ is a probability measure, it now follows that T ψ (y k ) -T ψ (y 0 ) L p (µ) → 0 as y k → y 0 , and the lemma follows by letting ǫ ↓ 0.

Corollary 2.10. Let µ be a Euclidean regular probability measure, and let r ∈ (0, 1). Then for any f ∈ L p (µ), and ϕ k an approximate identity sequence

(ϕ k ∈ C ∞ c (R n ) with ´ϕk (x) dx = 1 and suppϕ k ↓ {0}), f r * ϕ k -f r L p (µ) → 0 as k → ∞. Proof. Fix ǫ > 0 and let ψ ∈ C c (R n ) be such that f -ψ L p (µ) < ǫ. The standard 3 term inequality in this case is f r * ϕ k -f r L p (µ) ≤ (f r -ψ r ) * ϕ k L p (µ) + ψ r * ϕ k -ψ r L p (µ) + ψ r -f r L p (µ) . (2.6)
Following the proof of Lemma 2.9, we have f r -ψ r L p (µ) ≤ r -n/p C µ (1/r, 0) 1/p ǫ, and from condition (1.2) of Definition 1.9 this is a uniformly bounded constant times ǫ for r away from 0. Also, note that

f r * ϕ k (x) = ˆfr (x -y)ϕ k (y) dy = ˆf (rx -ry)ϕ k (y) dy = r -n ˆf (rx -u)ϕ k (u/r) du;
that is to say, f r * ϕ k = r -n (f * φk ) r , where we set φk = (ϕ k ) 1/r . Hence,

(f -ψ) r * φk L p (µ) = r -n ((f -ψ) * φk ) r L p (µ) ≤ r -n r -n/p C µ 1 r , s k r 1/p Vol(supp φk ) 1/p φk L p ′ (R n ) • f -ψ L p (µ)
by Proposition 2.7, where s k = sup{|w| ; w ∈ suppϕ k }. Since C µ (1/r, s r ) is increasing in s, this constant is uniformly bounded as k → ∞. What's more, cf. Remark 2.8, the product Vol(supp φk ) 1/p φk L p ′ (R n ) can also be made constant with k (for example by choosing ϕ k (x) = k n ϕ(kx) for some fixed unit mass C ∞ c test-function ϕ). The result is that both the first and last terms in (2.6) are uniformly small as k → ∞. Thus, we need only show that ψ r * ϕ k → ψ r in L p (µ). The quantity in question is the pth root of

ˆ ˆψr (x -y)ϕ k (y) dy -ψ r (x) p µ(dx) = ˆ ˆKk [ψ r (x -y) -ψ r (x)]ϕ k (y) dy p µ(dx), (2.7)
where we have used the fact that ϕ k is a probability density; here K k denotes the support of ϕ k . Since ψ r is bounded, we may make the blunt estimate that the quantity in (2.7) is

≤ ˆsup y∈K k |ψ r (x -y) -ψ r (x)| p ˆKk ϕ k (y) dy p µ(dx) = ˆsup y∈K k |ψ r (x -y) -ψ r (x)| p µ(dx).
Since ψ r is continuous and K k is compact, there is a point y k ∈ K k such that the supremum is achieved at y k :

sup y∈K k |ψ r (x -y) -ψ r (x)| p = |ψ r (x -y k ) -ψ r (x)| p .
As k → ∞, the support K k of ψ k shrinks to {0}, and so y k → 0. The function |ψ r (x -y k ) -ψ r (x)| p is continuous in x, and so converges to 0 pointwise as y k → 0. It therefore follows from the dominated convergence theorem that ψ r * ϕ k -ψ r L p (µ) → 0, completing the proof.

We will now use Proposition 2.7 and Corollary 2.10 to prove our main approximation theorem: that L p E (µ) is dense in L p (µ) through log-subharmonic functions.

The Proof of Theorem 1.12

Proof of Theorem 1.12. The basic idea of the proof is as follows: approximate a function f ∈ LSH ∩ L p (µ) by (f * ϕ) r , and let ϕ run through an approximate identity sequence and r tend to 1. We show that the dilated convolution (f * ϕ) r is in LSH ∩ L p E (µ), and that these may be used to approximate f in L p -sense.

Part 1: (f * ϕ) r is in LSH ∩ L p E (µ). Let ϕ ∈ C ∞ c (R n
) be a non-negative test function. Lemma 1.19 shows that f * ϕ is C ∞ and LSH. It is elementary to verify that the cone C ∞ ∩ LSH is invariant under dilations g → g r ; hence the dilated convolution (f * ϕ) r is C ∞ and LSH. For fixed r < 1, Proposition 2.7 shows that (f * ϕ) r is in L p (µ), since f ∈ L p (µ). We must now apply the differential operator E. Note that (f * ϕ) r is C ∞ , and so

E[(f * ϕ) r ](x) = x • ∇[(f * ϕ) r ](x) = ˆrx • ∇ϕ (rx -y)f (y) dy.
Decomposing rx = (rx -y) + y, we break this up as two terms

E[(f * ϕ) r ](x) = ˆ(rx -y) • ∇ϕ (rx -y)f (y) dy + ˆy • ∇ϕ(rx -y)f (y) dy.
(

The first term is just (f * Eϕ) r (x), and since Eϕ is also C ∞ c (R n ), Proposition 2.7 bounds the L p -norm of this term by the L p -norm of f . Hence, it suffices to show that the second term in (2.8) defines an L p (µ)-function of x. We now proceed analogously to the proof of Proposition 2.7. Changing variables u = rx -y for fixed x in the internal integral and then using Hölder's inequality,

ˆRn ˆRn y • ∇ϕ (rx -y)f (y) dy p ρ(x) dx = ˆRn ˆK (rx -u) • ∇ ϕ(u)f (rx -u) du p ρ(x) dx ≤ ˆRn ˆK |rx -u| p |f (rx -u)| p du ˆK |∇ϕ (u)| p ′ dy p/p ′ ρ(x) dx,
where K = suppϕ. Note that ∇ϕ p ′ < ∞ is a constant independent of f . So we must consider the double integral, to which we apply Fubini's theorem,

ˆRn ˆK |rx -u| p |f (rx -u)| p du ρ(x) dx = ˆK ˆRn |rx -u| p |f (rx -u)| p ρ(x) dx du.
Now we change variables v = rx -u for fixed u in the internal integral, to achieve

ˆK ˆRn |v| p |f (v)| p ρ v + u r r -n dv du. (2.9)
Finally, we utilize the assumption that ρ is exponential type p, and so there is a constant C(p, r, K) so that

|v| p ρ( v+u r ) ≤ C(p, r, K)ρ(u) for u ∈ K. Hence the integral in (2.9) is bounded above by C(p, r, K)r -n Vol(K) times the finite norm ´|f | p dµ, which demonstrates that E[(f * ϕ) r ] is in L p (µ).
Part 2: (f * ϕ) r approximates f in L p (µ). Let ϕ k be an approximate identity sequence. Note by simple change of variables that (f * ϕ k ) r = f r * (r n ϕ k ) r , and that (r n ϕ k ) r is also an approximate identity sequence. Since f r ∈ L p (µ), by Lemma 2.5, it follows from Corollary 2.10 that

(f * ϕ k ) r → f r , k → ∞, in L p (µ). We must now show that f r → f in L p (µ) as r ↑ 1. For this purpose, once again fix ǫ > 0 and choose a ψ ∈ C c (R n ) so that f -ψ L p (µ) < ǫ. Then f -f r L p (µ) ≤ f -ψ L p (µ) + ψ -ψ r L p (µ) + ψ r -f r L p (µ) .
(2.10)

The first term is < ǫ, and changing variables the last term is

ψ r -f r p L p (µ) = ˆ|ψ(rx) -f (rx)| p ρ(x) dx = r -n ˆ|ψ(u) -f (u)| p ρ(u/r) du ≤ r -n C µ 1 r , 0 ˆ|ψ -f | p dµ.
Here we have used the fact that µ is Euclidean regular. Note that, by condition (1.2) of Definition 1.9, the constant appearing here is uniformly bounded by, say, C, for r ∈ ( 1 2 , 1]. Thence, the last term in (2.10) is bounded above by C 1/p ǫ and is also uniformly small. Finally, the middle term tends to 0 as r ↑ 1 since ψ r → ψ pointwise and the integrand is uniformly bounded. Letting ǫ tend to 0 completes the proof.

3 The Intrinsic Equivalence of (sLSI) and (sHC)

In this section, we prove Theorem 1.14: if a measure µ is sufficiently Euclidean regular (satisfying the conditions of Definition 1.9), and if µ is invariant under rotations, then µ satisfies a strong log-Sobolev inequality precisely when it satisfies strong hypercontractivity. It will be useful to fix the following notation. Notation 3.1. Let c > 0 be a fixed constant, let µ be a measure on R n , and let f be a function on R n . 1. For r ∈ (0, 1], let q = q(r) denote the function

q(r) = r -2/c .
Note that q ∈ C ∞ (0, 1], is decreasing, and q(1) = 1.

Define a function α

f,µ : (0, 1] → [0, ∞) by α f,µ (r) ≡ f r L q(r) (µ) = ˆ|f (rx)| q(r) µ(dx) 1/q(r)
.

When the function f and measure µ are clear from context, we denote α f,µ = α.

We begin with the following general statement. Lemma 3.2. Suppose µ is a Euclidean regular probability measure. Let q 0 > 1, and let f ≥ 0 be in L q 0 (µ) ∩ C ∞ (R n ). Let ǫ ∈ (0, 1), and suppose there are functions h 1 , h 2 ∈ L 1 (µ) such that for all r ∈ (ǫ, 1],

|f (rx) q(r) log f (rx)| ≤ h 1 (x), |f (rx) q(r)-1 Ef (rx)| ≤ h 2 (x) a.s.[x]. (3.1)
Then there is ǫ ′ ∈ (ǫ, 1) such that α = α f,µ is differentiable on (ǫ ′ , 1], and for r in this domain,

α ′ (r) = 2 crq(r) f r 1-q(r) q(r) f r q(r) q(r) log f r q(r) q(r) -ˆf (rx) q(r) log f (rx) q(r) µ(dx) + cq(r) 2 ˆf (rx) q(r)-1 Ef (rx) µ(dx) . (3.2)
Remark 3.3. Note that (1/q(r)) c/2 = r. Hence, if f ∈ LSH and µ satisfies the strong hypercontractivity property of (sHC) (with p = 1) we have α(r) ≤ f 1 = α(1) for r ∈ (0, 1]. The conditions of Lemma 3.2 guarantee that α is differentiable; hence, we essentially have that α ′ (1) ≥ 0. Equation (3.2) shows that α ′ (1) is closely related to the expression in (sLSI), and indeed this is our method for proving the logarithmic Sobolev inequality in what follows.

Proof. Set β(r, x) = f (rx) q(r) , so that α(r) q(r) = ´β(r, x) µ(dx). Note, β(r, x) = f r (x) q(r) . The function q(r) is continuous and q(1) = 1, so there is ǫ ′ > 0 so that q(r) < q 0 for r ∈ (ǫ ′ , 1); and hence f q(r) ∈ L 1 (µ).

(We increase ǫ ′ if necessary so 0 < ǫ < ǫ ′ .) As µ is Euclidean regular, Lemma 2.5 shows that f q(r) r is also in L 1 (µ), and so β(r, •) ∈ L 1 (µ) for all r ∈ (ǫ ′ , 1). Since f ∈ C ∞ , we can check quickly that β(•, x) is as well; using the fact that q ′ (r) = -2 c r -2/c-1 = -2 cr q(r), and that ∂ ∂r f (rx) = 1 r Ef (rx), logarithmic differentiation yields ∂ ∂r β(r, x) = q(r) -2 cr f (rx) q(r) log f (rx) + 1 r f (rx) q(r)-1 Ef (rx) .

From the hypotheses of the Lemma, we therefore have

∂ ∂r β(r, x) ≤ q(r) r 2 c h 1 (x) + h 2 (x)
for almost every x ∈ R n , for r ∈ (ǫ ′ , 1]. As q(r)/r is uniformly bounded on (ǫ ′ , 1], we see that | ∂ ∂r β(r, x)| is uniformly bounded above by an L 1 (µ) function. It now follows from the Lebesgue differentiation theorem that α(r) q(r) = ´β(r, x) µ(dx) is differentiable on a neighbourhood of 1, and d dr α(r) q(r) = ˆ∂ ∂r β(r, x) µ(dx)

= -2 cr q(r) ˆf (rx) q(r) log f (rx) µ(dx) + 1 r q(r) ˆf (rx) q(r)-1 Ef (rx) µ(dx).

(3.4)

Consequently α(r) is differentiable in a neighbourhood of 1. Again using logarithmic differentiation,

α ′ (r) = α(r) d dr log α(r) = α(r) d dr 1 q(r)
log α(r) q(r) , and again using the fact that q ′ (r) = -2 cr q(r), d dr 1 q(r) log α(r) q(r) = 2 crq(r) log α(r) q(r) + 1 q(r) α(r) -q(r) d dr α(r) q(r) = α(r) -q(r) q(r) 2 cr α(r) q(r) log α(r) q(r) + d dr α(r) q(r) .

Combining with (3.4), we therefore have α ′ (r) = α(r) 1-q(r) q(r) 2 cr α(r) q(r) log α(r) q(r) -2 cr q(r) ˆf (rx) q(r) log f (rx) µ(dx)

+ 1 r q(r) ˆf (rx) q(r)-1 Ef (rx) µ(dx) (3.5)
Simplifying (3.5), and using the definition α(r) = f r q(r) , yields (3.2), proving the lemma.

We therefore seek conditions on a function f (and on the measure µ) which guarantee the hypotheses of Lemma 3.2 (specifically the existence of the Lebesgue dominating functions h 1 and h 2 ). Naturally, we will work with LSH functions f . We will also make the fairly strong assumption that µ is rotationally-invariant. Notation 3.4. Let f : R n → R be locally-bounded. Denote by f the spherical average of f . That is, with ϑ denoting Haar measure on the group O(n) of rotations of R n ,

f (x) = ˆO(n) f (ux) ϑ(du).
If µ is rotationally-invariant, then ´f dµ = ´f dµ for any f ∈ L 1 (µ). As such, we can immediately weaken the integrability conditions of Lemma 3.2 as follows.

Lemma 3.5. Suppose µ is a Euclidean regular probability measure that is invariant under rotations of R n . Let q 0 > 1 and let f ≥ 0 be in

L q 0 (µ) ∩ C ∞ (R n ). Denote by f 1 , f 2 : (0, 1] × R n → R the functions f 1 (r, x) = f (rx) q(r) log f (rx), f 2 (r, x) = f (rx) q(r)-1 Ef (rx). (3.6)
Fix ǫ ∈ (0, 1), and suppose that there exist functions h 1 , h 2 ∈ L 1 (µ) such that, for r ∈ (ǫ, 1], | fj (r, x)| ≤ h j (x) for almost every x, j = 1, 2. (Here fj (r, •) refers to the rotational average of f j (r, •), as per Notation 3.4.) Then the conclusion of Lemma 3.2 stands: for some ǫ ′ ∈ (ǫ, 1), the function α = α f,µ is differentiable on (ǫ ′ , 1], and its derivative is given by (3.2).

Proof. Following the proof of Lemma 3.2, only a few modifications are required. Defining β(r, x) as above, α(r) q(r) = ´β(r, x) µ(dx); since µ is rotationally-invariant, this is equal to ´β(r, x) µ(dx) where β refers to the rotational average of β in the variable x. Evidently β(r, •) is µ-integrable for sufficiently large r < 1 (since β is). To use the Lebesgue differentiation technique, we must verify that ∂ ∂r β(r, x) exists for almost every x and is uniformly bounded by an L 1 (µ) dominator. Note that β(r, x) is locally-bounded in x for each r, and so for fixed x it is easy to verify that indeed

∂ ∂r β(r, x) = ˆO(n) ∂ ∂r β(r, ux) ϑ(du).
Using (3.3), we then have

∂ ∂r β(r, x) = q(r) ˆO(n) - 2 cr f (rux) q(r) log f (rux) + 1 r f (rux) q(r)-1 Ef (rux) ϑ(du).
That is, using (3.6), ∂ ∂r β(r, x) = q(r) -2 cr f1 (r, x) + 1 r f2 (r, x) . Hence, from the assumptions of this lemma,

∂ ∂r β(r, x) ≤ q(r) r 2 c h 1 (x) + h 2 (x)
and so, since q(r)/r is uniformly bounded for r ∈ ( 1 2 , 1], it follows that α(r) q(r) = ´β(r, x) µ(dx) is differentiable near 1, with derivative given by

ˆ∂ ∂r β(r, x) µ(dx) = q(r) - 2 rc ˆf 1 (r, x) µ(dx) + 1 r ˆf 2 (r, x) µ(dx).
Now using the rotational-invariance of µ again, these integrals are the same as the corresponding non-rotated integrands ´fj (r, x) µ(dx), yielding the same result as (3.4). The remainder of the proof follows the proof of Lemma 3.2 identically.

Remark 3.6. The point of Lemma 3.5 -that it is sufficient to find uniform Lebesgue dominators for the rotational averages of the terms in (3.1) -is actually quite powerful for us. While a generic subharmonic function in dimension ≥ 2 may not have good global properties, a rotationally-invariant subharmonic function does, as the next proposition demonstrates. We will exploit this kind of behaviour to produce the necessary bounds to verify the conditions of Lemma 3.5 and prove the differentiability of the norm.

Proposition 3.7. Let f : R n → R be subharmonic and locally-bounded. Then f is also subharmonic; moreover, for fixed

x ∈ R n , r → f (rx) is an increasing function of r ∈ [0, 1]. Proof. Fix u ∈ O(n). Since f is locally-bounded, subharmonicity means that ffl B(x,r) f (t) dt ≥ f (x) for every x ∈ R n , r ∈ (0, ∞). Changing variables, we have B(x,r) f (αt) dt = u•B(x,r) f (t) dt = B(ux,r) f (t) dt ≥ f (ux).
Hence, f • u is subharmonic for each u ∈ O(n). The local-boundedness of f means that the function u → f (ux) is uniformly bounded in L 1 (O(n), ϑ) for x in a compact set, and hence it follows that f is subharmonic.

Hence f is a rotationally-invariant subharmonic function. Fix x ∈ R n and r ∈ [0, 1]. Then rx is in the ball B(0, |x|), and since f is subharmonic, the maximum principle asserts that f (rx) is no larger than the maximum of f on ∂B(0, |x|). But f is constantly equal to f (x) on ∂B(0, |x|) by rotational-invariance, and so f (rx) ≤ f (x), proving the proposition. Proposition 3.7 makes it quite easy to provide a uniform Lebesgue dominating function for the function f 1 in Lemma 3.5. Proposition 3.8. Suppose µ is a rotationally-invariant probability measure on R n . Let q 0 > 1, and let f ≥ 0 be subharmonic and in L q 0 (µ). Define f 1 as in (3.6): f 1 (r, x) = f (rx) q(r) log f (rx). Set g 1 (x) = f (x) q 0 , and set h 1 = g1 + 1; i.e. h 1 (x) = 1 + ´O(n) f (ux) q 0 ϑ(du). Then h 1 ∈ L 1 (µ) and there is an ǫ ∈ (0, 1) and a constant C > 0 so that for all r ∈ (ǫ, 1], | f1 (r, x)| ≤ Ch 1 (x) for almost every x. Remark 3.9. By the rotational-invariance of µ, ´h1 dµ = ´g 1 dµ + 1 = ´g1 dµ + 1 = ´f q 0 dµ + 1 < ∞, and so h 1 is a uniform L 1 (µ) dominator verifying the first condition of Lemma 3.5.

Proof. Choose some small δ ∈ (0, 1). First note from simple calculus that, for u ≥ 1, u -δ log u ≤ 1 eδ . Now, choose ǫ ∈ (0, 1) so that qǫ < q 0 -δ; then q(r) < q 0 -δ for r ∈

(ǫ, 1]. Consequently, if f (y) ≥ 1, we have 0 ≤ f (y) q(r) log f (y) ≤ f (y) q 0 -δ log f (y) ≤ 1 eδ f (y) q 0 .
On the other hand, for 0 ≤ u ≤ 1, |u q(r) log u| ≤ 1 eq(r) ≤ 1 e (again by simple calculus). Thus, since f ≥ 0, in total we have |f (y) q(r) log f

(y)| ≤ 1 e max 1 δ f (y) q 0 , 1 ≤ 1 eδ [f (y) q 0 + 1]. (3.7)
Set C = 1 eδ . With y = rx, the left-hand-side of (3.7) is precisely f 1 (r, x). Averaging (3.7) over O(n) and recalling that g 1 (y) = f (y) q 0 , we have

| f1 (r, x) ≤ C[ g1 (rx) + 1].
Recall that if ϕ is convex and f is subharmonic then ϕ • f is also subharmonic. Thus, since q 0 > 1 and f is subharmonic, g 1 is also subharmonic, and hence from Proposition 3.7, g1 (rx) ≤ g1 (x). This proves the proposition.

We must now bound the second term f2 (r, •) uniformly for r in a neighbourhood of 1. The following Lemma is useful in this regard. Lemma 3.10. Let k be a C ∞ non-negative subharmonic rotationally-invariant function. Then for x ∈ R n and r ∈ (0

, 1], E k(rx) ≤ r 2-n E k(x). (3.8)
Proof. Since k is rotationally-invariant, there is a function h : [0, ∞) → R so that k(x) = h(|x|). The Laplacian of k can then be expressed in terms of derivatives of h; the result is ∆ k(x) = h ′′ (|x|) + (n -1) 1 |x| h ′ (|x|).

(3.9)

Hence, since k is subharmonic and smooth, it follows that for t > 0, t h ′′ (t) + (n -1)h ′ (t) ≥ 0.

(3.10)

One can also check that, in this case, E k(x) = |x|h ′ (|x|). Now, define F (r) = r n-2 E k(rx) = r n-2 r|x|h ′ (r|x|).

Then F is smooth on (0, ∞) and F (1) = |x|h ′ (|x|) = E k(x). We differentiate, yielding Equation (3.10) with t = r|x| now yields that F ′ (r) ≥ 0 for r > 0. Hence, F (r) ≤ F (1) for r ≤ 1. This is precisely the statement of the lemma.

Proposition 3.11. Let q 0 > 1 and let µ be a rotationally-invariant probability measure on R n . Let f ≥ 0 be subharmonic, C ∞ , and in L q 0 E (µ). Define f 2 as in (3.6): f 2 (r, x) = f (rx) q(r)-1 Ef (rx). Set g 3 (x) = (f (x) q 0 -1 + 1)|Ef (x)|, and set h 2 = g3 . Then there is an ǫ ∈ (0, 1) and a constant C > 0 so that for all r ∈ (ǫ, 1], | f2 (r, x)| ≤ Ch 2 (x) for almost every x; moreover, h 2 ∈ L 1 (µ).

Proof. Fix ǫ ∈ (0, 1) small enough that q(r) < q 0 for all r ∈ (ǫ, 1]. Define g 2 (r, y) = f (y) q(r)-1 Ef (y). and note that f 2 (r, x) is given by the dilation f 2 (r, x) = g 2 (r, rx). Since E is a first-order differential operator, we can quickly check that g 2 (r, y) = 1 q(r) E(f q(r) )(y).

We now average both sides over O(n). Set k = f q(r) , which is C ∞ , and let u ∈ O(n). Then we have the following calculus identity: In other words, g2 (r, y) = 1 q(r) E( f q(r) )(y). As in the proof of Proposition 3.8, the function k = f q(r) is subharmonic, and rotationally invariant. Hence, we employ Lemma 3.10 and have g2 (r, rx) = 1 q(r) E k(rx) ≤ 1 q(r) r 2-n E k(x) = r 2-n g2 (r, x).

E(k • u)(y
Since r 2-n is uniformly bounded for r ∈ (ǫ, 1], it now suffices to find a uniform dominator for g2 (r, x).

We therefore make the estimates: since q(r) < q 0 we have |g 2 (r, x)| = f (x) q(r)-1 |Ef (x)| ≤ max{f (x) q(r)-1 , 1}|Ef (x)| ≤ max{f (x) q 0 -1 , 1}|Ef (x)| ≤ f (x) q 0 -1 + 1 |Ef (x)|. As usual, by rotational invariance of µ, ´g 3 dµ = ´g3 dµ, and so to show h 2 ∈ L 1 (µ) we need only verify that g 3 ∈ L 1 (µ). To that end, we break up g 3 (x) = f (x) q 0 -1 |Ef (x)| + |Ef (x)|. By assumption, f ∈ L q 0 E (µ) and so |Ef | ∈ L q 0 (µ); as µ is a finite measure, this means that |Ef | ∈ L 1 (µ) and hence the second term is integrable. For the first term, we use Hölder's inequality:

ˆf q 0 -1 |Ef | dµ ≤ f q 0 -1 q ′ 0 Ef q 0 = f q 0 -1 q 0 Ef q 0 .
Both terms are finite since f ∈ L q 0 E (µ), and hence g 3 ∈ L 1 (µ), proving the proposition.

Combining Lemma 3.5 and Propositions 3.8 and 3.11, we therefore have the following.

Theorem 3.12. Let q 0 > 1 and let µ be a probability measure of Euclidean type q 0 , that is invariant under rotations of R n . Suppose that µ satisfies strong hypercontractivity of (sHC) with constant c > 0. Let f ∈ L q 0 E (µ) ∩ LSH ∩ C ∞ . Then the strong log-Sobolev inequality, (sLSI), holds for f :

ˆf log f dµ -ˆf dµ log ˆf dµ ≤ c 2 ˆEf dµ.
Proof. Under the conditions stated above, the results of the preceding section show that the function α = α f,µ is differentiable on (ǫ ′ , 1] for some ǫ ′ ∈ (0, 1). Since µ satisfies strong hypercontractivity, Proposition 1.16 shows that the function α is non-decreasing on (0, 1]. It therefore follows that α ′ (r) ≥ 0 for r ∈ (ǫ ′ , 1] (here α ′ (1) denotes the left-derivative). Hence, from (3.2) we have, for r ∈ (ǫ ′ , 1],

f r q(r) q(r) log f r q(r) q(r) -ˆf (rx) q(r) log f (rx) q(r) µ(dx) + cq(r) 2 ˆf (rx) q(r)-1 Ef (rx) µ(dx) ≥ 0.

At r = 1, this reduces precisely to (sLSI), proving the result.

Theorem 3.12 is part (2) of Theorem 1.14. The proof of (1) is essentially the same.

F

  ′ (r) = |x| d dr r n-1 h ′ (r|x|) = |x|(n -1)r n-2 h ′ (r|x|) + |x|r n-1 h ′′ (r|x|)|x| = |x|r n-2 r|x|h ′′ (r|x|) + (n -1)h ′ (r|x|) .

  ) = y • ∇(k • u)(y) = y • u ⊤ ∇k(uy) = (uy) • ∇k(uy) = (Ek)(uy).For fixed y the function u → (Ek)(uy) is uniformly bounded and so we integrate both sides to yieldEk(y) = ˆO(n) (Ek)(uy) ϑ(du) = ˆE(k • u)(y) ϑ(du) = E ˆk • u(y) ϑ(du) = E( k)(y).

  That is to say,|g 2 (r, x)| ≤ g 3 (x) for r ∈ (ǫ, 1]. Hence, | g2 (r, x)| = ˆO(n) g 2 (r, ux) ϑ(du) ≤ ˆO(n) |g 2 (r, ux)| ϑ(du) ≤ ˆO(n) g 3 (ux) ϑ(du) = g3 (x) = h 2 (x),thus proving the estimate.

  Proposition 2.3. Let p ∈ [0, ∞), let µ 1 be a Euclidean exponential type p measure on R n 1 , and let Let µ 2 be a Euclidean exponential type p measure on R n 2 . Then the product measure µ 1 ⊗ µ 2 is Euclidean exponential type p on R n 1 +n 2 .

* Partly supported by ANR-09-BLAN-0084-01 † Partly supported by NSF Grant DMS-1001894 and NSF CAREER Award DMS-1254807 1


D Germain

G Fromentin

G Poulachon

S Bissey-Breton

A FORCE MODEL FOR SUPERFINISH TURNING OF PURE COPPER WITH ROUNDED EDGE TOOLS AT LOW FEED RATE

Keywords: Cutting forces, edge geometry, superfinish turning, pure copper, scale effect

This paper presents a model for force prediction of superfinish turning operation on pure copper. The model is divided in two parts. The first part computes the forces acting on the rake face of the tool. The second part computes the forces on the clearance face that are much more important in superfinish machining than in conventional machining.

INTRODUCTION

The modelling of cutting forces has been widely explored among the last decades in order to achieve optimum quality in machining, improve cutting tools and machine-tool capability. It is also still a helpful data for workpiece deformation or surface integrity predictive models. When preliminary tests cannot be performed before production starts, a physically significant model may provide enough information for the qualification of the tool geometry or even the machine-tool without producing any chip. Superfinish machining involves low feed rates, thus, uncut chip thickness is thin in front of the edge radius. Several strategies, like the mechanistic models based on the model of (Kienzle 52), exists to predict the forces but often requires a large batch of experimental data to calibrate their constants. This paper presents an analytical model for the prediction of forces in cutting of pure copper considering the scale effects encountered in superfinishing operations in the case of orthogonal machining.

DEVELOPED APPROACH

This work is based on an analytical and mechanical study of a continuous chip formation process. Let us consider the simple case of orthogonal machining. In this case, the forces are all in the same plane containing feed and cutting velocity. The work material is a pure copper with elasto-plastic mechanical behaviour. As shown on figure 1 the tool has a rounded edge and a positive rake angle.

Forces on the rake face

The primary shear zone is assumed without any curvature and in plane strain state with a constant shear strength along the shear plane (Thomsen 65). Two resultant forces are assumed; one acting on the rake face (R) due to the action of the chip on the tool and one on the clearance face (C) due to the contact with the machined surface. Those resultants are then decomposed along the cutting direction (c) and the thrust direction (t), and expressed in equation (1). The forces F c,R and F t,R acting on the rake face are the result of the plastic deformation of the work material in the primary shear zone oriented by the shear angle φ . These forces can be computed, assuming the chip equilibrium, from the following relation,

F c = F c,R + F c,C , F t = F t,R + F t,C . (1)
F shear = τ s b h sin φ , F comp = τ s b h sin φ tan (φ -γ + λ R ) , (2)
where τ s is the equivalent shear stress, h is the uncut chip thickness, b is the width of cut, λ R = arctan µ R is the friction angle and γ is the rake angle. The formulation given by (Merchant 44), frequently used in numerical simulations, assume the shear band as infinitely thin. Assuming a perfect plastic material and a Coulomb's friction law, the shear angle is obtained by the minimisation of the cutting energy. The model proposed by (Lee 51) is also frequently used but some discrepancy where found using theses two models that are special cases of the Zworykin law (Zvorykin 93) given in the following equation.

φ = C Z1 +C Z2 (γ -λ R ) (3)
Due to strain rate ε > 10 3 and true strain ε ≈ 1, classical mechanics are not sufficient. (Thomsen 65) assumed a constant shear stress equal to the shear strength along the shear plane. (Shaw 80) assumed a normal and frictional stress distribution analogous to the one of Zorev. The model proposed by Johnson and Cook is the most used in numerical simulation (Pujana 07). This model is a thermo-viscoplastic law considering strain hardening and its application range is closed to those encountered in machining. Because of the lack of data on temperature, the model used to predict the equivalent stress state in the shear band is expressed by the following relation,

τ s = 1 √ 3 σ 1 ε m if ε ≤ 1 1 √ 3 [(1 -m) σ 1 + m σ 1 ε] if ε > 1 , (4)
where σ 1 is the stress for a deformation ε = 1 and m the stress hardening index. The deformation ε is usually estimated in the middle of the shear band as

ε = 1 2 cos γ e f f √ 3 sin φ cos φ -γ e f f , (5)
from the visualisation method. Equation ( 5) is involving the effective rake angle γ e f f . This angle is usually considered in micro-machining (Outeiro 05) where the uncut chip thickness is inferior or close to the edge radius r β as shown on figure 1. The root of the shear band is assumed to be localised at the stagnation point S and the stress is maximum. Then, the stress decreases along the rake face contact length to reach zero when the tool and the chip are no longer in contact as shown on figure 1b. The same type of distribution is assumed on the clearance face with a continuity with the rake face. Some results in the literature are suggesting the normal stress distribution on the rake face given in equation ( 6), where L c,R is the tool/chip contact length and x ∈ [0, L c,R ].

σ n,R (x) = σ max,R 1 - x L c,R n R (6)
The experimental results available in the literature reveal that the exponent n R , conditioning the curvature of the distribution, cannot be inferior to 1 and the the stress σ max,R is not always restricted to the tip of the tool (Barrow 82; Arsecularatne 97). Integrating equation (6) leads to

n R = σ max,R b L c,R F N,R -1, (7)
where F N,R is the normal force acting on the rake face due to the action of the chip. The tool/chip contact length L c,R is dependant of the chip thickness h c (8). As shown on figure 2, the more the chip is thick, the less it can bend, which increases its contact length on the rake face.

h c = h cos (φ -γ) sin φ (8)
The length L c,R is then obtained for a positive rake angle by the folowing equation

L c,R = h c -ζ 1 γ + ζ 2 , (9)
where ζ 1 and ζ 2 are empirical constants. The stress σ max,R is obtained from the Hencky's theorem p + 2 τ ψ = constant (Oxley 77). The normal stress at the stagnation point S is then computed from

σ max,R = 2 τ s tan (φ + λ R -γ) -τ s 1 + 2 π 4 -φ + τ s sin 2 (γ -φ ) . ( 10
)
Assuming a constant friction coefficient µ, the frictional stress is expressed from

τ f ,R (x) = τ p if µ σ n,R ≥ τ p µ σ n,R (x) if µ σ n,R < τ p , (11)
were τ p = σ y √ 3 and σ y the yield stress.

Forces on the clearance face

The forces acting on the clearance face are usually neglected in force modelling because of the importance of the forces generated in the primary shear zone. In superfinish machining, as in micro-machining, the separation of the work material is affected by the sharpness of the edge. The two main approaches considers a stagnation point, as (Albrecht 60), or a dead metal zone acting as a stable built-up edge indenting the work material (Abdelmoneim 74). In this work, the forces generated on each face of the tool are independent and the work material is divided around a stagnation point. To compute the stresses on the clearance face, a continuity with the stresses on the rake face is assumed. The normal stress at the tool tip is still given by equation (10). The contact length between the clearance face of the tool and the machined surface is estimated from the dimension of the ploughed layer h min and computed from equation ( 12). The parameter h min is computed from the position of the stagnation point S. Its localisation is obtained from visualisation of the machined area and is given by the angle θ s = 5 π 36 -γ as visible on figure 1.

L c,C = π 2 + α -θ S r β + h min sin α with h min = r β (1 -sin θ S ), ( 12
)
where α is the clearance angle. Then, the same type of distribution as equation ( 6) is applied along L c,C as described on figure 1b. Finally, equations ( 6) and ( 11) are used with parameters related to the clearance face.

EXPERIMENTAL PROCEDURE

To calibrate the model, experimental data are required. Machining tests were performed on a SOMAB T400TD CNC lathe. ISO K20 Tungsten carbide tools were used. The input parameters are the rake angle γ, the clearance angle α, the feed rate f , the cutting speed V c and the edge radius r β that varies slightly on each tool in a range from 8 to 12 micrometres. The edge radii are measured using a stereoscopic method from SEM pictures. Orthogonal cutting tests were conducted on copper discs and tubes at constant cutting speed and feed rate. The parameters are listed in table I. This test configuration simplifies the cutting process as a two dimensional problem. The sample thickness is set to be approximately ten times the feed rate to achieve the plane state of both strain and stress hypothesis in the main shear zone (Grzesik 08). In addition, a high resolution CMOS camera (figure 3) records images of the chip formation area in order to measure geometrical parameters such as the contact area between the tool and the machined surface. For all tests, forces, axial position, feed rate and spindle speed were measured. Chip thickness is also measured to compute the shear angle φ according to the geometrical relation tan φ = (h cos γ) / (h c -h sin γ). A micro quantity lubrication device was used.

IDENTIFICATION OF THE MODEL

To use this model, some parameters have to be identified. For a large ratio h r β , the effects of the clearance face and edge radius are minimum and can be neglected. Then, in equation (1), it is allowed to assume F c,C = 0 and F t,C = 0. The constants C Z1 and C Z2 are obtained from curve fitting of equation (3) on experimentally observed shear angles φ . In the case of a large ratio h r β , the stress τ s can be expressed from F c,R and F c,C . The figure 4 illustrates this hypothesis showing a rise of the apparent shear stress below a ratio of 5. The parameters σ 1 and m can be obtained from equation (4) by curve fitting along ε. In the case of a computed exponent n R < 1 this one is set to 1 and a plateau at σ max,R is generated until the force F N,R is reached. The corresponding exponent n C on the clearance face is set to 1.

RESULTS AND DISCUSSION

As expected, the forces increase significantly with the reduction of the rake angle and slightly with the clearance angle. Small uncut chip thickness h against edge radius r β has shown an important scale effect on force measurements for configurations with the ratio h r β ≤ 5. The observations made by the camera helped to evaluate the contact with the workpiece. Despite the initial blur and the averaging effect of the post treatment, the images revealed a larger clearance contact area with tools of low rake angles. The model does not fit perfectly the experimental data as visible on figure 5. For the cutting force on figure 5a, the model cannot reach the experimental force level at the lowest depth of cut. The force prediction for a rake angle of 20°is also underestimated. The modelling of the thrust force on figure 5b presents also a lack of accuracy for the lowest depth of cut but fits relatively well the other configurations. To explain such divergence, several remarks have to be made. First of all, the force measurements are not perfectly reliable. For example, the force signals measured for the tools with a rake angle of 20°do not follow the same trend than for the other tools. Secondly, the friction on both rake and clearance faces is assumed as constant along the contact lengths L c,R and L c,C . The variation of pressure between the separation point and the end of the contact length probably affects its value. Thirdly, the visualisation of the interference volume between the workpiece and the tool cannot ensure the modelling of the angle θ S that rules the variable h min , especially for the lowest depth of cut that are too blurred. At last, it can be argued that the force components acting on the clearance face are not completely reliable. In addition, the distributions of the normal and tangential stresses are identified for a straight surface and then computed on a curved area.

As maximum stress computed on the rake face remains steady along a certain range, a similar behaviour might exist on the clearance face.

CONCLUSIONS

The purpose of this study is to describe a model for the prediction of cutting forces at the scale of superfinishing operations on pure copper. This work material is very ductile which increases the phenomenon of ploughing. The model considers the forces acting on both rake and clearance faces. The forces computed on the rake face provide interesting results but the ploughing forces need to be improved. The observation with a camera is a helping device but is a little weak to observe the interaction of the tool with the machined surface. Developments have to be done on the estimation of the forces acting on the clearance face.

Figure 1 :

 1 Figure 1: Orthogonal cutting configuration.

Figure 2 :

 2 Figure 2: Relation between chip thickness h c and contact length L c,R .

Figure 3 :

 3 Figure 3: Experimental device.

Figure 4 :

 4 Figure 4: Apparent shear stress.

  Figure 5: Modelled (lines) and experimental (dots) forces.

December 8-10, 2010