Abstract
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Introduction
Let n, k ∈ N with n ≥ 2 and 1 ≤ k < n. A density ψ on R k is given by ψ(x) := e h(|x|) for x ∈ R k where h is a smooth convex even function on R. Then ν is the log-convex measure ν := ψL k on the L k -measurable sets in R k . We employ the decomposition R n x = (x , y) with x ∈ R n−k and y ∈ R k . Let γ be a positive function in C ∞ (R n−k ). The Riemannian metric on R n associated to the differential expression
is denoted by g. The gradient associated to the metric g applied to a sufficiently smooth function u is written ∇ g u while its norm is denoted |∇ g u| g . Let ρ ∈ C(R n−k ) be a positive function. We introduce the product measure µ := ρL n−k ⊗ ν on the L n -measurable sets in R n .
The symmetric rearrangement E of a L k -measurable set E in R k with finite ν-measure is the open ball B k (0, r) centred at the origin with r ≥ 0 chosen so that ν(E) = ν(B k (0, r)). The (k, n)-Steiner symmetral E s of a L n -measurable set E in R n with finite µ-measure is the set E s := {x = (x , y) ∈ R n : ν(E x ) < +∞ and y ∈ (E x ) } where E x stands for the x -slice of E. The (k, n)-Steiner symmetrisation of a suitable L nmeasurable function u on R n is denoted by u s .
Let f : R × R n → [0, ∞) be a function with the properties (A.1) f is continuous; (A.2) the mapping R n → R; z → f (s, z) is convex for each s ∈ R;
(A.3) f (s, 0) = 0 for each s ∈ R;
(A.4) the mapping R k → R; w → f (s, z , w) is radial for each s ∈ R and z ∈ R n−k .
In particular, each function w → f (s, z , w) is increasing in |w|.
The auxiliary function j : R n × R n → R n ; (x, z) → (z , γ(x )w) (z = (z , w)) (1.2) has the property that g x (z, z) = |j(x, z)| 2 for each x, z ∈ R n .
Let 1 ≤ p < +∞ and put
R n |u| p dµ < +∞ and
where the norm of the gradient with respect to the metric g is given by
with obvious notation for ∇ x u resp. ∇ y u (see below for details). Let W In Theorem 7.6 we show that the Sobolev space W 1,p g (R n , µ) (1 ≤ p < +∞) is closed under symmetrisation · s (cf. [11] Lemma 2.4 for example). We now state our main theorem. 
The topic of Pólya-Szegö inequalities for Steiner symmetrisation has been studied intensively in the literature. Such inequalities provide an indispensable tool for studying variational problems in mathematical physics (see [28] 7.3 (1) and 7.4 for example). We discuss briefly the context of the above result. Suppose that the function f 1 : R n−1 × R × R → [0, +∞) with argument (x , s, z) is continuous, even in both s and z and convex in z. A Pólya-Szegö inequality for functionals of the form R n f 1 (x , u, |∇u|) dx with k = 1 and ψ = 1 acting on nonnegative Lipschitz functions with compact support is contained in [9] Theorem 1 and [25] Theorem 2.31 (to mention a special case); in fact, these results yield a version of Theorem 1.1 in the case k = 1 and ψ = 1 for functionals of the form R n f (u, |∇ g u| g )ρ(x ) dx acting on nonnegative Lipschitz functions with compact support with f as above. Moreover, the results in [9] , [25] hold for metrics of the form g x (z, z) = h x (z , z ) + γ(x ) 2 w 2 with h x diagonal (and more generally).
Now suppose that the function f 2 : R n → [0, +∞) is convex, vanishes at zero and is even in y := x n . A Pólya-Szegö inequality for functionals of the form R n f 2 (∇u) dx with k = 1 and ψ = 1 acting in W 1,p (R n ) (1 ≤ p < +∞) is contained in [17] Theorem 2.1 (see also Remark 4.6). A Pólya-Szegö inequality for arbitrary codimension 1 ≤ k < n has been obtained in [13] Theorem 2.1 for functionals of the form R n f 2 (∇u) dx where f 2 is convex in z, vanishes at zero and is radial in w. Yet again consider a continuous function f 3 : R n−1 ×R×R n → [0, +∞); (x , s, z) → f 3 (x , s, z) with the properties that f 3 is even in s and w, f 3 (x , 0, 0) = 0 and is convex in z. Under a boundedness assumption a Pólya-Szegö inequality for functionals of the form R n f 3 (x , u, ∇u) dx with k = 1 and ψ = 1 is obtained in [10] Theorem 4.4. We remark that it is not possible to derive (a version of) Theorem 1.1 from this last result by writing f 3 (x , s, z) = ρ(x )f (s, z , γ(x ) −1 w) in view of the contrasting convexity assumptions. We mention also that if the metric g takes the form ds 2 = |dx | 2 + γ(y) 2 dy 2 then a Pólya-Szegö inequality for functionals of the form R n f 3 (x , u, j(∇ g u)) dµ holds with k = 1 and ψ = 1 according to [19] Theorem 3.1 provided that the positive even function γ −1 on R is strictly convex (see also [9] Theorem 1). As far as we are aware, the Pólya-Szegö inequality for Steiner symmetrisation with respect to a log-convex measure ν has not been discussed in the literature. The analogous problem for (1, n)-Steiner symmetrisation for a log-concave measure ν has been treated in [7] . As an application of Theorem 1.1 we obtain a Pólya-Szegö inequality in the hyperbolic plane. We identify the vector z = (x 1 , x 2 ) ∈ R 2 with the complex number z = x 1 + ıx 2 ∈ C. The upper half-plane
is endowed with the Riemannian metric g associated with the differential expression
and we write the corresponding volume measure as µ g . We introduce orthogonal geodesics in H 2 : P := {z ∈ H 2 : z = ıt for t > 0}; Q := {z ∈ H 2 : |z| = 1}.
For z ∈ H 2 let θ(z) stand for the angle between ı and z measured in a clockwise direction. Then each point on the ray P τ := {θ = arctan sinh τ } for τ ∈ R has hyperbolic distance |τ | from P ( [6] (7.20.3)). The hypercyclic region Q r (r > 0) is defined by Q r := {d(·, Q) < r} where d stands for the hyperbolic metric (see [6] §7.2 for example). For each τ ∈ R define a measure υ τ := (1/x 2 )H 1 P τ . The symmetric rearrangement E of a H 1 -measurable set E in P τ with finite υ τ -measure is the set P τ ∩ Q r where r ≥ 0 is chosen in such a way that the υ τ -measure of the two sets match. The Steiner symmetral E s of a µ g -measurable set E in H 2 with finite µ g -measure is formed by disintegrating E along each P τ and applying the symmetrization . to each slice. As before we write u s to signify the corresponding symmetrised function.
Let T resp. N be unit tangent resp. normal vector fields along P in H 2 such that {N, T } is positively oriented and T points in the direction of the positive imaginary axis. Denote by X resp. Y the parallel transport of N resp. T along a geodesic emanating from P in the direction N . The result below follows as an application of Theorem 1.1.
Further explanation is contained in Section 8. A version of Theorem 1.2 holds on n-dimensional hyperbolic space H n (n ≥ 2) but for a restricted class of integrands (see [3] , [27] ). The hyperbolic metric in case n ≥ 3 may be written in R × H n−1 in the form
where |dy| 2 H n−1 signifies the metric in H n−1 (see the proof of Theorem VI.4.1 in [15] ). The case of Steiner (n − 1, n)-symmetrisation in hyperbolic space with n > 2 (for example) does not fall into the framework studied here.
We briefly summarise the contents of each section. Section 2 focuses on isoperimetric properties of the log-convex measure ν on R k ; in particular, Lemma 2.2 provides a lower bound for the ν-measure of the h-neighbourhood (h > 0) of a measurable set. This is derived using an equivalence in [8] Theorem 2.1 where ν appears as a probability measure. Section 3 investigates properties of the metric space (R n , d) associated with the Riemannian metric (1.1). Lemma 3.4 (a counterpart of Lemma 2.3) plays a key role in showing that the symmetrisation · s is smoothing. This latter property is derived in Proposition 4.2. The proof adopts the approach taken in the proof of [7] Lemma 4.17. Section 5 establishes formulae for the approximate differential of the distribution function for a nonnegative Lipschitz function on R n with compact support and proceeds along the lines of [17] Lemma 4.1 where the case k = 1, ψ = 1 is treated. Lower semicontinuity of the Sobolev functionals considered in Theorem 1.1 is derived in Section 6 along with a useful approximation result for integrands satisfying the conditions (A.1)-(A.4). The proof of our main result is contained in Section 7 and combines elements of the proofs of [17] 
for any L k -measurable set E in R k with ν(E) < +∞. According to [14] Theorem 1.1 (see also [20] , [29] ) for any L k -measurable set E with ν(E) < +∞ and locally finite perimeter in R k ,
an explanation of terminology follows.
Given an open set Ω in R k we say that u ∈ L 1 (Ω) has bounded variation and write u ∈ BV(Ω) if the distributional derivative of u is representable by a finite Radon measure Du (cf. [2] Definition 3.1 for example) with total variation |Du|. A function u ∈ L 1 loc (Ω) is locally of bounded variation u ∈ BV loc (Ω) if u| Ω ∈ BV(Ω ) for each open relatively compact set Ω in Ω. An L k -measurable set E in Ω is said to have finite resp. locally finite perimeter relative to Ω if χ E ∈ BV(Ω) resp. χ E ∈ BV loc (Ω). Finally, the weighted perimeter
The reduced boundary F E of E is defined as follows (cf. [2] Definition 3.54). Let Ω be the largest open set in R k such that E has locally finite perimeter relative to Ω. Then
exists in R k and |ν E (x)| = 1 ;
and this is a Borel set (cf.
[2] Theorem 2.22 for example).
The function F is smooth on [0, ∞) and f(r) := F (r) = kω k r k−1 e h(r) for r ≥ 0. Note that
where I h (E) stands for the open h-neighbourhood of E relative to the usual metric in R k .
Lemma 2.1. Let E be a finite union of open balls in R k . Then
(ii) E has finite perimeter in R k and
On the other hand,
and hence
(ii) The set E has finite perimeter in virtue of [2] Proposition 3.38. Now ∂E is countably
Proof. The proof runs as in the proof of [8] Theorem 2.1 in which the counterpart of ν is a probability measure. We reproduce the proof here for the convenience of the reader. For r, h ≥ 0 and σ ≥ 1 set
and write R h for R 
Fix σ > 1. We claim that if E is a finite union of open balls in R k then
The claim follows once it is shown that ∆ = (0, +∞). To this end we establish (a) ε ∈ ∆ for all ε > 0 sufficiently small;
By Lemma 2.1 (ii) and the definition of ν + ,
as ε ↓ 0. The mapping ε → R σ ε (ν(E)) is C 1 and Taylor expansion gives
) then this inequality also holds with h replaced by h + ε provided ε is small as the function h → ν(I h (E)) is non-decreasing and the function
) for small ε > 0 as in the derivation of (a) above. By the semigroup property (2.4),
for small ε > 0; that is, h + ε ∈ ∆ for small ε > 0. This shows (b) and hence the claim (2.5). On letting σ ↓ 1 we obtain by continuity of F that
whenever E is a finite union of open balls in R k .
Now let E be an open set in R k with ν(E) < +∞. We may assume that ν(I h (E)) < +∞ for small h > 0. We may also suppose that E takes the form
by the above. On the other hand, ε>0 I ε (I h (E)) = I h (E). So ν(I h (E)) ≥ R h (ν(E)). Now let 0 < h < h with h small. Then I h (E) ⊂ I h (E) and hence ν(I h (E)) ≥ ν(I h (E)) ≥ R h (ν(E)). On letting h ↑ h we obtain ν(I h (E)) ≥ R h (ν(E)). Let E be an arbitrary L k -measurable set with ν(E) < +∞. By inner regularity, there exists an increasing sequence (E n ) of closed sets contained in E such that
The result follows on taking the limit n → ∞ using the continuity of R h .
Proof. Let E be as in the statement and r > 0. We may assume that ν(I r (E)) < +∞. By equimeasurability (2.1) and Lemma 2.2,
and the statement follows.
Let A, B be L k -measurable sets in R k with finite ν-measure. The following properties are readily verified:
Let Λ be an arbitrary index set and
stands for the infimum of lengths of piecewise C 1 parametrised curves in R n connecting x 1 to x 2 ; d(·, ·) is a metric on R n . Given x ∈ R n and r > 0 we write B(x, r) := {y ∈ R n : d(y, x) < r} for the open ball with centre x and radius r.
The completeness property below entails an equality of Sobolev spaces in Section 6.
Proof. Given x 0 = (x 0 , y 0 ) ∈ R n and r > 0 we first show that B(x 0 , r) ⊂ {x = (x , y) ∈ R n : |x − x 0 | < r and |y − y 0 | < R}
Let (x h ) be a Cauchy sequence in (R n , d). Given r > 0 we can find k ∈ N such that x h ∈ B(x k , r) for h ≥ k. From the inclusion above we see that B(x k , r) is relatively compact in (R n , d). By extracting a convergent subsequence if necessary using the Bolzano-Weierstrass theorem we may suppose that (x h ) converges in (R n , d). It can then be seen that (x h ) converges in (R n , d).
Lemma 3.2. The metric d on R n has the following properties:
(iii) B(x + z, r) = z + B(x, r) for any x ∈ R n , z = (0, q) with q ∈ R k and r > 0;
In (ii), SO(k) stands for the subgroup of SO(n) that fixes R n−k × {0}.
A similar argument gives the reverse inequality. For (ii) let c = (x , y) be a piecewise C 1 parametrised curve connecting x 1 = (x 1 , 0) to x 2 + z = (x 2 , q). Then the curve Rc is piecewise C 1 , connects
. To see (v) use (iv) for the lower bound; for the upper bound, consider piecewise C 1 parametrised curves connecting x 1 to x 2 of the form c = (
in an obvious notation. (vi) follows in a similar way to the proof of the upper bound in (v). Given E ⊂ R n and x ∈ R n−k the x -slice of E is defined to be the set
Proof. (i) Suppose B(x 1 , r) x 2 = ∅ and that |x 1 − x 2 | ≥ r. We may find y ∈ R k such that
rotation and hence is a ball centred at the origin in
Let E be an L n -measurable set in R n with the property that
The symmetric rearrangement E s of E is the set defined by
It follows directly that (E s ) x = (E x ) for each x ∈ R n−k . We write I r (E) for the r-neighbourhood of E (r > 0) in the metric space (R n , d).
Assume that for each r > 0 the set I r (E) has the property (3.2). Then I r (E s ) ⊂ I r (E) s for each r > 0.
Proof. Let E be any L n -measurable set in R n and assume that for each r > 0 the set I r (E) has the property (3.2). Then for any r > 0 and x ∈ R n−k ,
by Lemma 3.3. By (2.9) and Lemma 2.3,
making use of the set equality above with E s in place of E. The result follows.
The Minkowski µ-content of the boundary of a L n -measurable set E in the metic space (R n , d) is defined by analogy with (2.3).
Corollary 3.5. Let E be an L n -measurable set in R n . Assume that for each r > 0 the set I r (E) has the property (3.2). Then µ
Proof. Let E be as in the statement and h > 0. Using property (3.2), Lemma 3.4 and equimeasurability (2.1),
and the result follows from the definition of the Minkowski µ-content.
The distance between sets
Proof. We use the criterion that for r > 0,
; we may assume that r > 0. The criterion entails that I r (A) ⊂ B and hence that I r (A) s ⊂ B s . Note that I r (A) inherits the property (3.2) from B. By Lemma 3.4,
Lemma 3.7. Let r > 0 and S(r) the set S(r) := {x = (x , y) ∈ R n : |x | < r and |y| < r}.
Then there exists a finite positive constant c such that
Proof. Let x 1 , x 2 ∈ S(r) and put z := (x 2 , y 1 ). By Lemma 3.2 (v) and (vi),
This exists thanks to Lemma 3.1 and [21] Corollary 2.105 and Theorem 2.103. We claim that Ran(c) ⊂ C(r + R) where
For otherwise,
contradicting the above estimate. Therefore,
Lemma 3.8. The following are equivalent:
(i) u is a Lipschitz function on R n with compact support;
(ii) u is a d-Lipschitz function on R n with compact support.
Proof. Assume (i) and that u has Lipschitz constant K. Choose r > 0 such that supp[u] ⊂ S(r) (note that B(0, r) ⊂ S(r)). Let x 1 , x 2 ∈ S(r). By Lemma 3.7,
Now suppose that x 1 ∈ S(r) and x 2 ∈ R n \ S(r). Let c be a minimal geodesic in (R n , d) connecting x 1 to x 2 as above. There exists z ∈ Ran[c] ∩ ∂S(r). Then
This proves (ii). The converse implication is similar.
Rearrangement of functions
Let u be a real-valued L k -measurable function on R k with the property that for each t > 0 the set {|u| > t} satisfies the condition ν({|u| > t}) < +∞ . Put m u (t) := ν({|u| > t}) for t ≥ 0. The function m u : [0, ∞) → [0, ∞] is non-increasing, right-continuous and m u (t) → 0 as t → ∞. Define its right-continuous inverse u
with the understanding that inf ∅ = +∞. The -rearrangement of u is then defined by u (
Given a real-valued function u on R n and x ∈ R n−k we set u x (y) :
The s-rearrangement u s of u is then defined by the relation
The modulus of continuity of a real-valued function u on (R n , d) is defined by
Lemma 4.1. Let u be a nonnegative real-valued function on R n and t, τ > 0. Then ω u (t) > τ if and only if there exist s 1 , s 2 > 0 with
Proof. Assume that ω u (t) > τ . Then there exists x, y ∈ R n such that d(x, y) < t and |u(x)−u(y)| > τ . We may suppose that 0 ≤ u(x) < u(y) and u(y) − u(x) > τ . So we can find
n with u(x) > s 2 and y ∈ R n with u(y) ≤ s 1 such that d(x, y) < t. So |u(x) − u(y)| > s 2 − s 1 > τ and ω u (t) > τ . Proposition 4.2. Let u be a nonnegative real-valued L n -measurable function on R n with the property that
Proof. Let t > 0. We may assume that ω u s (t) > 0. Choose τ > 0 such that ω u s (t) > τ . By Lemma 4.1 there exist s 1 , s 2 > 0 with
and likewise for s 2 . By Lemma 3.6 we deduce that d({u > s 2 }, R n \ {u > s 1 }) < t and again by Lemma 4.1 that ω u (t) > τ . Item (i) then follows. Part (ii) is a ready consequence. As for (iii), (ii) v * is a Lipschitz function on (0, +∞) with compact support.
Proof. (i) follows using Lemma 2.3 and a similar argument to that used in the proof of Proposition
where K is a Lipschitz constant for v . In case s ∈ (0, R] and t > R use the fact that |v
We note the following version of Cavalieri's principle in passing which follows from the layer cake representation (cf.
[26] Theorem 1.13).
Proof. By Tonelli's theorem, u, v satisfy (4.2). By the former result and [18] Theorem 3,
Proof. As in the proof of [18] Corollary 1 put G(x, y) := −g(x − y); then G satisfies (i)-(ii) in Theorem 4.5.
Some derivation formulae
Let Ω be an open set in R n for some n ∈ N and u an R-valued Lipschitz function on Ω. We introduce the sets
Additionally, we put
item (iv) follows from this combined with (iii). Given a function φ : [0, +∞) → R we define the function φ
Lemma 5.2. Suppose that φ is a Lipschitz function on [0, +∞) with compact support. Then
• is a Lipschitz function on R k and has compact support.
and choose z ∈ ∂B k (0, R) such that z lies on the line segment joining x to y. Then
the other cases are straightforward. Parts (ii) and (iii) are elementary.
Lemma 5.3. Suppose that φ is a nonnegative Lipschitz function on (0, +∞) with compact support. Then
Proof. This is more or less a specialisation of [16] 
As any open set G in (0, +∞) is a countable union of open intervals we derive that
as the intersection is empty. So Dh a = 0 and by [2] Theorem 3.28, h = 0 L 1 -a.e. on (0, +∞).
Let
according to the generalised coarea formula (see [2] (2.72) for example). We note that the set F {u > t} is countably (n − 1)-rectifiable for each t ∈ R by [2] Theorem 3.59. Moreover, F {u > t} ⊂ {u = t} for each t ∈ R and 
by Fubini's theorem and v ν((0, +∞)) = ν({v > 0}) < +∞; so m v ∈ BV((0, +∞)) and
The result follows by the generalised coarea formula [2] (2.72) with g = χ R k \Zv ψ/|∇v| applied to the second term. 
Proof. By Corollary 4.3, v * is a nonnegative Lipschitz function on (0, +∞) with compact support. By equi-measurability and Lemma 5.4,
.
that is,
which leads to the first equality in the statement. The second equality follows from the definition of the weighted perimeter.
Lemma 5.7. Let v be a nonnegative Lipschitz function on R k with compact support. Then
for every t > 0.
Proof. From equi-measurability and Lemma 5.5,
for each t ∈ (0, M ). From Lemma 5.6 we derive that for L 1 -a.e. t ∈ (0, M ),
since the function t → ν({v > t} ∩ Z v ) is non-increasing with negative derivative L 1 -a.e. (cf.
[2] Corollary 3.29). The statement then follows for t ∈ (0, M ); for t ≥ M both sides vanish. 
and use the dominated convergence theorem. Define Lipschitz projection operators
Given an L n+1 -measurable set E in R n × (0, +∞) we set
We write Ω for R n × (0, +∞).
Proposition 5.9. Let E be a set of finite perimeter in Ω and g : Ω → [0, ∞] a Borel measurable function. Then
Proof. This is a consequence of the generalised coarea formula [2] Theorem 2.93.
In particular there exists a Borel set
Proof. This result is a minor variation of [5] Theorem 2.4 (see also [30] ). Let u be a nonnegative Lipschitz function on R n with compact support. The subgraph of u is the set
By the area formula (e.g. 
for H n -a.e. (x, t) ∈ F S u ∩ Ω and the latter has full measure in ∂S u ∩ Ω. By [2] Theorem 3.59,
So x ∈ π n−k (S u ) + if and only if M (x ) > 0 and (x , t) ∈ π n−k,1 (S u ) + if and only if M (x ) > 0 and t ∈ (0, M (x )).
Lemma 5.11. Let u be a nonnegative Lipschitz function on R n with compact support. Then for
In particular there exists a Borel set
We have omitted the dependence of the integrand on x , t for the sake of legibility.
By [22] Chapter 4 Section 1.5 Theorems 2 and 4, A has full
. Let G 1 be the set in Theorem 5.10 with S u in place of E. Put
and choose a Borel set G 3 with |G 2 ∆G 3 | = 0. Set
for (x , t) ∈ R n−k × (0, +∞). Suppose that (x , t) ∈ G Su and y ∈ F (S u ) (x ,t) ∩ (F S u ∩ Ω) (x ,t) ∩ A (x ,t) such that ν Su y (x , y, t) = 0. Then y ∈ (D u ) x and ∇ y u(x , y) = 0 so that y ∈ R k \ W x . Consequently, the set G Su has the properties listed in the lemma.
Since S u has finite perimeter in Ω,
where we have used Proposition 5.9 and properties (i)-(iii) above. The final estimate follows in a similar way.
Lemma 5.12. Let u be a nonnegative Lipschitz function on R n with compact support. Then
and for L n−k -a.e. x ∈ π n−k (S u ) + the approximate differential is given by
. By Fubini's theorem and Lemma 5.8,
This shows that m u ∈ BV(R n−k × (0, +∞)).
The above formula also holds for X ∈ C c (R n−k ×(0, +∞), R n−k+1 ) using a mollification argument. In fact, it holds for any bounded Borel function X : R n−k × (0, +∞) → R n−k+1 with compact support. To see this, define the finite Borel measure 
. . , n − k + 1 and λ({X = Y }) < ε. We obtain the estimate
from which we obtain equality of the terms on the left-hand side.
Let X ∈ C c (R n−k × (0, +∞), R n−k ) and G Su the set in Lemma 5.11. We note that F S u ∩ Ω is a Borel set and that ν Su : F S u ∩ Ω → S n is a Borel map. By the previous observation, Proposition 5.9 and Lemma 5.11,
We deduce that
By the Caldéron-Zygmund theorem (
Note that
By Tonelli's theorem the identity in (iii) holds for L n−k -a.e. x ∈ π n−k (S u ) + and for L 1 -a.e. t ∈ (0, M (x )). The proof of (ii) is similar.
Lemma 5.13. Let u be a nonnegative Lipschitz function on R n with compact support. For L n−ka.e. x ∈ π n−k (S u ) + it holds that
Proof. By Cavalieri's principle Lemma 4.4 we infer that π n−k (S u )
for L 1 -a.e. t ∈ (0, M (x )). In light of Lemma 5.1 we may suppose that
is a sphere on which |∇ y u s | is constant and positive. Thus
and (i) follows. For (ii) note that for any R ∈ SO(k), u s = u s • R on R n . Further, the set D u s is invariant under SO(k) and ∇ x u s = (∇ x u s ) • R on D u s for any R ∈ SO(k). Otherwise, the proof proceeds as before.
Lemma 5.14. Let u be a Lipschitz function with compact support on R n . Put
Proof. Put C := {x ∈ R n : x ∈ D u , ∇u(x) = 0 and M (x ) = 0}. For x ∈ R n−k ,
and by Tonelli's theorem,
Assume the latter. By [2] Corollary 2.23 we may find a Lebesgue point x = (x , y) ∈ C for χ C . Then ∇ y u(x) = 0. By considering cones in R n−k with vertex at x we can find a linearly independent set {v 1 , . . . , v n−k } of vectors in R n−k such that ∇ x u(x) · v h = 0 for h = 1, . . . , n − k. We conclude that ∇u(x) = 0, a contradiction. if r ≤ τ < 2r, 0 if 2r ≤ τ, and put w := θp. Then
for small δ > 0. Properties (i), (ii) and (iv) follow immediately; while (iii) follows using uniform convergence of convolution (see [2] 2.1 for example).
Lower semicontinuity
Given u ∈ W 1,1 loc (R n ) the gradient associated to the Riemannian metric (1.1) is defined by
which we write as a tangent vector at x to R n . The auxiliary function j introduced in (1.2) has the property that
for each x, z ∈ R n .
Lemma 6.1. Let a : R → R and b : R n → R be bounded measurable functions with a ≤ 0 on R. Then the functional
Proof. This result is adapted from [12] Proposition 4.27. There exists an increasing sequence (K h ) of compact sets in R and a sequence of continuous functions (a h ) on R with a h ≤ 0 and 
is an increasing sequence of functions on R × R n with limit f . Let u ∈ W 1,1 loc (R n ). By the monotone convergence theorem,
By [12] Proposition 4.2.2,
for any u ∈ W 
loc (R n ). Put α h := χ K h a h and β h := χ K h b. By Lusin's theorem and a mollification argument we may write
for each u ∈ W 
follows by Fatou's lemma. We suppress the dependence on h. Let B x resp. B y be primitives for β x resp. β y . Then B x resp. B y are (vector-valued) Lipschitz functions on R n and B x (u) ∈ W 
Thus shows that the functional
is continuous on W 
is well-defined on W 
where
Note that each a h ≤ 0 on R by property (A.3). By [2] Lemma 2.35, for u ∈ W 1,1
in the supremum I ⊂ N is a finite set and (B h ) h∈I are pairwise disjoint relatively compact open sets in R n . By [12] Proposition 1.1.2 it suffices to show that each functional
But this is the content of Lemma 6.1. (ii) for each h ∈ N and (s,
By [12] Theorem 2.2.4 and Remark 2.2.5 there exist
where
By (B.3) a h ≤ 0 on R and
The function f
h satisfies (B.1)-(B.4). Finally, set
for h ∈ N and (s, z) = (s, z , w) ∈ R × R n−k × R k . Note that for s ∈ R, z 1 , z 2 ∈ R n and 0 < λ < 1,
using the fact that each function f (2) j is convex and even in t. It can then be seen that each function g h satisfies conditions (A.1)-(A.4) . The estimates in (ii) and (iii) follow from (6.2) and (6.3). Item (iv) follows readily.
A Pólya-Szegö inequality
and is differentiable L n -a.e. and the differentials coincide L n -a.e. on R n ;
(ii) |∇ g u| g ∞ ≤ K.
Proof. (i) Let r > 0 and x 1 , x 2 ∈ B(0, r). By Lemma 3.7,
that is, u is Lipschitz on B(0, r). The statement follows by [2] Proposition 2.13 and Theorem 2.14.
(ii) Assume first that u ∈ C ∞ (R n ). Let x, z ∈ R n with g x (z, z) = 1 and c : R → R n be a geodesic in (R n , d) parametrised by arc-length such that c(0) = x and c (0) = z. In the notation of [21] ,
from which the result follows. Now let u be an arbitrary d-Lipschitz function on R n . Let (ρ ε ) ε>0 be a family of mollifiers and set
by extracting a subsequence if necessary.
function with the properties (A.1)-(A.4). Let u be a Lipschitz function on R
n with compact support. Then u s is a Lipschitz function on R n with compact support and
Proof. Let u be a nonnegative Lipschitz function on R n with compact support. By Lemma 3.8 and Proposition 4.2 u s is a Lipschitz function on R n with compact support.
Let us assume firstly that
By Tonelli's theorem and Lemma 5.1,
By Lemma 5.7 the above also holds for u s : that is,
In particular,
. From (A.1) and these remarks,
By the coarea formula [2] Theorem 2.93,
for L n−k -a.e. x ∈ R n−k with M (x ) > 0 and a similar formula holds for u.
By Lemma 5.13 we may write: for
using Lemma 5.1 and (7.2). We now apply the isoperimetric inequality (2.2) making use of property (A.4). We then use Lemma 5.12 and apply Jensen's inequality (e.g. [2] Lemma 1.15) to the R n -valued random variable Y := (∇ x u, γ(x )|∇ y u|ω) making use of the property (A.2). We obtain: for
Combining (7.4) and (7.5) we obtain that for
for L 1 -a.e. t ∈ (0, M (x )). We then deduce using (7. 3) that for
. By [2] Proposition 2.13, ∇u ∞ < +∞ and
The image in R × R n of the set K under the mapping x → (u(x), j∇ g u(x)) is then essentially bounded. On the other hand, the function x → g(u, j∇ g u) vanishes on R n \ K by property (A.3). The upshot of these considerations is that the mapping x → f (u, j∇ g u) is essentially bounded on R n and has compact support by continuity of f . The same holds with u s in place of u.
By Tonelli's theorem, Lemma 5.14 and (7.6),
We now turn to the general case. Let u be a nonnegative Lipschitz function on R n . Suppose that the support of u is contained in B(0, r) for r > 0. By Lemma 5.16 there exists a sequence (u h ) of nonnegative Lipschitz functions on R n with support contained in B(0, 2r) which satisfy (7.1), converge to u in W 1,1 (R n ) and with u h ∞ ≤ 2 u ∞ and ∇u h ∞ ≤ 2 ∇u ∞ for each h ∈ N. Define
for each h ∈ N. The essential range of each φ h is contained in the compact set
is uniformly essentially bounded on R n . The functions u, u h vanish on R n \ B(0, 2r) and likewise ∇u, ∇u h . This entails that f (u h , j∇ g u h ) = 0 on R n \ B(0, 2r) by (A.3) and the same holds with u h replaced by u. In short, each function f (u h , j∇ g u h ) has support in B(0, 2r). By choosing a subsequence if necessary we may assume that u h → u and ∇u h → ∇u L n -a.e. on R n as h → ∞. By the dominated convergence theorem,
By the non-expansivity result Corollary 4.6, lower semi-continuity Theorem 6.2 and (7.7),
and this proves the theorem. Sobolev spaces. Given u ∈ W 1,1 loc (R n ) we write
R n |u| p dµ < +∞ and
so u is weakly differentiable with respect to y j and we may identify ∂ yj u with γv. There is a similar argument for 
for each h ∈ N. For each h ∈ N we can find ε h > 0 such that
by choice of (ε h ).
For any multi-index α with |α| = 1,
We may then estimate
by choice of (ε h ). Thus we can find a sequence of functions µ) -Cauchy sequence. This shows the reverse inclusion.
Proof. Suppose that K is a compact set in (R n , d). Let (x h ) be a sequence of points in K. Then r := sup h d(0, x h ) < +∞. Fix h ∈ N and let c = (x , y) : [α, β] → R n be a minimal geodesic in (R n , d) connecting 0 to x h . Choose R > r. We claim that Ran(c) ⊂ C(r + R). For otherwise,
contradicting the above estimate. This means that
We infer that K is compact in (R n , d). So u has compact support in (R n , d). The result then follows from Lemma 7.3 and Proposition 7.1.
Proof. We argue as in [24] B(x, r) is compact in (R n , d) for each x ∈ R n and r > 0. Define
n -a.e. on R n . The same is true for the function θ(r − h)
for each h ∈ N and θ(r−h) has compact support B(x, h+1). By Lemma 7.4,
by an argument as in the proof of Lemma 7.3. For each h ∈ N,
It can then be seen with the help of the dominated convergence theorem that (u h ) converges to
for each h ∈ N by Theorem 7.2. By [1] Proposition 2.1 and (12),
for each δ > 0. By the Dunford-Pettis theorem (e.g. [2] Theorem 1.38) the family F := {|∇ g u h | g : h ∈ N} is equiintegrable; in particular,
Let Ω be a relatively compact open set in R n . Then
It follows then by definition that the family F (s)
By the Dunford-Pettis theorem, choosing a subsequence if necessary, we may suppose that
that is, u s is locally weakly differentiable and u s ∈ W 1,1 loc (R n ). By Theorem 6.2 and Theorem 7.2,
for each 1 ≤ j ≤ n because for some finite c > 0 and any h ∈ N,
by Theorem 7.2. By the Banach-Alaoglu theorem we may suppose that ( 
(A.6) there exists α ∈ C b (R), β ∈ C b (R, R n ) and a finite constant λ > 0 such that
g (R n , µ) by Lemma 7.5. By selecting a subsequence if necessary we may assume that (u h ) converges to u µ-a.e. on R n . Let ε > 0. As the family {|∇ g u h | g : h ∈ N} is equiintegrable there exists a L n -measurable set A with µ(A) < +∞ such that
By the properties (A.5) and (A.6),
By the dominated convergence theorem we infer that
By Theorem 6.2 and non-expansivity Lemma 4.6,
for each h ∈ N by adapting the above argument slightly. By the monotone convergence theorem f h can be replaced by f .
We now consider the case 1 < p < +∞. Let u ∈ W 1,p g (R n , µ). Guided by [17] Remark 4.6 we define u ε := max{u−ε, 0} for ε > 0. By [31] Corollary 2.1.8 u ε ∈ W 1,1 g (R n , µ) and j∇ g u ε = χ {u>ε} j∇ g u. We note that (u ε ) s = (u s ) ε for each ε > 0. Define f ε (s, z) := f (ε + s, z) for (s, z) ∈ R × R n . Then f ε satisfes (A.1)-(A.4). By the monotone convergence theorem and the foregoing result, 
A Pólya-Szegö inequality in the hyperbolic plane
We work in the hyperbolic plane H 2 equipped with the metric g as mentioned in the Introduction. The distribution function of the measure υ τ (τ ∈ R) is given by F τ (r) := υ τ (P τ ∩ Q r ) for τ ∈ R and r ≥ 0. Given a measurable set E in P τ define E by E := P τ ∩ Q r where r ≥ 0 is chosen to satisfy F τ (r) = υ τ (P t ∩ E).
Let τ ∈ R and u be a real-valued H 1 -measurable function on P τ with the property that for each t > 0 the set {|u| > t} satisfies the condition υ τ ({|u| > t}) < +∞ . Put m u (t) := υ τ ({|u| > t}) for t ≥ 0. Define its right-continuous inverse u with the understanding that inf ∅ = +∞. The -rearrangement of u is then defined by u (x) := (u * • F τ )(d(x, Q)) for x ∈ P τ . We suppress the dependence on τ for the sake of legibility.
Given a real-valued function u on H 2 and τ ∈ R we set u τ := u| Pτ . Let u be a real-valued L 2 -measurable function on H 2 with the property that: for L 1 -a.e. τ ∈ R, υ τ ({|u τ | > t}) < +∞ for each t > 0.
The s-rearrangement u s of u is then defined by analogy with (4.3).
We equip R 2 with the Riemannian metric g associated to the differential expression ds 2 = (dx ) 2 + cosh 2 (x )(dy) 2 and denote the corresponding metric by d. The measure µ g is given by
for any L 2 -measurable set E in R 2 . The symmetric rearrangement of a L 1 -measurable set E in R with |E| < +∞ is the set E = (−|E|/2, |E|/2). For a real-valued L 2 -measurable function u on R 2 with the property |{|u x | > t}| < +∞ for each t > 0 for L 1 -a.e. x ∈ R, we form the s-rearrangement denoted u s as in previous sections.
It can be verified directly that ι : R 2 → H 2 ; (x , y) → (e y tanh(x ), e y sech (x ))
is an isometry in the sense of [21] Proof. By the generalised area formula [2] Theorem 2.91,
in the notation of [2] . Let x = (t, y) ∈ {x = t} and e 2 := (0, 1). In the notation of [21] J 1 d {x =t} ι x = |T x ι · e 2 |. As ι is an isometry, cosh t = g x (e 2 , e 2 ) = g ιx (T x ι · e 2 , T x ι · e 2 ) = [1/(ιx) 2 ]|T x ι · e 2 | = (x 2 • ι)
Lemma 8.2. Let τ ∈ R. Then the following statements hold.
(i) Let E be a H 1 -measurable set in P τ with υ τ (E) < +∞. Then |(ι −1 E) τ | < +∞ and (ι −1 E ) τ = ((ι −1 E) τ ) .
(ii) Let u be a H 1 -measurable function on P τ such that υ τ ({|u| > t}) < +∞ for each t > 0 and put v := (u • ι)(τ, ·) on R. Then v = (u • ι)(τ, ·) on R.
Moreover, (iii) let u be a real-valued L 2 -measurable function on H 2 with the property that υ τ ({|u τ | > t}) < +∞ for each t > 0 for L 1 -a.e. τ ∈ R and put v := u • ι. Then v s = u s • ι on R 2 .
Proof. (i) By definition, E = P τ ∩ Q r where υ τ (E) = F τ (r). As ι −1 Q = R × {0} and ι is an isometry, by (ii). Let e 1 , e 2 be the standard basis vectors for R 2 and e 1 resp. e 2 the vector fields given by e 1 (x) := e 1 for x ∈ R 2 and likewise for e 2 . Let X := ι e 1 and Y := ι (sech(x )e 2 ) be the images of the vector fields e 1 resp. sech(x )e 2 under ι (cf. [21] 1.64). As ι is an isometry it is straightforward to calculate that g(X, X) = 1, g(X, Y ) = 0, g(Y, Y ) = 1 on H 2 with the help of (8.1).
We now give an intrinsic characterisation of the vector fields X, Y .
Lemma 8.3. Let T resp. N be unit tangent resp. normal vector fields along P in H 2 such that {N, T } is positively oriented and T points in the direction of the positive imaginary axis. Given a > 0 define c : R → H 2 by c(t) := ι(t, a) (t ∈ R). Then X resp. Y is the parallel transport of N resp. T along c.
Proof. Let D resp. D stand for the Levi-Civita connections associated to the Riemannian metrics g resp. g. A computation leads to the equations 

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