Self-similarity in homogeneous stationary and evolution problems

We analyse self-similarity properties related to linear elliptic and evolutionary problems involving homogeneous operators in several spaces including measures. We employ these techniques to analyse in particular 2mth-order diffusion equations and the associated fractional problems.


Introduction
In this paper, motivated by the study of 2m-order parabolic equations (m ∈ N) and fractional diffusion problems, we consider general evolution problems defined by homogeneous operators and address general properties related to self-similarity as we now describe.
Homogenous operators are those that interact in a special form with the dilations of functions in R N , defined by For example, a differential operator L that involves only derivatives of order m ∈ N, L = |β|=m a β ∂ β , with constant coefficients a β ∈ C, satisfies for a sufficiently smooth function φ, The semigroup of solutions to {S(t)} t>0 , turns out to be homogenous of degree σ in the sense that for t > 0, R > 0 and φ ∈ X It was proved in [7,8] that if the space X is moreover homogeneous of degree ν ∈ R, that is, its norm satisfies φ R X = R ν φ X , φ ∈ X, R > 0, (1.2) then both semigroups and resolvent operators must satisfy some sharp estimates derived only from homogeneity. In this paper, we further analyse these types of semigroups and resolvent operators along the following lines. In Sect. 2 we show that under quite general conditions, 2m-order parabolic equations have an associated kernel with suitable Gaussian bounds, that is, , t > 0, x, y ∈ R N that allows to represent solutions as in large spaces of initial data X which include Radon uniform measures, that is φ = μ such that see Sect. 3. Then, in Sect. 4, we prove that from the kernel for S(t), the resolvent operator R(λ, L) inherits a Green's function so that see Proposition 4.1. If moreover the semigroup commutes with spatial translations, then the representation by the kernel is actually a convolution, that is k(t, x, y) = K (t, x − y) and the same is true for the Green function, G λ (x − y), see Proposition 4.3. This holds for parabolic problems as in (1.3) with constant coefficients. On the other hand, for a homogeneous semigroup of degree σ = 0, we prove the kernel is self-similar in the sense that and the Green's function is also self-similar, see Proposition 4.5. With these pieces of information, similar results are then proved for the associated fractional semigroups and operators, see Sect. 4.4, where moreover we obtain bounds on the fractional kernel and the fractional Green's function from the semigroup kernel alone. In Sect. 5, we study self-similar solutions for (1.1) and prove the existence of suitable naturally associated self-similar variables, which provide an alternative form of describing the semigroup. First, in Theorem 5.2 we prove that homogeneous distributions in X of degree −β give rise to β-self-similar solutions of (1.1) and characterise all possible β-self-similar solutions, which must have the form where the self-similar profile satisfies a suitable elliptic problem. Indeed, we show that β must be an eigenvalue of this elliptic operator and the profile must be an associated eigenfunction. The existence of self-similar variables is then shown in Theorem 5. 6 where we show that, in self-similar variables, the semigroup is equivalent to an Ornstein-Uhlenbeck-type semigroup. Finally, in Sect. 6, we employ the previous general tools to analyse the heat as well as higher-order diffusion equations and their corresponding fractional problems. We also include some discussion on the heat equation with a Hardy potential.
The final Appendix contains a discussion involving dilations and homogeneous distributions, which play an important role in the results of Sect. 5. In particular we characterise the homogenous elements of several function spaces appearing in the paper.

Elliptic operators and semigroups in uniform spaces
In this section, we show that under quite general conditions a linear 2m-order parabolic equations (1.3) have an associated kernel with Gaussian bounds, see Theorem 2.2.
We denote by ∂ j the partial derivative with respect to the variable x j , where j = 1, . . . , N and x = (x 1 , . . . , x N ) is a point in R N . Following [1, p. 635] we let D j := −i∂ j , and given a multi-index α ∈ N N , we also set D α = (−i) |α| ∂ α .
Definition 2.1. The differential operator A in (2.1) is (M, θ 0 )−uniformly elliptic if the principal symbol A 0 (x, ξ) = |α|=2m a α (x)ξ α , x, ξ ∈ R N satisfies inf x,ξ ∈R N ,|ξ |=1 with θ 0 < π 2 and max |α|<2m a α L ∞ (R N ) ≤ M. Consider the locally uniform spaces L p 3) defines an analytic (but not strongly continuous) semigroup of solutions {S(t)} t≥0 . The solutions are smooth for t > 0 and enter W 2m,q U (R N ) for any p ≤ q < ∞. Also the semigroup extends to initial data in L 1 U (R N ). Furthermore, if the lower-order terms satisfy a α ∈ BUC(R N ), |α| < 2m, then the semigroup is analytic and strongly continuous inL p U (R N ). Then, we have the following result.

Diffusion semigroups in the space of uniform measures
Consider now the space of uniform measures M U (R N ) as in (1.4). Clearly, Then, we have the following result. Observe that this result, in particular, improves the estimates obtained in [6].
Now for z ∈ R there exists at most two k ∈ Z such that z ∈ [k − 1, k]. Removing these intervals and using the fact that g δ t is an even function, decaying and with a maximum g δ Step 2 Using (2.3) and Fubini for every z ∈ R N , and changing variables and Fubini again Now for fixed w ∈ R N we have, as a function of y ∈ R N , X B(z−y,1) (w)=X B(z−w,1) (y) and Step 3 Now, by interpolation, for any z ∈ R N and 1 ≤ p ≤ ∞ we get which leads to (3.2).
Step 4 Now we prove, for 1 ≤ p < ∞ and u 0 ∈ L p U (R N ), In fact from (2.3) we get for each 1 ≤ p < ∞ and x ∈ R N , t > 0, Now we proceed as in Step 1 in the proof of Theorem 4.3 in [6] and split the integral as the sum of integrals in the cubes and use properties of the function G bt (z) as in Step 1 above to get which proves (3.4).
Step 5 Now we show that for each 1 ≤ p < ∞ and u 0 ∈ L p U (R N ) we have Indeed, again the Gaussian bounds give for y ∈ R N , t > 0, where we have used the generalised Minkowski's inequality (see [20, §18.1, formula (4)]). This proves (3.5).
Restricting to Lebesgue spaces and to the space of measures of bounded total variation, M BTV (R N ) that is, satisfying μ BTV := |μ|(R N ) < ∞, we get the following result.
On the other hand, from properties of convolution, we have for 1 Now, we show the semigroup in Theorem 3.1 attains the initial data in the sense of measures, if the kernel is symmetric. This will hold in the example in Sect. 6.

Proposition 3.3. Assume that k(t, x, y) in (3.1) is symmetric in x, y ∈ R N .
Then, the semigroup {S(t)} t≥0 in M U (R N ) defined in Theorem 3.1 is continuous at t = 0 + in the sense that given any μ ∈ M U (R N ) Proof. From (3.1), we have From the bound (2.3), Step 1 in the proof of Theorem 3.1 and Lemma 3.4 we can change the order of integration and using the symmetry of the kernel we get [6, (4.6)], and using Lemma 3.4 and Lebesgue's dominated convergence theorem, we get the result. Now we prove the lemma used above.

Lemma 3.4.
Given φ ∈ C c (R N ) and T > 0 there are constants C, γ > 0 such that Proof. If 1 < p ≤ 2, x, y ∈ R N are fixed and x = 0, then the function We next consider a ball B R of radius R around zero such that suppφ ⊂ B R and using the inequality above with arbitrarily fixed 0 < δ < 1 we obtain

Kernels, Green functions and self-similarity
We now analyse semigroup kernels and Green functions.

Semigroup kernels and Green functions
Motivated by (2.2) and the results in Sect. 3, in this section we consider semigroups Observe that if D(R N ) ⊂ X , then the kernel is uniquely determined. Also from the results in [15,21] we can assume for some M > 0 and ω ∈ R and the semigroup curves solve where −L is the generator of {S(t)} t≥0 , and has a domain D(L) ⊂ X .

Corollary 4.2.
Assuming the bound above, we have the following estimate for the Green's function for Re(λ) < −a, and splitting the integral according to whether |x−y| t 1 σ is larger or smaller than 1, we get where we used the fact that g(z) ≤ cz −m for any m ≥ N and |z| ≥ 1 and g is bounded for |z| ≤ 1. Then, as |x − y| → 0, taking m = N the first term is of order and that for any 0 < α < 1 to be chosen below, z = |x−y| σ and then ασ σ −1 = 1 and we get the result.

Invariance under translations: convolution kernels
Now we prove the following result that applies to (1.3) if the operator has constant coefficients.

(i) Then, the domain D(L) is invariant under translations and τ y L = Lτ y in D(L). Conversely, if L commutes with translations as above, then τ y S(t) = S(t)τ y in D(L).
(ii) Assume furthermore that the semigroup has a kernel k(t, x, y). Then, for some function k 0 (t, z). Hence, k is a convolution kernel.
iii) Finally, for Re(λ) < −ω, the Green function of L satisfies For the converse, by the uniqueness for the equation Taking z = −x or z = −y gives (4.4).
iii) The result for the Green function follows from (4.2).

Homogeneous operators and semigroups: self-similar kernels
Notice that for σ = 0 this implies that L commutes with dilations.
(ii) We say that a family {T (t)} t>0 of linear mappings in X is a scaling family of degree (α, β) on X , with α, β ∈ R, if for every φ ∈ X , R > 0 and t > 0 we have (iii) If a scaling family {T (t)} t>0 is also a semigroup (which implies necessarily β = 0), then we say that {T (t)} t>0 is a homogeneous semigroup of degree α ∈ R.
It was proved in [8] that homogenous semigroups of degree σ have homogenous generator of the same degree. If additionally X is a homogeneous space of degree ν ∈ R as in (1.2), then For homogenous operators, L, of degree σ it was proved in [7] that the resolvent set ρ(L) is invariant by multiplication by positive numbers, that is sρ(L) = {sλ, λ ∈ ρ(L)} = ρ(L) for all s > 0, and the resolvent operator R(λ, For kernels and Green functions, we get the following self-similarity result. If moreover (4.3) holds, then we can assume a = 0.
In particular, if σ = 0 then the Green function is homogeneous of degree −N .
(ii) Using now the Green function as in Proposition 4.3, the self-similarity of the resolvent (4.6) implies that Hence, we get (4.9).

Fractional diffusion
In this section, we show the results above apply in particular to the case of fractional diffusion. Our goal is to obtain as much information as possible on the fractional diffusion problem from information on the original semigroup. So we consider {S(t)} t≥0 a bounded semigroup in a Banach space X of functions in R N , that is, Denote by −L the generator of the semigroup with domain D(L) in X . From the general results compiled in the Appendix in [8], we have the fractional semigroup {S α (t)} t>0 associated with the fractional evolution equation This applies in particular for the case satisfies (4.5). For uniform spaces, see Proposition 6.1 and Sect. 6.2. Then, we show below that the fractional semigroup also has a kernel and there exists a Green function for L α . Corollary 4.6. Assume X , {S(t)} t≥0 and L as in (4.10), and the semigroup has a kernel. Then, for α ∈ (0, 1) the fractional semigroup {S α (t)} t≥0 has kernel, called fractional kernel, k α (t, x, y).
For semigroups that commute with translations, we get the following result.

Proof.
Step 1 We first prove that for α ∈ (0, 1 2 ] there exists C > 0 such that Indeed, from (4.13), using f t,α ≥ 0, (4.3) and Lemma 4.10, we have Taking the change of variable s = ξ 1−σ and forc α a multiple of c α , we get Now, the integral above is a multiple of t|x − y| −N −σ α (see (4.15)) and we get the result.

Proposition 4.14. Assume that k(t, x, y) in (3.1) is symmetric in x, y ∈ R N and a = 0 in (2.3).
Then, the semigroup {S α (t)} t≥0 is continuous at t = 0 + in the sense that given any and it is analytic and satisfies S α (t) L(M U (R N ),L 1 U (R N )) ≤ c for all t > 0. Proof. Observe that for φ ∈ C c (R N ), using [24, (20'), p. 264], Now if B R is a ball in R N containing the support of φ ∈ C c (R N ) then, using (3.2), where c R,N is a number of balls in R N of radius 1 necessary to cover B R . Since from (3.7) lim t→0 + R N φ S(ξ t 1 α )μ dy = R N φ dμ for every ξ > 0, using (4.14) and Lebesgue's dominated convergence theorem from (4.17), we get the result.
To prove that {S α (t)} t≥0 is analytic, observe that uniform estimates on the semigroups and the time derivative can be obtained using (4.12) as in [24, proof of Theorem 1, pp. 263-264]. Then, [15, Proposition 2.1.9] concludes the result, by observing that the weak convergence to the initial data proved above is enough to reproduce the proof. Finally, the estimate on S α (t) L(M U (R N ),L 1 U (R N )) follows from [8, Lemma 4.4].

Self-similar solutions and self-similar variables
Now we pay attention to self-similar solutions of homogeneous semigroups. Our main goal below is to determine those β ∈ R for which β-self-similar solutions exist.

. (i) A β-self-similar solution of the semigroup is a function
(ii) A strong β-self-similar solution of the semigroup is a β-self-similar solution of the semigroup that is a strong solution of That is, u(t) ∈ D(L), u is differentiable in X for every t > 0 and satisfies the differential equation above.
Notice that for σ = 0 a β-self-similar solution is a homogeneous function of degree −β for each t > 0.
Then, we prove the following result concerning self-similar solutions.

then u(t) = S(t)φ is a β-self-similar solution of the semigroup and satisfies (5.1) with = S(1)φ. Conversely, if the semigroup is injective and u(t)=S(t)φ is a β-self-similar solution of the semigroup, then φ is homogeneous of degree −β. (ii) A function (0, ∞) t → u(t) ∈ X is a β-self-similar solution of the semigroup if and only if it satisfies
for some ∈ X such that (iii) If u is β-self-similar solution of the semigroup as in (5.1) and (5.2) and ∈ D(L) then u is a strong β-self-similar solution, R → R is differentiable in X , x∇ ∈ X and that is, β σ is an eigenvalue of the operator L − 1 σ x∇ and is a corresponding eigenfunction.
This happens for any semigroup with a smoothing effect. (iv) Conversely, assume there exists ∈ D(L) such that R → R is differentiable in X , x∇ ∈ X and satisfies (5.3). Then, u defined in (5.1) is a strong β-self-similar solution of the semigroup.
Proof. (i) Notice that for each φ ∈ X and β ∈ R The converse is also true if the semigroup is injective. That u satisfies (5.1) follows by the argument below.
On the other hand, notice that for a function as in (5.1), Conversely, (5.

(iii) If
∈ D(L), then τ → S(τ ) is differentiable in X and then R → R is differentiable in X , which by Proposition A.1 yields x∇ ∈ X . Also, from (5.1), using Proposition A.1 and , hence must satisfy (5.3).
For a semigroup with a smoothing effect, since S(τ ) ∈ D(L) for τ > 0, (5.2) implies that ∈ D(L). (iv) From the assumptions on we get that u in (5.1) satisfies u(t) ∈ D(L) for t > 0, it is differentiable in X for t > 0 and, from the computation in part (iii) above, is a strong self-similar solution of the semigroup. Remark 5.3. (i) Observe that from the semigroup property, either S(t) is injective for all t > 0 or not injective for any t > 0. (ii) For strongly continuous semigroups such that the curves t → S(t)φ are analytic (in particular for analytic semigroups), we obtain that S(t) is injective for all t > 0.
The next result shows that, in general, in homogeneous spaces self-similar solutions either have constant norm or decay to zero.

Corollary 5.4. Assume X is a homogeneous space of degree ν ∈ R and {S(t)} t≥0 is a homogeneous semigroup of degree σ = 0 in X . (i) Assume the semigroup is injective. A β-self-similar solution of the type u(t) = S(t)φ with φ ∈ X , can only exist for β = −ν. In such a case, the norm of u(t) is constant in time. (ii) If there is a β-self-similar solution of the semigroup as in Definition 5.1, then for
Hence, if β = −ν then all β-self-similar solutions have constant norm. Otherwise, the β-self-similar solution converges to 0 in X as t → ∞. Furthermore, if the self-similar solution above is a strong one, then additionally Proof. For part (i), from Theorem 5.2 we know that self-similar solutions u(t) = S(t)φ correspond to φ ∈ X homogeneous of degree −β, that is φ R = R −β φ. Since X is a homogeneous space, by Lemma A.2, we get β = −ν. In such a case the norm of u(t) does not change in time as we prove (5.4). For part (ii), from Theorem 5.2 we know that self-similar solutions are of the form (5.1), which implies the first part in (5.4), since X is a homogeneous space. Also

The same occurs for all φ ∈ X if S(t) is a semigroup with a smoothing effect. In particular, β-self-similar solutions of S(t) correspond to stationary solutions of the semigroup T β (s). More generally, if v(s, x) is a solution of the Ornstein-Uhlenbeck semigroup above, then
and if u is a strong solution then u t (t) = has a smoothing effect), we have that τ → S(τ )φ ∈ X is differentiable and therefore s → v(s) e − s σ = e β σ s S(e s −1)φ ∈ X is differentiable. Differentiating both sides we have Thus, we get The rest follows easily. For (iii) the estimate of the norm of u(t) is immediate. Then, with s = log(1 + t), which leads to the estimate of the norm of the derivative.

Some examples
We now apply the results to some relevant homogeneous operators.

Higher-order diffusion
Consider m ∈ N and a homogenous of degree 2m elliptic operator with constant real coefficients, A 0 = |α|=2m a α D α as in Sect. 2 and the associated parabolic equation (6.1) Proposition 6.1. Equation (6.1) defines a homogenous semigroup of degree 2m in M U (R N ), S(t), that has a convolution kernel that satisfies for some b > 0 and estimates (2.3) and those in Theorem 3.1 and Proposition 3.2 with a = 0. Also k 0 is even and hence the kernel is symmetric. (i) Except for multiples, k(t, x) is the only N -self-similar solution of (6.1) originating in M U (R N ), and has constant L 1 (R N ) norm, which are actually in the Morrey spaces M β (R N ), that give rise to β-self-similar solutions as in (5.1), that is,

iii) For β ∈ R, in self-similar variables, (6.1) is equivalent to the Ornstein-Uhlenbeck equation
in such a way that (iv) For m = 1 and A 0 = − , that is for the heat equation, k 0 (x) = (4π) −N /2 e −|x| 2 /4 and for each k ∈ N and Proof. The results in Sects. 2 and 3 apply to L = A 0 , so we have a well-defined semigroup with a kernel and Gaussian bounds in M U (R N ). Hence, by Proposition 4.5 we get a = 0. Now observe that ifφ(x) = φ(−x) then (S(t)φ)(x) = (S(t)φ)(−x) which easily gives that k 0 is even. Therefore, the kernel is symmetric. In particular, Proposition 3.3 applies and this and the analyticity of the semigroup curves in Theorem 3.1 and part (ii) in Remark 5.3 implies that the semigroup is injective in M U (R N ).
For part (i), according to Theorem 5.2 and Corollary 5.5, the initial data φ = δ ∈ M BTV (R N ) ⊂ M U (R N ), which is homogeneous of degree −N , give rise to a Nself-similar solution as in (5.1), with constant M BTV (R N ) (or L 1 (R N )) norm. Since the semigroup is injective, except for multiples, this is the only such N -self-similar solution. Since the kernel is N -self-similar and satisfies Eq. (6.1) pointwise, we get and (x) = S(1)δ = k 0 (x). Also, using translations and we also get the elliptic equation for k 0 and part (i) is proved. Observe that an expression for k 0 in terms of Bessel functions can be found in [12]. Part (ii), follows from Corollary 5.5 and Theorem 5.2, while part (iii), in turn, follows from Theorem 5.6. Finally, if m = 1 and A 0 = − , as k(t, z) = 1 (4π t) N 2 e − |z| 2 4t we get the expression for k 0 (x). Also, the equation in self-similar variables can be written as with ρ(x) = exp( |x| 2 4 ) so it can be naturally studied in L 2 is compact, then it has an increasing sequence of eigenvalues which are explicitly given by μ k = N +k−1 2 , k = 1, 2, . . ., of which the first one is simple with positive eigenfunction ) and the eigenspace of μ k is spanned by {D α 1 , |α| = k − 1}, see [10,11,14,23].

Higher-order fractional diffusion
Given m ∈ N and α ∈ (0, 1) such that mα is not an integer note that L = (− ) mα is homogenous of degree 2mα and consider Notice the results below apply when (− ) m is replaced by any other elliptic operator with constant real coefficients, A 0 = |α|=2m a α D α as in Sect. 2.
Proposition 6.2. Equation (6.2) defines a homogenous semigroup of degree 2mα in M U (R N ), S α (t), that has a convolution kernel that satisfies for some even function k 0,α that satisfies

in self-similar variables, (6.2) is equivalent to the Ornstein-Uhlenbeck equation
in such a way that Proof. Observe that we proved in Proposition 6.1 that, for α = 1, Theorem 3.1 and Proposition 3.2 hold with a = 0 and therefore we have (4.10) in those spaces, and we can apply the results in Sect. 4.4 to get all the stated properties of the kernel. Proposition 4.14 applies and, using part (ii) in Remark 5.3, the semigroup in the statement is analytic and injective in M U (R N ). The rest follows as in the proof of Proposition 6.1.
In particular, for m = 1, α ∈ (0, 1) from the results in Sect. 4.4 we get the following result. Theorem 6.3. For 0 < γ < 1, the fractional diffusion equation has a nonnegative, convolution and N -self-similar kernel that satisfies Proof. First, that k γ (t, x, y) ≥ 0 follows from (4.13) since the heat kernel is positive and f t,α (s) ≥ 0. Also that k γ (t, x, y) = k γ (t, x − y) follows from Corollary 4.7. Finally, that the fractional kernel is self-similar follows from Corollary 4.8. Fix γ ∈ (0, 1) and take 2 < m ∈ N such that N < 2m. Now observe that the powers of the operator − in, say, L p (R N ), 1 < p < ∞ satisfy , see [16,Theorem 5.4 ). Therefore, and self-similarity of the kernel gives in turn , i.e. the semigroup generated by −(− ) m satisfies the assumptions in Proposition 4.9 with σ = 2m. Hence, we take α = γ m < 1 2 and Proposition 4.9 gives the upper bound. The lower bounds can be seen in [4,5]. ds.
Changing variables as s = ξ −1 gives, up to the constant, Using estimates of the kernel as the ones in Theorem 6.3, it was proved in [5] that the semigroup can be extended to the optimal class of measures satisfying . This space contains homogeneous functions φ(x) = c|x| γ for 0 ≤ γ < 2α of degree γ , so they give rise to (−γ )-self-similar solutions, [5, Section 7.2].
As a consequence, we get the following result. Corollary 6.5. For 0 < γ < 1 and Re(λ) < 0, the fractional Green's function of (− ) γ is positive if λ < 0, of convolution type and self-similar and satisfies the estimates from above in Corollary 4.13 with σ = 2.
Moreover, for γ = 1 2 and λ < 0 we have the estimate from below of the form Proof. That the fractional Green's function satisfies G γ,λ (x, y) = G γ,λ (x −y) follows from Corollary 4.7 and that it is self-similar follows from Corollary 4.8. From the bounds in Theorem 6.3, we can estimate the Green's function as in the proof of Corollary 4.13 and we get the result. Also, if λ < 0, from (4.11) we get G γ,λ (x, y) > 0 since the fractional heat kernel is positive by Theorem 6.3.

Heat equation with Hardy potential
the critical Hardy constant, and the evolution equation An associated number that plays an important role in the analysis is [3]. Then, we have the following result.
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Appendix A. Dilations and homogeneous distributions
We address here various aspects involving dilations and homogeneous distributions. For functions in R N , the dilations are given by and then the family of linear operators {D R } R>0 forms a multiplicative group, that is For distributions (either in D (R N ) or S (R N )) the dilation φ R is defined by duality as for all test functions ϕ.
This applies, in particular, to Radon measures. Also, the group {D R } R>0 extends to distributions in M loc (R N ), D (R N ), or S (R N ). Then, we have the following result.
Proposition A.1. For any distributions φ ∈ D (R N ), the dilation curve Proof.
Step 1 We start by proving the result for test functions ϕ ∈ D(R N ). That is we which converges, as h → 0, to ∂ R ϕ R (x) pointwise for x ∈ R N . Also, by the mean value theorem, since ϕ has compact support, we get R ϕ(h) → ∂ R ϕ R uniformly in R N . Now, we prove the convergence is also in D(R N ). For this, for any multi-index α Hence, from the argument above, this converges as h → 0, uniformly in R N , to On the other hand, it is not difficult to show that Therefore, ∂ α R ϕ(h) → ∂ α ∂ R ϕ R as h → 0, uniformly in R N and the claim is proved.
Step 2 Now it is standard to show that we can apply the chain rule to prove that Step 3 Now for a distribution φ ∈ D (R N ) and any ϕ ∈ D(R N ), R > 0 and h ∈ R, as h → 0, where we used Step 2 for R → 1 R N ϕ 1

R
. To conclude notice that using Step 1, Also, using Step 1 above we can alternatively write Using the first expression above, we get Using the second expression we get ∂ R φ R , ϕ = 1 R N +1 x∇φ, ϕ 1 R = 1 R (x∇ϕ) R , ϕ and the result is proved.
Below, we turn our attention to homogeneous distributions, which in Sect. 5 play an important role in producing self-similar solutions to the evolution problem (1.1).
Recall that a function h(x) in R N is homogeneous of degree σ ∈ R, if h(Rx) = R σ h(x), for R > 0, x ∈ R N . For example, h(x) = |x| σ or h(x) = x α for some multiindex such that |α| = σ . Analogously, a distribution is homogeneous of degree σ ∈ R if φ R = R σ φ. For example, the Dirac delta is homogeneous of degree −N , since for a compactly supported function δ R , ϕ = δ, 1 Observe, for 2m < N or odd N with N ≤ 2m, for the polyharmonic operator (− ) m the fundamental solution 1 |x| N −2m is homogeneous of degree −(N − 2m) see [17, p. 7].
Lemma A.2. Assume X is a linear space of functions or distributions in R N invariant by rescaling and · X is a homogenous norm in X of degree ν ∈ R. Then, if φ ∈ X \{0} is homogeneous of degree σ , then σ = ν.
The next result characterises homogeneous distributions. According to the result below, homogeneous distributions are the eigendistributions of the operator G 0 = −x∇. Proof. The result follows by Proposition A.1 differentiating φ R = R σ φ at R = 1. For the converse, given ϕ ∈ D(R N ), setting f (R) = φ R , ϕ one gets, again by Proposition A.1, which gives f (R) = f (1)R σ and then φ R = R σ φ.
The following results determine homogeneous distributions in some of the spaces appearing before. Observe that the function φ(x) = 1 |x| β , which is homogeneous of degree σ = −β and locally integrable for β < N , captures all the possibilities in the Lemma. For 1 ≤ p < ∞, the space L Clearly the function φ(x) = 1 |x| β is homogeneous of degree σ = −β and belongs to L p U (R N ) as soon as − N p < σ ≤ 0 (and thus to M p, (R N ) for = pβ ∈ [0, N )).
If μ ∈ M BTV (R N ) is homogeneous of degree σ , then Lemma A.2 gives σ = −N and μ is a multiple of δ.
If σ > 0, then for every compact set and ε > 0 there exists a finite covering of K with balls of radius not exceeding ε. Thus, |μ|(K ) ≤ c i r N +σ i ≤ cε σ i r N i . Minimising over all finite coverings of K , we get ε −σ |μ|(K ) ≤ cH N ,ε (K ). As ε → 0, the righthand side converges to the Lebesgue measure of K and therefore |μ|(K ) = 0 for any compact set. As μ is regular, we get μ = 0. Hence, σ ≤ 0 and then μ ∈ M (R N ) for = −σ .