Quantitative H\"older Estimates for Even Singular Integral Operators on Patches

In this paper we show a constructive method to obtain $\dot{C}^\sigma$ estimates of even singular integral operators on characteristic functions of domains with $C^{1+\sigma}$ regularity, $0<\sigma<1$. This kind of functions were shown in first place to be bounded (classically only in the $BMO$ space) to obtain global regularity for the vortex patch problem [5, 2]. This property has then been applied to solve different type of problems in harmonic analysis and PDEs. Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but $\dot{C}^{\sigma}$ in each side. This $\dot{C}^{\sigma}$ regularity has been bounded by the $C^{1+\sigma}$ norm of the domain [8, 14, 16]. Here we provide a quantitative bound linear in terms of the $C^{1+\sigma}$ regularity of the domain. This estimate shows explicitly the dependence of the lower order norm and the non-self-intersecting property of the boundary of the domain. As an application, this quantitative estimate is used in a crucial manner to the free boundary incompressible Navier-Stokes equations providing new global-in-time regularity results in the case of viscosity contrast [12].


Introduction
In this paper we deal with characteristic functions of domains The main interest is to study singular integral operators of Calderón-Zygmund type applied on this kind of functions and obtained as follows Above, pv stands for principal value and the kernel K is homogeneous of degree´n, given by Kpxq " Ωpxq |x| n , Ωpλxq " Ωpxq @λ ą 0, ż |x|"1 Ωpxqdσpxq " 0.
In the classical theory of singular integrals, the function T p1 D qpxq belongs to the BM O space [19]. In particular, for odd kernels, it is not difficult to show that as x approaches to a point on BD the function T p1 D qpxq is not bounded and diverges to infinity logarithmically. On the other hand, when the kernel is even, Ωpxq " Ωp´xq, a new geometric cancellation was found in [5,2] which shows that T p1 D qpxq belongs to L 8 . This L 8 bound is given in terms of the C 1`σ norm of the domain, 0 ă σ ă 1. The regularity of the boundary together with the fact that the kernel has mean zero on half spheres cancel the singularity on the boundary of the domain. The motivation was to show preservation of C 1`σ regularity for domains moving by the 2D Euler equations; i.e. global in time existence for the vortex patch problem [5,2]. From the harmonic analysis point of view, Calderón-Zygmund operators with smooth and even kernel have been studied specifically as they satisfy stronger inequalities than general ones. In [15], it is shown that the following pointwise inequality holds for even, higher-order Riesz Transforms where T˚is the maximal singular integral and M is the Hardy-Littlewood maximal operator. It yields a stronger estimate than the classical Cotlar's inequality [20].
The extra cancellation providing L 8 bounds has been extensively used in different PDEs problems. Considering the Beltrami equation, it guarantees that the solutions are bi-Lipschitz [14]. For the Muskat problem, modeling the evolution of incompressible immiscible fluids in porous media or Hele-Shaw cells, this bound yields lack of squirt singularities [6] (also known in the literature as splat singularities). For multidensional aggregation equations with a Newtonian potential, it provides propagation of C 1`σ regularity up to the blow-up [1]. In the two dimensional inhomogeneous Navier-Stokes equations modeling the evolution of incompressible fluids of different densities, this L 8 bound provides global-in-time regularity for higher order norms (W 2,8 and C 2`σ ) of the moving free boundary between the fluids [10]. In [11], a combination of parabolic and elliptic estimates together with this L 8 bound are used to propagate the same higher order norms but for Boussinesq temperature fronts. See also [3] for recent developments in contour dynamics for non-linear transport equations. In all these results the singular integral operators are given with Ω a polynomial function.
In this work, we go further in order to control higher regularity for functions given by (1.1) with even kernel. Despite the fact that these functions are discontinuous on BD, it is possible to obtain C σ regularity in D and in R n D. In [14,16], this regularity has been shown together with qualitative bounds of the form (1.2) }T p1 D q} C σ pDq ď CP p}D} C 1`σ q, with P a polynomial and C ą 0 a constant depending on n, σ, and the geometry of the domain D. Above, the function T p1 D q is extended continuously on D. The latter paper also characterizes the regularity of the domains in terms of odd singular integrals operators on BD. It uses harmonic analysis techniques and Clifford algebras as a generalization of the field of complex numbers to higher dimensions. In this paper, we show that the bound above can be improved to make it linear in the higher regularity norm of the boundary. Moreover, the dependency on the arc-chord condition is made explicit: C σ pR n Dq ď Cp1`|BD|qP p}D}˚`}D} Lip qp1`}D} 9 C 1`σ q.
Above, C " Cpn, σq, Lip stands for Lipschitz, }D}˚measures the non self-intersecting property of the boundary BD, P is a polynomial function, and |BD| denotes the pn´1q-dimensional surface area. The higher order norm is homogeneous, given for a function by See below for more details about the notation. During the review process of this article the referee pointed out work [8], in which the author also proves an estimate similar to the one above. However, our proof is different, working at the level of the interface via contour dynamics methods. An important motivation of these estimates comes from a classical two dimensional fluid mechanics problem. Concretely, the dynamics of two incompressible immiscible fluids evolving by the inhomogeneous Navier-Stokes equations. In that problem, the viscosity can be understood as a patch function and the gradient of the velocity is related to the viscosity by combinations of second and fourth-order Riesz transforms, T " B j B k p´∆q´1pI´∇p´∆q∇¨q, j, k " 1, ..., n, I the identity.
In [12], global-in-time well-posedness for the evolution of C 1`σ interfaces between the two fluids is proved. Global-in-time regularity was recently shown in [17] for H 5{2 Sobolev regularity of the interface instead of C 1`σ and by using striated regularity. In the argument of the proof in [12], the estimate (1.3) is used in an important manner. In particular, we emphasize the importance of the quantitative bound of the non self-intersection condition }D}˚. This quantity has to be controlled globally, since it is known that free-boundary incompressible Navier-Stokes can developed finitetime pointwise particle collision on the free interface [4,7].
The rest of the paper is structured as follows. Section 2 contains the statement of the main results: Theorems 2.2 and 2.3. It describes how the operators (1.1) can be studied in terms of odd operators on the boundary, yielding Theorem 2.3 as a corollary of Theorem 2.2. The rest of the paper, Section 3, is the proof of Theorem 2.2. To study the Hölder regularity of the operators involved, the proof distinguishes three situations: when the two points are on the boundary (Subsection 3.1), near the boundary (Subsection 3.2), and far from the boundary (Subsection 3.3). The deciding cut-off is defined in terms of the non self-intersecting condition and the 9 C 1`σ regularity of the domain. In the second scenario, we need to consider further whether the separation between the points occurs mostly in normal or tangential direction. The nearly normal direction case is decomposed in purely normal (Case 1) and tangential (Case 2) differences, each one estimated through delicate splittings of the singular integrals. The nearly tangential case is reduced, using a fixed point argument, to the purely normal plus on the boundary cases. Finally, the third situation is less singular.

Main Result
We consider higher-order Riesz transform operators of even order 2l, l ě 1. That is, we deal with Calderón-Zygmund operators given by and P 2l pxq is a homogeneous harmonic polynomial of degree 2l in R n . We want to study the Hölder regularity of the operator T applied to the characteristic function 1 D pxq of a C 1`σ domain D, We recall that the operators (2.1) have explicit Fourier multipliers, Remark 2.1. Any homogeneous polynomial P k of degree k can be written as P k pxq " p k pxq| x| 2 p k´2 pxq, where p k is a homogeneous harmonic polynomial of degree k and p k´2 is homogeneous of degree k´2 (Sec. 3 in Chapter 3 of [18]). Thus the restriction to harmonic polynomials in (2.2) involves no loss of generality.
Using Euler's homogeneous function theorem and integration by parts, the regularity of (2.3) can be studied through the associated operators where the kernel kpxq is given by kpxq " Q 2l´1 pxq |x| n`2l´2 , and Q 2l´1 is a homogeneous harmonic polynomial of degree 2l´1.
Since one of the main motivations of these results are physical, we show the techniques in dimension three. An analogous approach provides the proof in any dimension.
We will denote by D a non-self-intersecting bounded domain of class C 1`σ . It is defined as follows. Denote by V j , j " 1, ..., J, the neighborhoods that provide local charts of the boundary BD in such a way that for any x P BD there exists a V j Ă R 2 such that x " Zpαq with α P V j , with well-defined normal vector. To measure the non-self intersection and the regularity of the parameterization, we define The Lipschitz norm is given by There exists also a well-defined normal vector given by For convenience, we take the parametrization so that N is pointing towards the interior of the surface. Now we are in position to state our main theorem: Assume D is a bounded domain of class C 1`σ , 0 ă σ ă 1. Then, the operator (2.5) maps boundedly 9 C σ pBDq into 9 C σ pDq Y 9 C σ pR n Dq. Moreover, the following bound holds }Spf q} 9 with P a polynomial function depending on S, and C " Cpn, σq.
As indicated before, we can write (2.3) as a sum of terms of the form (2.5) with f " N j . Therefore, we have the following result: Assume D is a bounded domain of class C 1`σ , 0 ă σ ă 1. Then, the Calderón-Zygmund operator (2.1) applied to the characteristic function of D, (2.3), defines a piecewise 9 C σ function, T p1 D q P 9 C σ pDq Y 9 C σ pR n Dq. Moreover, it satisfies the bound }T p1 D q} 9 C σ pDqY 9 C σ pR n Dq ď Cp1`|BD|qP p}D}˚`}D} Lip qp1`}D} 9 C 1`σ q, with P a polynomial function depending on T , and C " Cpn, σq.

Proof of Theorem 2.2
Without loss of generality, we show the proof for the following case  kpxq " x 1 x 2 x 3 |x| 5 . The developed techniques can be applied to any other odd homogeneous polynomial and any dimension. We choose this case to show more clearly the crucial steps. The case of kpxq of degree one is more direct. If the degree is greater than three the approach is the same but technically longer. With the kernel chosen, we provide a constructive and direct method, showing the main difficulties and cancellations.
3.1. Regularity on the boundary. First, consider an atlas of the surface BD and, on a given chart, fix a cut-off η ą 0 and define the ball A η " ty P BD : |x´y| ă ηu.
Consider any h P R 3 such that then x`h P A η , and we will generally write x " Zpαq, x`h " Zpβq.
We will also use the middle coordinate point Then, The second term is away of the singular part and thus more regular, The first term is given by Z´1pAηq`k pZpαq´Zpγqq´kpZpβq´Zpγqq˘f pZpγqq|N pγq|dγ.
For simplicity of notation, we will denote gpγq " f pZpγqq|N pγq|, so we have that The nonlinear kernels in I are neither odd nor given by a derivative. We decompose the I term as follows ż Z´1pAη q´k pB α Zpξqpα´γqq´kpB α Zpξqpβ´γqq¯gpγq dγ.
We estimate I 2 first. Since the kernel is odd, we first isolate the singularity from the boundary. We notice that thus, we take a smooth cut-off χpγq defined as χpγq " 1 for |α´γ| ď η 2}BαZ} L 8 , χpγq " 0 for |α´γ| ě 3η 4}BαZ} L 8 , and radial centered at α. Introducing the cut-off in I 2 , we have (3.6) I 2 " I 2,1`I2,2 , with I 2,1 " ż Z´1pAη q´k pB α Zpξqpα´γqq´kpB α Zpξqpβ´γqq¯χpγqgpγq dγ, Along the paper, we will need to control the singularity in the kernel kpB α Zpξqpα´γqq. We will use that it is comparable to |α´γ|´2. In fact, since the domain is regular and of class C 1`γ , it holds that for some ε ą 0. Therefore, We now see that we can take ε ą 0 explicit by using the arc-chord quantity. For all γ P R 2 zt0u, and ξ P R 2 , Therefore, Hence we can take and thus we have the bound This yields the following bound for the kernel kpB α Zpξqpα´γqq, We notice that with p m a homogeneous polynomial of degree m. Thanks to (3.5) and the cut-off function, in I 2,2 it holds that 1 2 |α´γ| ď |β´γ| ď 3 2 |α´γ|. Together with (3.9) and (3.5), this gives that Then, the term I 2,2 can be estimated directly since the kernel is not singular in its domain, The support of χ allows to rewrite I 2,1 as follows The classical splitting to show that the singular integral goes from 9 C σ to 9 C σ (see Lemma 4.6 in [13] for example) provides the desired bound: Combining the bounds above, we have obtained that We proceed to estimate I 1 (3.4). We split it as follows: (3.14) Using the following sets we decompose I 1,1 further, |Zpαq´Zpγq| 5´1 |Zpβq´Zpγq| 5¯. Then, we have that The estimate for K 1 follows immediately Since U 1 is a ball, we can write K 2 as follows Similarly as in (3.11), the difference between the denominators is given by It is clear that J 1,2 satisfies the same bound, hence By writing Taking into account that on U 2 it holds that 1 2 |α´γ| ď |β´γ| ď 3 2 |α´γ|, the bounds for J 3 , J 4 , and J 5 follows C σ˘| h| σ . Now, recalling the splitting for I 1 (3.14), it is clear that the bound above works as well for I 1,2 , I 1,3 , and I 1,4 , hence it is valid for I 1 . Combining it with the bound for I 2 (3.13) in (3.4) and recalling (3.2) and (3.3), we finally have that (3.19) |Spf qpxq´Spf qpx`hq| ď Cp1`|BD|qP p}F pZq}

3.2.
Regularity near the boundary. Consider two points x P D and x`h P D (or analogous situation in R 3 D, R 3 D) and, without loss of generality, suppose that We can write (3.21) x " Zpαq`δÑ pαq, and we notice that, as in (3.7), we have τ2 . It will be sometimes convenient to write the point x`h in the following form (3.24) x`h " Zpα`λq`µÑ pα`λq, where by assumption (3.20) (3.25) µ|Ñ pα`λq| ě δ|Ñ pαq|.
We must first make sure that such a λ and µ always exist for given δ and h satisfying (3.23). More specifically, we want to find λ " pλ 1 , λ 2 q and µ satisfying (3.25) solutions of (3.24), Let us denote λ " pλ 1 , λ 2 , µq T , h " ph τ1 , h τ2 , h n`δ q T . Then, upon projecting the equations onto B α1 Zpαq, B α2 Zpαq, andÑ pαq, the system reads as follows Denote the limit matrix byM pαq, As a fixed point equation, the equation reads SinceM´1h " h, and taking into account (3.7), we see that Brouwer's Fixed Point Theorem [9] yields the existence of a solution to (3.26) in the ball of Finally, the third equation in (3.26) shows that the condition (3.25), i.e., (3.20), implies that h n ě´δˇˇ|Ñ pαq|´|Ñ pα`λq| |Ñ pα`λq|ˇˇ´3 C´1 1 |λ|, which in particular gives that, for suitable C 1 big enough, Next, we distinguish two cases: 3.2.1. Regularity in nearly normal direction: Assume that We can write (3.30) Spf qpx`hq´Spf qpxq " Spf qpx`hq´Spf qpZpαq`pδ`h n qÑ pαqq Spf qpZpαq`pδ`h n qÑ pαqq´Spf qpZpαq`δÑ pαqq, so that the first two terms correspond to a difference in the tangential direction and the last two to a difference in the normal direction. We estimate each of these terms separately. Note that the above splitting is valid since (3.28) and the assumption |h τ | ď 1 4 |BαZ| inf }BαZ} L 8 δ guarantees that h n`δ ě δ{2, hence the point Zpαq`pδ`h n qÑ pαq belongs to D. We can thus assume in the following subsection, Case 1, that h n ě 0; otherwise, interchange the roles of δ and δ`h n .
The last term satisfies that The fact that h n`δ ě δ{2 and the choice of the cutoff for δ (3.23) allow us to obtain that (3.34) D ě 1 2 |B α Z| 2 inf´| α´γ| 2`p h n`δ q 2¯. Next, we proceed to estimate each of these terms I 1,i , i " 1, . . . , 6. For I 1,1 we have that (3.35) We introduce the bound for the denominator (3.34) in I 1,1 to obtain that γ.
We have that and hence, recalling that we are dealing with the case where (3.29) holds, we obtain that Analogously, we find that and thus The terms I 2 and I 3 (3.61) follow similarly and share the same bound while for I 4 we find that We proceed to estimate I 5 . We recall the notation (3.62) and define for which we have the lower bound, also valid for u h (3.62), Then, we perform the following splitting ppα´γ`h τ q¨B α Z j pαq`δÑ j pαqq`u´5 h´v´5 h˘d γ, ppα´γq¨B α Z j pαq`δÑ j pαqq`u´5 h´v´5 h˘d γ, ppα´γq¨B α Z j pαq`δÑ j pαqq`u´5 h´v´5 h´u´5 0`v´5 0˘d γ.
We bound I 5,1 as follows We have that Introducing this bound back, we obtain that (3.71) The same bound holds for the terms I 5,2 and I 5,3 . We are left with I 5,4 . We split it further as follows h "´|h τ¨Bα Zpαq| 2´2 ph τ¨Bα Zq¨pB α Zpαqpα´γqq, which, together with (3.63), provides that Therefore, we obtain We proceed to estimate J 2 . We further split this term for 1 ď k ď 5, and γ.
The rest of K l are bounded similarly. Hence J 2,1 is controlled. The remaining terms J 2,k , k " 2, ..., 6, in (3.75) are controlled in a similar manner to J 2,1 , and all of them are bounded with the same bound (we omit the details to avoid repetition, as the estimates follow along the lines below (3.44)). Therefore, This concludes the proof of Case 2. Together with (3.51) (Case 1) and recalling the splitting (3.30), the estimate for two points near the boundary in nearly normal direction, i.e., assuming (3.29), is done.

3.3.
Regularity far from the boundary. Consider two points x and x`h in D (or analogously both in R 3 D) such that they are sufficiently far from the boundary. That is, recalling (3.23), we consider now that mintdpx, BDq, dpx`h, BDqu ě L, |h| ď L 2 , where L " 1 6´| Then, for any y P BD and s P p0, 1q, |x´y`sh| ě |x´y|´|h| ě |x´y| 2 , and thus the mean value theorem gives that |kpx`h´yq´kpx´yq| ď C |h| |x´y| 3 , |kpx`h´yq´kpx´yq| ď C |x´y| 2 .