Waves of maximal height for a class of nonlocal equations

# Waves of maximal height for a class of nonlocal equations with homogeneous symbols

Gabriele Bruell Institute for Analysis, Karlsruher Institute of Technology (KIT), D-76128 Karlsruhe, Germany  and  Raj Narayan Dhara Department for Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway
###### Abstract.

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order , where . Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol . Thereby we recover its unique highest -periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.

###### Key words and phrases:
Highest wave; singular solution; nonlocal equation with homogeneous symbol; fractional KdV equation
###### 2010 Mathematics Subject Classification:
35B10, 35B32, 35B65, 35S30, 45M15
Date: July 20, 2019

## 1. Introduction

The present study is concerned with the existence and regularity of a highest, periodic traveling-wave solution of the nonlocal equation

 ut+Lrux+uux=0, (1.1)

where denotes the Fourier multiplier operator with symbol , . Equation (1.1) is also known as the fractional Korteweg–de Vries equation. We are looking for -periodic traveling-wave solutions , where denotes the speed of the right-propagating wave. In this context equation (1.1) reduces after integration to

 −μϕ+Lrϕ+12ϕ2=B, (1.2)

where is an integration constant. Since the symbol of is homogeneous, any bounded solution of the above equation has necessarily zero mean; in turn this implies that the integration constant is uniquely determined to be

 B=14π∫π−πϕ2(x)dx.

Of our concern is the existence and regularity of highest, traveling waves for the fractional Korteweg–de Vries equation (1.1), where . In the case when , (1.1) can be viewed as the nonlocal form of the reduced Ostrovsky equation

 (ut+uux)x=u.

For the reduced Ostrovsky equation, a highest, periodic, peaked traveling-wave solution is known explicitly [Ostrovsky1978] and its regularity at each crest is exactly Lipschitz continuous. Recently, the existence and stability of smooth, periodic traveling-wave solutions for the reduced Ostrovsky equation, was investigated in [GP, HSS]. In [GP2], the authors prove that the (unique) highest, -periodic traveling-wave solutions of the reduced Ostrovsky equation is linearly and nonlinearly unstable. We are going to investigate the existence and precise regularity of highest, periodic traveling-wave solutions of the entire family of equations for Fourier multipliers , where . Based on the nonlocal approach introduced for the Whitham equation [EW], we adapt the method in a way which is convenient to treat homogeneous symbols, and prove the existence and precise Lipschitz regularity of highest, periodic, traveling-wave solutions of (1.1) corresponding to the symbol , where . The advantage of this nonlocal approach relies not only in the fact that it can be applied to various equations of local and nonlocal type, but in particular, that it is suitable to study entire families of equations simultaneously; thereby providing an insight into the interplay between a certain nonlinearity and varying order of linearity. The main novelty in our work relies upon implementing the approach used in [EW, EJC, A] for equations exhibiting homogeneous symbols. For a homogeneous symbol, the associated convolution kernel can not be identified with a positive, decaying function on the real line. Instead we have to work with a periodic convolution kernel. The lack of positivity of the kernel can be compensated by working within the class of zero mean function, though. Moreover, we affirm that starting with a linear operator of order strictly smaller than in equation (1.1) a further decrease of order does not affect the regularity of the corresponding highest, periodic traveling-wave.

### 1.1. Main result and outline of the paper

Let us formulate our main theorem, which provides the existence of a global bifurcation branch of nontrivial, smooth, periodic and even traveling-wave solutions of equation (1.1), which reaches a limiting peaked, precisely Lipschitz continuous, solution at the end of the bifurcation curve.

###### Theorem 1.1 (Main theorem).

For each integer there exists a wave speed and a global bifurcation branch

 s↦(ϕk(s),μk(s)),s>0,

of nontrivial, -periodic, smooth, even solutions to the steady equation (1.2) for , emerging from the bifurcation point . Moreover, given any unbounded sequence of positive numbers , there exists a subsequence of , which converges uniformly to a limiting traveling-wave solution that solves (1.2) and satisfies

 ¯ϕk(0)=¯μk.

The limiting wave is strictly increasing on and exactly Lipschitz at .

It is worth to notify that the regularity of peaked traveling-wave solutions is Lipschitz for all . The reason mainly relies in the smoothing properties of the Fourier multiplier, which is of order strictly bigger than , see Theorem 4.6.

The outline of the paper is as follows: In Section 2 we introduce the functional-analytic setting, notations, and some general conventions. Properties of general Fourier multipliers with homogeneous symbol and a representation formula for the corresponding convolution kernel are discussed in Section 3. Section 4 is the heart of the present work, where we use the regularity and monotonicity properties of the convolution kernel to study a priori properties of bounded, traveling wave solutions of (1.1). In particular, we prove that an even, periodic traveling-wave solution , which is monotone on a half period and whose maximum equals the wave speed, is precisely Lipschitz continuous. Eventually, in Section 5 we investigate the global bifurcation result. By excluding certain alternatives for the bifurcation curve, we conclude the main theorem. In Section 6 we apply our result to the reduced Ostrovsky equation, which can be reformulated as a nonlocal equation of the form (1.2) with Fourier symbol . We recover the well known explicit, even, peaked, periodic traveling-wave given by

 ϕ(x)=2π2−x218,forμ=π29

on and extended periodically. Moreover, we prove that any periodic traveling-wave is at least Lipschitz continuous at its crests; thereby excluding the possibility of periodic, traveling-waves exhibiting a cusp at its crests. Let us mention that the Fourier multiplier for the reduced Ostrovsky equation can be written as a convolution operator, whose kernel can be computed explicitly, see Remark 3.7. Furthermore, relying on a priori bounds on the wave speed coming from a dynamical system approach for the reduced Ostrovsky equation in [GP], we are able to obtain a better understanding of the behavior of the global bifurcation branch.

## 2. Functional-analytic setting and general conventions

Let us introduce the relevant function spaces for our analysis and fix some notation. We are seeking for -periodic solutions of the steady equation (1.2). Let us set , where we identify with . In view of the nonlocal approach via Fourier multipliers, the Besov spaces on torus form a natural scale of spaces to work in. We recall the definition and some basic properties of periodic Besov spaces.

Denote by the space of test functions on , whose dual space, the space of distributions on , is . If is the space of rapidly decaying functions from to and denotes its dual space, let be the Fourier transformation on the torus defined by duality on via

 Ff(k)=^f(k):=12π∫Tf(x)e−ixkdx,f∈D(T).

Let be a family of smooth, compactly supported functions satisfying

 suppφ0⊂[−2,2],suppφj⊂[−2j+1,−2j−1]∩[2j−1,2j+1] forj≥1,
 ∑j≥0φj(ξ)=1for allξ∈R,

and for any , there exists a constant such that

 supj≥02jn∥φ(n)j∥∞≤cn.

For and , the periodic Besov spaces are defined by

 Bsp,q(T):={f∈D′(T)∣∥f∥qBsp,q:=∑j≥02sjq∥∥ ∥∥∑k∈Zeik(⋅)φj(k)^f(k)∥∥ ∥∥qLp<∞},

with the common modification when 111One can show that the above definition is independent of the particular choice of . If and , then

 Ws,p(T)⊂Bsp,q(T)⊂Lp(T)for anyq∈[1,∞].

Moreover, for , the Besov space consisting of functions satisfying

 ∥f∥Bs∞,∞=supj≥02sj∥∥ ∥∥∑k∈Zeik(⋅)φj(k)^f(k)∥∥ ∥∥∞<∞

is called periodic Zygmund space of order and we write

 Cs(T):=Bs∞,∞(T).

Eventually, for , we denote by the space of -Hölder continuous functions on . If and , then denotes the space of -times continuously differentiable functions whose -th derivative is -Hölder continuous on . To lighten the notation we write for .

As a consequence of Littlewood–Paley theory, we have the relation for any with ; that is, the Hölder spaces on the torus are completely characterized by Fourier series. If , then is a proper subset of and

 C1(T)⊊C1−(T)⊊C1(T).

Here, denotes the space of Lipschitz continuous functions on . For more details we refer to [T3, Chapter 13].

We are looking for solutions in the class of -periodic, bounded functions with zero mean, the class being denoted by

 L∞0(T):={f∈L∞(T)∣f has zero% mean}.

In the sequel we continue to use the subscript to denote the restriction of a respective space to its subset of functions with zero mean.

If and are elements in an ordered Banach space, we write () if there exists a constant such that (). Moreover, the notation is used whenever and . We denote by the nonnegative real half axis and by the set of natural numbers including zero. The space denotes the set of all bounded linear operators from to .

## 3. Fourier multipliers with homogeneous symbol

The following result is an analogous statement to the classical Fourier multiplier theorems for nonhomogeneous symbols on Besov spaces (e.g. [BCD, Proposition 2.78]):

###### Proposition 3.1.

Let and be a function, which is smooth outside the origin and satisfies

 |∂aσ(ξ)|≲|ξ|−m−afor allξ≠0,a∈N0.

Then, the Fourier multiplier defined by

 Lf=∑k≠0σ(k)^f(k)eik(⋅)

belongs to the space .

###### Proof.

In view of the zero mean property of , the proof can be carried out in a similar form as in [AB, Theorem 2.3 (v)], where it is show that a function belongs to if and only if

 ∑k≠0^f(k)(ik)−meik(⋅)∈Bs+m∞,∞(T).

The above proposition yields in particular that

 Lrf:=∑k≠0|k|−r^f(k)eik(⋅),r>1,

defines a bounded operator form to for any ; thereby it is a smoothing operator of order .

We are interested in the existence and regularity properties of solutions of

 −μϕ+Lrϕ+12ϕ−12ˆϕ2(0)=0,r>1. (3.1)

The operator is defined as the inverse Fourier representation

 Lrf(x)=F−1(mr^f)(x),

where for and . In view of the convolution theorem, we define the integral kernel

 Kr(x):=2∞∑k=1|k|−rcos(xk),x∈T, (3.2)

so that the action of is described by the convolution

 Lrf=Kr∗f.

One can then express equation (3.1) as

 −μϕ+Kr∗ϕ−12ˆϕ2(0)=0,Kr:=F−1(mr).

In what follows we examine the kernel . We start by recalling some general theory on completely monotonic sequences taken from [Guo, Widder].

###### Definition 3.2.

A sequence of real numbers is called completely monotonic if its elements are nonnegative and

 (−1)nΔnμk≥0for anyn,k∈N0,

where and .

###### Definition 3.3.

A function is called completely monotone if it is continuous on , smooth on the open set , and satisfies

 (−1)nf(n)(x)≥0for anyx>0.

For completely monotonic sequences we have the following theorem, which can be considered as the discrete analog of Bernstein’s theorem on completely monotonic functions.

###### Theorem 3.4 ([Widder], Theorem 4a).

A sequence of real numbers is completely monotonic if and only if

 μk=∫10tkdσ(t),

where is nondecreasing and bounded for .

There exists a close relationship between completely monotonic sequences and completely monotonic functions.

###### Lemma 3.5 ([Guo], Theorem 5).

Suppose that is completely monotone, then for any the sequence is completely monotonic.

We are going to use the theory on completely monotonic sequences to prove the following theorem, which summarizes some properties of the kernel .

###### Theorem 3.6 (Properties of Kr).

Let . The kernel defined in (3.2) has the following properties:

• is even, continuous, and has zero mean.

• is smooth on and decreasing on .

• for any . In particular, is integrable and is -Hölder continuous with if , and continuously differentiable if .

###### Proof.

Claim a) follows directly form the definition of and . Now we want to prove part b). Set

 μk:=(k+1)−rfork∈N0.

Clearly is completely monotone on . Thus, Lemma 3.5 guarantees that is a completely monotonic sequence. By Theorem 3.4, there exists a nondecreasing and bounded function such that

 (k+1)−r=∫10tkdσr(t)for anyk≥0.

In particular

 |k|−r=∫10t|k|−1dσr(t)for anyk≠0.

The coefficients can be written as

 t|k|−1=∫Tf(t,x)e−ixkdxfork≠0,

where

 f(t,x)=∑k≠0t|k|−1eixk+a0(t)

for some bounded function . Thereby,

 |k|−r=∫T∫10f(t,x)dσr(t)e−ixkdxfor anyk≠0.

In particular, we deduce that

 ∫10f(t,x)dσr(t)=∑k≠0|k|−reixk=Kr(x).

Notice that we can compute explicitly as

 f(t,x)−a0(t) =∑k≠0t|k|−1eixk=2∞∑k=1tk−1cos(xk)=2∑k=0tkcos(x(k+1)) =2Re(eix∑k=0tkeixk)=2Re(eix∞∑k=0(teix)k).

Thus, for , we have that

 f(t,x)=2Re(eix11−teix)+a0(t)=2(cos(x)−t)1−t2cos(x)+t4+a0(t).

Consequently, on the interval , the kernel is represented by

 Kr(x)=∫10(2(cos(x)−t)1−tcos(x)+t2+a0(t))dσr(t).

From here it is easy to deduce that is smooth on and decreasing on , which completes the proof of b). Regarding the regularity of claimed in c), let be arbitrary. On the subset of zero mean functions of an equivalent norm is given by

 ∥Kr∥Wr−ε,10≂∥F−1(|⋅|r−ε^Kr)∥L1.

Thereby, is in if and only if the function

 x↦F−1(|⋅|r−ε^Kr)(x)=2∞∑k=1|k|r−ε−rcos(xk)=2∞∑k=1|k|−εcos(xk)

is integrable over . Now, this follows by a classical theorem on the integrability of trigonometric transformations (cf. [Boas, Theorem 2] ), and we deduce the claimed regularity and integrability of . The continuity properties are a direct consequence of Sobolev embedding theorems, see [Demengel, Theorem 4.57]. ∎

###### Remark 3.7.

The proof of Theorem 3.6 includes a general approach on the relation between the symbol and the monotonicity property of the corresponding Fourier multiplier. However, there exists even a more explicit expression of in terms of the Gamma function (cf. [PBM, Section 5.4.3]) given by

 Kr(x)=2Γ(r)∫∞0tr−1(etcos(x)−1)1−2etcos(x)+e2tdt,x∈[0,π],r>1.

Moreover, we would like to point out that if , , we have that

 K2n(x)=(−1)n−1(2n)!(2π)2nB2n(x2π),x∈[0,π],

where is the -th Bernoulli polynomial. If (which corresponds to the case of the reduced Ostrovsky equation), or , then , , and

 K2(x)=12(|x|−π)2−π26,K4(x)=π445−124(x2−2π|x|)2,x∈[−π,π].
###### Lemma 3.8.

Let . The operator is parity preserving on . Moreover, if are odd functions satisfying on , then either

 Lrf(x)>Lrg(x)for allx∈(0,π),

or on .

###### Proof.

The fact that is parity preserving is an immediate consequence of the evenness of the convolution kernel. In order to prove the second assertion, assume that are odd, satisfying on and that there exists such that . Using the zero mean property of and , we obtain that

 Lrf(x0)−Lrg(x0)=∫π−π(Kr(x0−y)−minKr)(f(y)−g(y))dy>0,

where denotes the minimum of on . In view of being nonconstant and for all , we conclude that

 Lrf(x0)−Lrg(x0)>0,

which is a contradiction unless on . ∎

## 4. A priori properties of periodic traveling-wave solutions

In the sequel, let be fixed. We consider -periodic solutions of

 −μϕ+Lrϕ+12ϕ2−12ˆϕ2(0)=0. (4.1)

The existence of solutions is subject of Section 5, where we use analytic bifurcation theory to first construct small amplitude solutions and then extend this bifurcation curve to a global continuum terminating in a highest, traveling wave. Aim of this section is to provide a priori properties of traveling-wave solutions . In particular, we show that any nontrivial, even solution , which is nondecreasing on the half period and attaining its maximum at is precisely Lipschitz continuous. This holds true for any , see Theorem 4.7.

We would like to point out that the subsequent analysis can be carried out in the very same manner for -periodic solutions, where is the length of a finite half period.

###### Lemma 4.1.

If is a nontrivial solution of (4.1), then

 ϕ(xM)+ϕ(xm)≥2(μ−∥Kr∥L1),

where and .

###### Proof.

If is a nontrivial solution of (4.1), then and

 μ(ϕ(xM)−ϕ(xm)) =Kr∗ϕ(xM)−Kr∗ϕ(xm)+12(ϕ2(xM)−ϕ2(xm)) ≤∥Kr∥L1(ϕ(xM)−ϕ(xm))+12(ϕ(xM)−ϕ(xm))(ϕ(xM)+ϕ(xm)),

which proves the statement. ∎

In what follows it is going to be convenient to write (4.1) as

 12(μ−ϕ)2=12μ2−Lrϕ+12ˆϕ2(0). (4.2)

In the next two lemmata we establish a priori properties of periodic solutions of (4.2) requiring solely boundedness.

###### Lemma 4.2.

Let be a solution of (4.2), then and

 ∥∥∥ddx(μ−ϕ)2∥∥∥∞≤2∥K′r∥L1∥ϕ∥∞for allx∈T.
###### Proof.

We can read of from (4.2), that the derivative of is given by

 ddx(μ−ϕ)2(x)=−2K′r∗ϕ(x).

Since and are integrable over (cf. Theorem 3.6), the convolution on the right hand side is continuous and the claimed estimate follows. ∎

###### Lemma 4.3.

Let be a solution of (4.2), then

 ∥ϕ∥∞≤2(μ+∥Kr∥L1)+2π∥K′r∥L1.
###### Proof.

If , there is nothing to prove. Therefore it is enough to assume that is a nontrivial solution. From Lemma 4.2 we know that is a continuously differentiable function. In view of being a function of zero mean and being continuous, we deduce the existence of such that

 (μ−ϕ)2(x0)=μ2.

By the mean value theorem, we obtain that

 (μ−ϕ)2(x)=[(μ−ϕ)2]′(ξ)(x−x0)+μ2

for some and

 ˆϕ2(0) =12π∫π−πϕ2(x)dx=12π∫π−π[(μ−ϕ)2]′(ξ)(x−x0)dx,

where we used that has zero mean. Again by Lemma 4.2 we can estimate the term above generously by

 ˆϕ2(0)≤2π∥K′r∥L1∥ϕ∥∞.

Using that solves (4.1), we obtain

 ∥ϕ∥2∞≤2(μ+∥Kr∥L1)∥ϕ∥∞+2π∥K′r∥L1∥ϕ∥∞.

Dividing by yields the statement.

From now on we restrict our considerations on periodic solutions of (4.2), which are even and nondecreasing on the half period .

###### Lemma 4.4.

Any nontrivial, even solution of (4.2) which is nondecreasing on satisfies

 ϕ′(x)>0andϕ(x)<μon(−π,0).

Moreover, if , then .

###### Proof.

Assuming that we can take the derivative of (4.2) and obtain that

 (μ−ϕ)ϕ′(x)=Lrϕ′(x).

Due to the assumption that on it is sufficient to show that

 Lrϕ′(x)>0on(−π,0) (4.3)

to prove the statement. In view of being odd with on , the desired inequality (4.3) follows from Lemma 3.8. In order to prove the second statement, let us assume that . Differentiating (4.2) twice yields

 (μ−ϕ)ϕ′′(x)=Lrϕ′′(x)+(ϕ′)2(x).

In particular, we have that

 (μ−ϕ)ϕ′′(0)=Lrϕ′′(0).

We are going to show that , which then (together with the first part) proves the statement. Using the evenness of and , we compute

 12Lrϕ′′(0) =12∫π−πKr(y)ϕ′′(y)dy =∫π0Kr(y)ϕ′′(y)dy =∫ε0Kr(y)ϕ′′(y)dy+∫πεKr(y)ϕ′′(y)dy =∫ε0Kr(y)ϕ′′(y)dy+Kr(ε)ϕ′(ε)−∫πεK′r(y)ϕ′(y)dy.

Notice that the first integral on the right hand side tends to zero if , so does the second term in view of being differentiable and continuous on . Concerning the last integral, we observe that

 12Lrϕ′′(0)=−limε→0+∫πεK′r(y)ϕ′(y)dy<0,

since and are negative on . ∎

We continue by showing that any bounded solution of (4.2) that satisfies is smooth.

###### Theorem 4.5.

Let be a bounded solution of (4.2). Then:

• If uniformly on , then .

• Considering as a periodic function on it is smooth on any open set where .

###### Proof.

Let uniformly on . Recalling Proposition 3.1, we know that the operator maps into for any . Moreover, if then the Nemytskii operator

 f↦μ−√12μ2−f

maps into itself for . From (4.2) we see that for any solution we have

 Lϕ−12ˆϕ2(0)<12μ2.

Thus,

 [Lrϕ↦√12μ2−Lϕ+12^ϕ2(0)]∘[ϕ↦Lrϕ−12ˆϕ2(0)]:Bs∞,∞0(T)→Bs+r∞,∞0(T), (4.4)

for all . Eventually, (4.2) gives rise to

 ϕ=μ−√μ2−2Lrϕ+^ϕ2(0).

Hence, an iteration argument in guarantees that . In order to prove the statement on the real line, recall that any Fourier multiplier commutes with the translation operator. Thus, if is a periodic solution of (4.2), then so is for any . The previous argument implies that for any , which proves statement (i). In order to prove part (ii) let be an open subset of on which . Then, we can find an open cover , where for any we have that is connected and satisfies . Due to the translation invariance of (4.2) and part (i), we obtain that is smooth on for any . Since is the union of open sets, the assertion follows. ∎

###### Theorem 4.6.

Let be an even solution of (4.2), which is nondecreasing on . If attains its maximum at , then cannot belong to the class .

###### Proof.

Assuming that , the same argument as in Lemma 4.2 implies that the function is twice continuously differentiable and its Taylor expansion in a neighborhood of is given by

 (μ−ϕ)2(x)=[(μ−ϕ)2]′(0)x+12[(μ−ϕ)2]′′(ξ)x2 (4.5)

for some where . Since attains a local maximum at , its first derivative above vanishes at the origin whereas the second derivative is given by

 12[(μ−ϕ)2]′′(ξ)=−K′r∗ϕ′(ξ).

We aim to show that in a small neighborhood of zero the right hand side is strictly bounded away from zero. Set . Using that and are even functions with and being negative on , we find that

 f(0)=−K′r∗ϕ′(0)=2∫π0K′r(y)ϕ′(y)dy=c>0

for some constant . Since is even (cf. Lemma 3.8) and continuous, there exists and a constant such that

 12[(μ−ϕ)2]′′(ξ)=f(ξ)≥c0,% for allξ∈(0,|x0|).

Thus, considering the Taylor series (4.5) in a neighborhood of zero, we have that

 (μ−ϕ)2(x)≳x2for|x|≪1,

which in particular implies that

 μ−ϕ(x)|x|≳1for|x|≪1.

Passing to the limit , we obtain a contradiction to . ∎

We are now investigating the precise regularity of a solution , which attains its maximum at .

###### Theorem 4.7.

Let be an even solution of (4.2), which is nondecreasing on . If attains its maximum at , then the following holds:

• and is strictly increasing on .

• , that is is Lipschitz continuous.

• is precisely Lipschitz continuous at , that is

 μ−ϕ(x)≃|x|for|x|≪1. (4.6)
###### Proof.
• Assume that is a solution which is even and nondecreasing on . Let and . Notice that by periodicity and evenness of and the kernel , we have that

 Kr∗ϕ(x+h) −Kr∗ϕ(x−h) =∫0−π(Kr(x−y)−Kr(x+y))(ϕ(y+h)−ϕ(y−h))dy.

The integrand is nonnegative, since for and for and by assumption that is even and nondecreasing on . Since is a nontrivial solution and is not constant, we deduce that

 Kr∗ϕ(x+h)−Kr∗ϕ(x−h)>0 (4.7)

for any . Moreover, we have that

 12(2μ−ϕ(x)−ϕ(y))(ϕ(y)−ϕ(x))=Kr∗ϕ(x)−Kr∗ϕ(y)

for any . Hence if and only if . In view of (4.7), we obtain that

 ϕ(x+h)≠ϕ(x−h)for anyh∈(0,π).

Thereby, is strictly increasing on . In view of Theorem 4.5, is smooth on .

• In order to prove the Lipschitz regularity at the crest, we make use of a simple bootstrap argument. We would like to emphasize that the following argument strongly relies on the fact that we are dealing with a smoothing operator of order , where . Let us assume that is not Lipschitz continuous and prove a contradiction. If is merely a bounded function, the regularization property of implies immediately the is a priori -Hölder continuous. To see this, recall that

 12(2μ−ϕ(x)−ϕ(y))(ϕ(y)−ϕ(x))=Lrϕ(x)−Lrϕ(y).

Using