MOLLIFICATIONS OF CONTACT MAPPINGS OF ENGEL GROUP Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 1, P. 5-19

YAK 514.765+517.57

DOI 10.46698/n0927-3994-6949-u

MOLLIFICATIONS OF CONTACT MAPPINGS OF ENGEL GROUP

S. G. Basalaev1

1 Novosibirsk State University, 1 Pirogova St., 630090 Novosibirsk, Russia E-mail: [email protected]

Abstract. The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as Holder continuity, differentiability almost everywhere in the sense of Pansu, Luzin N-property.

Key words: Carnot group, Engel group, quasiconformal mappings, bounded distortion. AMS Subject Classification: 30C65, 58C25.

For citation: Basalaev, S. G. Mollifications of Contact Mappings of Engel Group, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 5-19. DOI: 10.46698/n0927-3994-6949-u.

Mollification or convolution with a smooth kernel is one of the main tools of analysis that allow us to transfer results from spaces of smooth functions to more general function spaces. However, for contact mappings, that is mappings that preserve a fixed distribution in the tangent bundle, mollifications lead to an undesirable effect: the mollified mapping is not contact anymore. In [1, 2] it is shown that at least for mappings between two-step Carnot groups we can give estimates on the failure of the mollified mapping to be contact sufficient to prove convergence of horizontal differential forms

as e ^ 0 in Lv, here f # is a pull-back of the form with the Pansu differential d of the contact mapping f, v is the Hausdorff dimension of Carnot group. The formula (1.1) in turn allows to obtain a crucial fact of quasiconformal analysis

1. Introduction

(/T w ^ f

(1.1)

df = / #dw

(1.2)

© 2023 Basalaev, S. G.

in distributional sense. In the theory of mappings with bounded distortion developed by Yu. G. Reshetnyak [3, 4] the commutativity of outer differential and superposition allows to prove that the superposition u = v o f of n-harmonic function v on Rn with the bounded distortion mapping f is a weak solution of the quasilinear elliptic equation div A(x, Vu) = 0.

For 2-step Carnot groups in [1, 2] a weaker statement is proved: if dw = 0 in weak sense, then df = 0. In full the formula (1.2) is proved in [5] for mappings of 2-step Carnot groups and in the recent preprint [6] for mappings of Carnot groups of arbitrary step. The proof of (1.2) in [5] does not rely on the convergence of mollifiers (1.1) but relies essentially on 2-step structure of the group. In [6] the formula (1.2) is derived from (1.1).

In this paper we obtain for a contact mapping f on Engel group E (the simplest example of 3-step Carnot group) estimates on convergence (or sometimes divergence) of components of the "Jacobi matrix" of the mollified mapping ff. However, in order to obtain meaningful estimates we should compute this matrix not in the Euclidean basis but in the graded basis of left invariant vector fields. Precisely, if X\,... ,X4 is a graded basis of the algebra of left-invariant vector fields of Engel group, we are interested in the behavior as e ^ 0 of the coefficients af, in the decomposition

If we introduce the dual basis of 1-forms ..., £4 such that £i(X,} = 5,, i,j = 1,..., 4, the coefficients are af, = (Xjff}. By a form of the weight l we mean 1-form dual to the vector field of the degree l (and thus having the homogeneous degree —l). The main result of the paper is the following

Theorem 1.1. Let Q, Q' C E, f : Q ^ Q' be weakly contact of class W^'C^Q). If X is a left-invariant vector field of the degree k, £ is a left-invariant 1-form of the weight l, p ^ l, then on every compact K c Q we have

as e ^ 0 in LP/l(K).

Section 3 of this paper is dedicated to the proof of the theorem in Engel case. For Engel group the result is new and may have independent interest, it also serves as an alternative way to prove (1.1) and (1.2). The analogue of Theorem 1.1 for 2-step groups follows from Lemma 2.1 and Lemma 3.3 in [2]. To prove Theorem 1.1 on Engel group we modify the approach of [2] to obtain finer estimates on divergence of components of the mollified mapping.

Recall that a stratified graded nilpotent group or a Carnot group (see e.g. [7-9]) is a connected simply connected Lie group G such that its algebra of left-invariant vector fields g decomposes into the direct sum g = V1 ® V2 ® ■ ■ ■ ® VM of vector spaces Vk satisfying [Vi, Vk] = Vfc+i, k = 1,..., M — 1, and [V1, VM] = {0}. Left-invariant vector field L € g is the field of the degree k if L € Vk. The subspace V1 = HG is a horizontal space of g, its elements are the horizontal vector fields.

4

Xjff = dff (Xj} = £ af, X,, i = 1,..., 4.

2. Carnot Groups

Elements g € G in some "privileged" coordinate system may be identified with elements (xi,..., XM) € RdimVl x ■ ■ ■ x RdimVm = RN in a way that dilations Sx : RN ^ RN, A > 0, defined by ¿a(x1, x2,..., xM) = (Ax1, A2x2,..., AMxM) are group automorphisms. The homogeneous norm on G is a function || ■ || : G ^ R such that

1) ||0|| = 0, ||g|| > 0, when g = 0;

2) ||¿Ag|| = A||g|| for all A ^ 0, g € G;

3) there is Q ^ 1 such that ||gh|| < Q(||g|| + ||h||) for all g,h € G.

The homogeneous norm generates left-invariant homogeneous quasimetric d(g, h) = ||h-1 g||. In constract to metric it satisfies only the generalized triangle inequality

d(gi,g3) < Q(d(gi,g2) + d(g2,g3)), gi,g2,g3 € G,

where Q ^ 1 is a constant from the definition of the homogeneous norm.

The ball in this quasimetric with the centre g € G and the radius r denote by Br (g). The topology given by d coincides with the Euclidean topology of RN. The Lebesgue measure dx on Rn is the bi-invariant Haar measure on G and d(^x) = Av dx, where v = ^M=1 j dim Vj is the homogeneous dimension of the group.

Let Q C G be open. The space Lp(Q), p ^ 1, consists of measurable functions u : Q ^ R integrable in the p-th power. The norm on Lp(Q) is defined by

IMLfi = i^ j HgWdg^j .

When Q = G, we write ||u||p = ||u|p,G.

Let left invariant vector fields X1,... , Xn be the basis of the horizontal space HG. The Sobolev space W 1,p(Q), p ^ 1, is a space of functions u € Lp(Q) that have distributional derivatives Xju € Lp(Q) along the vector fields Xj, j = 1,..., n, that is such functions gj that

J gj (x)^(x) dx = — J u(x)Xj<^(x) dx, j = 1,..., n, n n

for all € C0°°(Q). The norm on W lp is ||u | W 1>p(Q)|| = ||u||p,n + || |Vffu| ||pn, where

VHu = (X1u,... ,Xnu). We say that u € Lfoc(Q) and v € W¿;f(Q), when u € Lp(K) and v € W 1,p(K) for each compact K C Q. A mapping f : Q ^ G f = (f1,..., fN) is in class W 1,p(Q) or W10',P(Q) if all its components fj are in the corresponding class.

A mapping f : Q ^ G of class W1o'c (Q) is (weakly) contact if Xjf (x) € Hf (x)G, j = 1,...,n, for a.e. X € Q. The formal horizontal differential dHf(x) : HxG ^ Hf(x)G of a contact mapping f is a linear mapping such that dHf (x)(Xj} = Xjf(x). It is proved in [10, 11] that the horizontal differential extends to the contact homomorphism of Lie algebras df (x) : TxG ^ Tf (x)G that we call the formal Pansu differential (P-differential) of f at X. The convolution of measurable functions u, v on Carnot group G is defined as

u . v(x) = I «(y)v(y-iX) dy = I u(xy-i)v(y) dy, X € G,

GG

if the integral converges.

Lemma 2.1 (Convolution Properties [8]). 1. Let p, q € [1, u € Lp(G), v € Lq(G). Then u* v £ Lr(G), where | + | = £ + 1, and the following Young inequality holds

llu * viir ^ iiuiipiiviiq.

2. If L is a left invariant vector field on G, u, v are smooth compactly supported functions, then

L(u * v) = u * (Lv), (Lu) * v = u * (Lv),

where L is the right-invariant vector field agreeing with L at the origin. For : G ^ R and e > 0 define

<fe(x) = ^<foS1/£(x), xeG,

where v is the homogeneous dimension of G. If ^ € L1(G), then fG <^f(x) dx = fG <^(x) dx.

Lemma 2.2. For every ^ € Cq°(G), left-invariant field L of the degree k and right-invariant field L such that L(0) = L(0) we have

Life = \{Lip)e, LLPs = -r(Ltp)e.

ek ek

< Since dilation 5a is an automorphism of G, that is 5a (x ■ y) = 5ax ■ 5Ay, for the left translation lx(y) = x ■ y we have 5a olx = -4aX o 5a. If L(0) € Vk(0), then D5aL(0) = AkL(0). Therefore, for left-invariant L € Vk

D5a(L(x)} = D5a o Dlx(L(0)} = Dl^ o D5a(L(0)} = AkL(5ax).

Thus,

¿Pefa) = ° <5l/e)(ic) = ^J • -p;(L<p)(61/ex) = -p;(L<p)e(x).

Obviously, the same argument holds for the right translation rx(y) = y ■ x. >

Lemma 2.3 (Properties of Mollifications [8]). Let ^ € Cq°(G) and fG <^(x) dx = a. Then

1. If u € L1oc(G) then u * ^ au a. e. as e ^ 0.

2. If u € Lp(G), p € [1, then ||u * — au||p ^ 0 as e ^ 0.

3. If u is bounded on G and is continuous on open set Q C G then u * ^ au uniformly on compact subsets of Q as e ^ 0.

3. Mollifications on Engel Group

Engel group E is a 4-dimensional 3-step Carnot group i.e. E = (R4, ■), and g(E) = V ® V2 ® V3, dim V1 = 2, dim V2 = dim V3 = 1. Such a group is unique up to an isomorphism. For convenience we use coordinate system (x, y, z, t) such that the group operation has the form

(x2y' \

x + x', y + y', z + z' + xy', t + t' + xz' H—— J .

Thus, the dilation is

5a(x, y, z, t) = (Ax, Ay, A2z, A3t), algebra of left-invariant vector fields is spanned by the graded basis

X = F = + Z = [X, Y] = rjz + a; dt, T = [X,Z]=dt,

and algebra of right-invariant vector fields — by the basis

X = dx + ydz + zdt, F = dy, Z = -[X ,F]= dz, T = -[X ,Z] = (3.1)

Further, we fix on Engel group infinitely smooth nonnegative function ^ : E ^ R supported in the unit ball such that JE ^(g) dg = 1.

For Q C E and u € L11oc(Q) define mollification u£ = u * where u is u extended on E \ Q by zero.

Lemma 3.1. There are functions Xj € Co°(B1(0)), i, j = 1, 2, such that

j Xij (g) dg = ¿ij, i,j = 1, 2,

(3.2)

where ¿ij is the Kronecker delta, and such that for every u € WOf (Q), compact K c Q, g € K and 0 < e < dist(K, dQ)

Xu£ = (Xu) * Xii,e + (Yu) * Xi2,e, Yu£ = (Xu) * X2i>e + (Yu) * X22,£- (3.3)

< We find functions Xj, i, j = 1,2, from the equations

X^ = X Xii + YXi2, Y^ = X X21 + Y X22-

Using the expressions of the right-invariant vector fields (3.1) we obtain

X^ = X^ - yZ^ - zTV = X^ - Z(y^) - T(z0) = X^ - YX(y^) + XY(y^) + XyX(z^) - XXY(z^) - YXX(z^) + XYX(z^)

= x

^ + Y (y^) - YX (z^) + 2 YX (z^)

- Y

X (y^) + X X (z^)

x2 ~ ~ ~ ~ /" X2

= Yip + rcZ^ + —Tip = + Z{xii) + T( yV = -XFX^yv) +^Y(yV>

= Y

-X

F(^)+2Fx(yV>) -

The expressions in square brackets are the desired functions , i,j = 1,2, for instance, X12 = —X(y0) — XXX(z^). On the one hand, the functions are of the form

Xij = 5i, ^ + X a», + F bj,, i, j = 1,2,

where 5, is the Kronecker delta, a,,6, € cq°(b1(0)). Since integrating by parts yields JE Xa,(g) dg = JE Y6», (g) dg = 0, the statement (3.2) of Lemma follows. On the other hand, from the expressions of x, and Lemma 2.2 it follows

Xlpe = ^(XV0£ = + ^(YXl2)£ = Xxil)£ + YX12,e,

and in the same way F^f = Xx21,f + FX22: f. This together with Lemma 2.1 leads to the statement (3.3) of Lemma. >

Lemma 3.2. Let u € W10'C(Q), p ^ 1, K c Q be a compact, X1, X2, X3 be horizontal left-invariant vector fields. Then X1 uf ^ X1u a. e. on K and

Xiu£ - Xiu||pK = o(1), ||X2Xiu£

X3 X2Xi u£

1

as e —> 0.

1

o

o

2

e

e

< Any horizontal left-invariant vector field is a linear combination of basic vector fields, e.g. X1 = aX + bY. For simplicity assume X1 = X, the argument for X1 = Y is the same and the general case is a linear combination of the two. By Lemma 3.1 we have Xu£ = (Xu) * Xu,£ + (Yu) * Xi2,e and fE xn = 1, /E X12 = 0. By properties of mollifications Xu£ ^ Xu as e ^ 0 a. e. and in LP(K). Next,

X2(Xu£) = (Xu) * X2(xu,e) + (Vm) * ^2(xi2>e) = -(Xu) * (X2Xn)£ + -(Yu) * (X2Xl2)£.

e e

By properties of mollifications, e.g., (Xu) * (X2xn)£ ^ 0 as e ^ 0 in LP(K). Analogously,

X3X2(Xue) = ^(Xu) * (X3X2Xn)e + Yu) * (x3x2xl2)£,

where each expression after the ^ term vanishes in LP(K) as e —> 0. >

In the next proof we use the following pointwise estimate for Sobolev functions:

Proposition 3.1 (see [12-14]). Let K c Q be a compact. For every u € W 1,P(Q) there is 0 < g € LP(K), such that

lu(y)- u(z)I < d(y, z)(g(y) + g(z))

for a.e. y, z € V and ||g||P,K ^ C||VHu||P;Q. Moreover, the constant C is independent of u.

Lemma 3.3. Let u € W 1,P(Q), v € LP(Q), Xi, X2 be horizontal left invariant vector fields, € Cq°(B(0,1)), and K c Q be a compact.

1. If p ^ 2, then for F£(x) = (uv) * ^>£(x) — u£(x)(v * <^£)(x) we have

\\Fs\\p/2,k = 0(e), WX^W^k = o(l), HXaXi^Hp/^ = oQ), (3-4)

as e ^ 0.

2. If p ^ 3, then for G£(x) = (u2v) * ^>£(x) — 2u£(x)(uv) * (x) + (u£)2(x)(v * )(x) we have

||Ge||PAK = O(e2), |XiG£|p/3,k = O(e), |X2XiG£|p/3,k = o(1), (3.5)

as e ^ 0.

< Let x £ K, 0 < e < £0 = | dist (x, <9Q), and K be an eo-neighborhood of K. For the summands in the expression of F£(x) we have

(uv) * ^>£(x) = J u(z)v(z)^>£(z-1x) dz = JJ u(z)v(z)^£(y-1x)^£(z-1 x) dydz,

K k xK

u£(x)(v * ^)(x) = JJ u(y)v(z)^£(y-1x)^£(z-1 x) dydz.

K xK

Thus,

(uv) * ^(x) — u£(x)(v * ^£)(x) = JJ (u(y) — u(z))v(z)^£(y-1x)^£(z-1x) dydz.

K xK

By Proposition 3.1 there is g € LP(K), such that

u(y) — u(z) < d(У, z)(g(y) + g(z))

and ||g||PK ^ C||Vhu|P,n. Since the term under the integral is nonzero only when d(x,y) ^ e and d(x, z) ^ e we get

| (uv) * (x) — u£(x)(v * )(x)|

^ 2Qe JJ (g(y) + g(z)) |v(z) | (y-1x) |^£ (z-1x)| dydz K xK

= 2Qe(g£(x)(|v| * |^|)(x) + (g|v|) * |^|(x)),

where Q is the constant from the generalized triangle inequality. Next, using Holder and Young inequalities we derive

||(uv) * — u£(v * ^£)||PAK < 4Qe|g|P>ii|v|P,i?H^^ < e(4QC1 )|vhu|P,n|v|P,n.

Since for the horizontal vector field X\ we have X\tpe = \(X\tp)e, it follows X1 ((uv) * — u£(v * )) (x)

= \ If iU(v) ~U(Z))V(Z)((XliP)e(y~1^),fs(z~1x)

K xK

+ ^£(y-1x)(X1^)£(z-1x)^ dydz,

and from that

X ((uv) * — u£(v * ^£^(x)|

< 2Q JJ (g(y)+ g(z))|v(z)|(|№ ^)£(y-1x)||^£(z-1x)| K xK

+ ^(y-1x) |(X1^)£(z-1x)^ dydz < 2^g*|(X1^)e| ■ |v|*|^| + (g|v|)*|^| ■ ||(X1 ^)£|1

+ g£ .|v|*|(X1^)£| + g|v|*|(X1 (x).

Again, by the Holder and the Young inequalities we derive

|Xi ((uv)

< 4Q||g||p,K |M|p;K ||(X1^)£|1 ll^k + 4Q||g||p,K |M|p;K ||^£ 111| (X1^)£ 11

< 4Q(||X1^|11 + 1 ||VHuHp>nHvHp>n.

The argument can be repeated for second derivatives giving

(uv) * ^ - u£(v * !p/2

K

< l^x^llill^lli + HXi^llilfelli + ||X2V||i||Xi^||i

+ IIVh u|| p,n llvllp,n

Thus, we obtain estimates uniform in e € (0, eo)

||X1 Ff ||p/2,K = ||X^(uv) * ^f — uf(v * <^f)) ||p/2,K ^ C||Vhu|P)n|v|P)n, ||eX2X1Ff ||p/2)K = HeX2X1 ((uv) * Pf — uf(v * Pf)) |p/2>K ^ C||Vhu|P;n|v|P;n.

Note that for smooth functions u, v we have

X1 ((uv) * — uf (v * ^>f)) — 0, eX2X1 ((uv) * — uf (v * )) — 0

in Lp/2(K) as e — 0. Since smooth functions are dense in LP(K) and W 1,P(K) we can choose sequences of smooth un — u in W 1,P(K) and vn — v in LP(K) as n — to. The operator X1Ff = X1Ff(u, v) is linear in both u and v, which implies

||X1Ff (u, v) ||p/2,k < ||X1Ff (u — u„, v) ||p/2,K + |№Ff (u„, v — v„) ||p/2,K + ||X1Ff (u„, vn) ||p/2,K ^ C'|Vh(u — ura)||p;n + C'||v — v„|p;n + ||X1 Ff(u„,vra)|p/2;K.

Therefore,

lim ||X1 Ff(u, v)||p/2,K ^ C'||Vh(u — un)||p,n + C'||v — vn||p,n — 0,

f^O '

as n — to. Analogously, eX2X1Ff — 0 in Lp/2(K) as e — 0. The estimates (3.4) follow.

The estimates (3.5) are obtained in a similar way. For the terms in the expression of Gf we have

,2"') * Pf (x) = i i i u2(z)v(z)^f (y-1x)^f (y-1x)^f (z-1x) dy1,

(u2v) * ^(x) = JJJ u2(z)v(z)^£ (y- (y2 1x)^£(z dyi dy2 dz,

K XK XK

u£(x) ■ (uv) * ^(x) = JU u(yi)u(z)v(z)^£(y-1 x)^£(y-1x)^£(z-1 x) dyi dy2 dz,

K xK xK

(u£)2(x) ■ (v * )(x) = JJJ u(y1)u(y2)v(z)^£(y-1x)^£(y21x)^£(z"1x) dy1 dy2 dz,

1)

K xK xK

thus

G(x) = JJJ (u(z) - u(y1^ (u(z) - u(y2^ v(z)^£(y21x)^£(y21x)^£(z-1x) dy1 dy2 dz.

K xî? xK

Similar to what is already proved one can obtain the bounds

IIGIIpAK < Ce2||Vnuy2>nyvyp>n,

< Ce||Vnu||p ,n ||v||P,n,

|x2x1g£|p/3,k < C||Vhu||p;nIM|p,n.

Moreover, since for smooth functions u, v we have X2X1G£ ^ 0 as e ^ 0 in lp/3(k), using for threelinear operator G£(u, u, v) arguments analogous to the ones given for F£(u, v) one can prove that ||X2X1G£||P/3;K ^ 0 as e ^ 0. This concludes the proof of the estimates (3.5). > The dual basis of 1-forms to the basis of left-invariant vector fields X, Y, Z, T is

1 2

dx, dy, ( = dz — xdy, r = dt — x dz + - x dy,

and satisfies the dual relations

dZ = —dx A dy, dr = —dx A

Lemma 3.4. Let f € W^'^Q), p ^ 2, be a contact mapping. Then on any compact K c Q

IIZ(Xf£)yp/2,K = O(e), z(Zf£) ^ Z<f(Z)), \\ayfe)\\P/2,K = O(e), ||Z(T/£)||P/2;X = oQ)

as e ^ 0 in Lp/2(K).

< Since f is contact, we have

0 = Z(Xf) = Xf3 — fi Xf2, 0 = Z(Yf) = Yf3 — fi Yf2,

It follows Xf3 = f1 Xf2, Yf3 = f1 Yf2. Next, for the mollification f£ by Lemma 3.1

z (Xf£) = Xf£ — f£ Xf£ = (Xf3) * Xii,£ + (Yf3) * Xi2,£ - f £ (Xf2) * Xii,£ — f£(Yf2) * Xi2,£ (3.6)

= [(fiXf2) * Xii,£ — f 1 (Xf2) * Xii,£] + [(fiYf2) * Xi2,£ — fi£(Yf2) * Xi2,^ •

From Lemma 3.3 we have ||Z(Xf£)||p/2,K = O(e) and similarly ||Z(Yf£)||p/2,K = O(e). Next, by the Cartan identity (see e.g. [15])

z(Zf£) = Z([X, Y]/£> = XZ(Yf£) — YZ(Xf£) — dZ(Xf£, Yf£),

z (df (Z )> = Z ([Xf, Yf ]) = —dZ (Xf, Yf )• .

From the representation (3.6) and Lemma 3.3 it follows that the first two terms in (3.7) vanish as e ^ 0 in Lp/2(K). Since dZ = —dx A dy, by Holder inequality

dZ (Xf£, Yf£) = Xf2 Yf£ — Yf2 Xf£ ^ Xf2 Yfi — Yf2 Xfi = dZ (Xf, Yf)

as e ^ 0 in Lp/2(K). Finally,

Z (Tf£) = Z ([X, Z]f£) = XZ (Zf£) — ZZ (Xf£) — dZ (Xf£, Zf£) = X (XZ (Yf£) — YZ (Xf£) — dZ (Xf£, Yf£)) — ZZ (Xf£) — dZ (Xf£, Zf£) = XXZ(Yf£) — 2XYZ(Xf£) + YXZ(Xf£) — XdZ(Xf£, Yf£) — dZ(Xf£, Zf£).

From (3.6) and Lemma 3.3 it follows that the first three terms are oQ) as e —> 0 in Lp/2(K). By Lemma 3.2 and the Holder inequality the last two terms are also oQ) as e —> 0 in Lp/2(K). The lemma is proved. >

Lemma 3.5. Let f € WlO'(f(Q), p ^ 3, be a contact mapping. Then on any compact K c Q

It (Xf £)||pAK = O(e2), |t (Zf £)||pAK = O(e), It (Yf £)|p/3,K = O(e2), t (Tf£) ^ t (df (T))

as e ^ 0 in Lp/3(K).

< Since f is contact, we have

0 = ((xf) = X/3 - fi X/2, 0 = r(xf) = X/4 - /i x/3 + i/2 x/2.

It follows xfs = /1I/2 and X/4 = X/2. Next, for the mollification f£ by Lemma 3.1

1 2

r(Xfe) = xfi - ft X/f + - (/f)2 X/f = xu*Xll,e + yf4*X12,s

1 2

-f!{xf3*Xll,e + yf3*x 12,e) + 2 if!) {Xf2*Xll,e + Y/2*X12,£)

+

Q/1^/2) * Xn,s - fl(fiXf2) * xn,s + \ {ft)2Xf2 * X11,

Q/l3^) *Xl2,e-fi{flYf2) *Xl2,e + \{fi)2Yf2*Xl2,e

(3.8)

Be Lemma 3.3 we have ||t(X/f )||P/3,K = O(e2) and similarly ||t(Y/f)|P/3,K = O(e2) as e — 0. Next, by the Cartan identity (see e.g. [15]) we have

t<Z/f> = t<[X, Y]/f> = Xt<Y/f) — Yt(X/f> — dr<X/f, Y/f>. (3.9)

From the representation (3.8) and Lemma 3.3 it follows that the first two terms are O(e) in Lp/3(K), as e — 0. Next, since dr = —dx A Z, we have

dr(X/f, Y/f > = —X/f Z<Y/f > + Y/f Z(X/f >. (3.10)

Applying Lemmas 3.2, 3.4 and the Holder inequality we obtain ||dr(X/f, Y/f)|P/3,K = O(e), as e — 0. Next,

r(T/f> = r<[X, Z]/f > = Xr(Z/f > — Zr<X/f > — dr(X/f, Z/f>.

Let us estimate each term. From (3.9) and (3.10) we have

Xt <Z/f > = XXr(Y/ f> — XYr <X>/f > (3 11)

+XX/f ^Y/^ + X/f XZ(Y/f) — XY/f ^X/^ — Y/f XZ<X/f).

From (3.8) and Lemma 3.3 it follows that the first two summands in (3.11) and also Zr(X/f) vanish in Lp/3(K). Applying Lemmas 3.2, 3.4 and the Holder inequality we can conclude that the last four summands, in (3.11) also vanish in Lp/3(K). Finally, using Lemmas 3.2, 3.4 and the Hoolder inequality we conclude

—dr(X/f, Z/f > = dx A Z(X/f, Z/f > = X/f Z<Z/f > — Z/f Z<X/f > — X/1 Z<d/(Z)> — 0

as e — 0 in Lp/3(K). The only thing left to note is that

r(d/(T)) = r([x/,d/(Z)]) = — dr(x/,d/(Z)) = dx A Z(x/,d/(Z)) = X/^d/(Z)).

Thus, r(T/f> — r(iT/(T)>, as e — 0 in Lp/3(K). The lemma is proved. > Define homogeneous weight of the basic left-invariant 1-form by

a(dx) = ^(dy) = 1, ct(z ) = 2, ct(t ) = 3.

< Proof of Theorem 1.1. For the forms of the weight 1 (i.e. looking like adx + bdy, a, b € R) the statement of the Theorem immediately follows from Lemma 3.2, for the forms of the weight 2 (cZ, c € R) it follows from Lemma 3.4, and for the forms of the weight 3 (cr, c € R) from Lemma 3.5. >

£

4. Applications

Let Q, Q' C E be open, f : Q ^ Q' be a contact mapping of the class W^'^Q), and w be a k-form on Q'. Define the pull-back f#w as

f #w(g)<6,..., a} = w(f (g)) (f (g)(ei),..., f (g)(efc)

g € Q, ... ,£fc € TgE. Note that f#(w1 A w2) = f#w1 A f#w2, and for basic left-invariant forms

f#dx = Xfi dx + Yfi dy, f = df <Z) Z,

f #dy = Xf2 dx + Yf dy, f #t = df <T) T.

The notion of the homogeneous weight can be extended on k-forms by the rule a(w1 A w2) = ct(w1) ■ a(w2). Next, we consider differential forms with terms of the maximal homogeneous weight. On Engel group such forms are

w1(g) = a(g) T

w2(g) = a(g) Z A t, (4.1)

w3(g) = (a1(g) dx + a2(g) dy) A Z A t, w4(g) = a(g) dx A dy A Z A t,

and have the weights a(w1) = 3, a(w2) = 5, a(w3) = 6, a(w4) = 7.

An analogue of the next theorem was proved for 2-step the Carnot groups in [2, Theorem 3.5] and for arbitrary the Carnot groups in [6, Theorem 4.3].

Theorem 4.1. Let Q, Q' C E be open, w be a k-form on Q' of the form (4.1) with the coefficents of the class C(Q') n L^(Q'), k = 1,..., 4, and f : Q ^ Q' be a contact mapping of the class WjO'cp(Q), p ^ a(w). Then

(fTw ^ f#w,

as e ^ 0 in Lp/f(w)(Q).

< It suffices to prove the theorem for the forms w(y) = a(y)£(y), where a € C(Q')nL^(Q'), and £ is a basic k-form. For basic 1-forms we have

(fTdx = df = Xff dx + Yf dy + Zff Z + Tff t, (f T dy = df} = Xf2 dx + Yf2 dy + Zff Z + Tff t, (fTZ = Z<Xf£) dx + Zf dy + Z(Zf£) Z + Zf T, (f£)*T = t<Xf£) dx + t<Yf£) dy + t<Zf£) Z + T<Tf) T.

Therefore, by Theorem 1.1 on each compact K c Q, as e ^ 0 we have

'1V / 1

(f)*dx = f*dx + o(l) dx + o(l) dy + Z + r in L?W>

(f£)*dy = f*dy + o(l) dx + o(l) dy + Z + r in

(/TZ = f*C + 0(e) dx + 0(e) dy + o(l) Z + o Q) r in L^2(K),

(f£)V = f#t + O(e2) dx + O(e2) dy + O(e) Z + o(1) t in Lp/3(K).

Thus, as e — 0

(/£)V — f#t in Lp/3(K), (/T(Z A t) — /#(Z A t) in Lp/5(K), (f£)*(dx A Z A t) — /#(dx A Z A t) in Lp/6(K), (/y (dy A Z A t) — /#(dy A Z A t) in Lp/6(K), (/e)*(dx A dy A Z A t) — /#(dx A dy A Z A t) in Lp/7(K).

From that for the basic k-form £ of the weight u(£) = u(w) we get (/e)*£ — /#£ in Lp/a(M\K) as e — 0. Sinse a(y) is continuous and bounded, the composition a o /e is uniformly bounded and converges to a o / a. e. on K as e — 0. Hence, by Lebesgue theorem (/e)*w — /#w in Lp/a(u) (K) as e — 0. >

Horizontal vector field on Q C E is a mapping V : Q — HE, V = v1X + v2Y. Weak (horizontal) divergence divHV of the horizontal vector field V € Lloc(Q) is a function h € L11oc(Q), such that for every p € C™(Q)

J Vp(g) dg = - J h(g)p(g) dg.

n n

It follows that divHV = Xv1 + Yv2 pointwise for V € C 1(Q) and distributionally for V € Wl0'CP(Q). If to the vector field V = v1X + v2Y we assign the dual 3-form

w = (v1 dy - v2 dx) A Z A t, (4.2)

then

dw = divH V dx A dy A Z A t = divH V dx A dy A dz A dt.

An analogue of the next theorem is proved for 2-step the Carnot groups in [5, Corollary 2.15] and for arbitrary the Carnot groups in [6, Theorem 4.24].

Theorem 4.2. Let Q, Q' C E be open, w be a horizontal 3-form on Q' of the form (4.2) with the coefEcents v\,v2 € IY1'°°(Q/). If / : Q —> Q' is a contact mapping of the class W£c7(fi), and 7(0) C Q', then

/ #dw = d/ #w

in the weak sense.

< Step 1. If w € C 1(Q'), then for each p € Cq°(Q) we have

j (/e)*dw • p = j d(/e)*w • p = (-l)k+1j (/e)*w A dp. n n n

By Theorem 4.1 as e — 0 we obtain

J /#dw • p = (-l)k+1 j /#w A dp = j d(/#w) • p. n n n

Step 2. Now let w = (v1 dy — v2 dx) AZ A t be as in the conditions of the theorem. Define Ve = v\X + v2Y and we = (vf dy — v2 dx) A Z A t. By step 1 for each p € C0^(Q) we have

J/ #dwe • p = (-l)k+1 j / #we A dp. (4.3)

nn

Since Vj are continuous and bounded, compositions Vjo/£ are uniformly bounded and converge to Vj o / a. e. on Q. Therefore, be Lebesgue theorem

J f #w£ A d^ ^ J f A d^

n n

as e —> 0. On the other hand

f #dw£ = f # (divH V£ dx A dy A Z A t) = divH V£ o f ■ det df ■ dx A dy A Z A t.

By the properties of convolutions divHV£(y) is bounded uniformly in e > 0 and

divHVe(y) = Xv£(y) + Yv£(y) ^ Xvi(y) + Yvi(y) = divHV(y)

as e ^ 0 for y € f (Q) \ E, where E is some null set. By the change of variables formula [11, Theorem 5.4] det ¿if = 0 a. e. on f-1(E). Therefore,

(Xv£ + Yv£) o f (x) ■ det df (x) ^ (Xvi + Yv2) o f (x) ■ det df (x)

for a. e. x € Q. Hence, by Lebesgue theorem

J f #dw£ ■ ^ = ^ p(x) divHV£(f (x)) det df (x) dx ^ J f #dw ■ ^ n n n

as e ^ 0. All in all, we can go to the limit as e ^ 0 in both sides of the equation (4.3). The theorem is proved. >

Theorem 4.2 extends on Engel group the theorems [5, Theorem 2.6, 2.14] proved for 2-step case. In [5, Remark 2.19] it is noted the these are the only theorems in the paper that rely on 2-step structure of the Carnot group. Thus, results of the paper [5] can be translated on Engel group without changes.

The mapping f : G 5 Q ^ G on a Carnot group G is the mapping with bounded distortion if f € W¿C (Q) and for some K > 0

|¿hf(x)|v < Kdetdf(x)

for a. e. x € Q. The least constant K is called the outer distortion coefficent and is denoted KO (f).

Corollary 4.1 [5, Theorem 4.10]. Let Q C E, f : Q ^ E be a mapping with bounded distortion. Then

1) f is locally Holder continuous;

2) f is Pansu differentiable a. e.;

3) f has Luzin N -property;

4) a certain change of variable formula holds: if D c Q is a compact, |dD| = 0, and u is a measurable function on E, then function u(y)^(y, f, D) is integrable on G iff is u(f (x)) J (x, f) integrable on D, moreover

J u(f (x))J(x, f) dx = y u(y)^(y,f, D) dy.

D G

Next, let

G(x) — i {detdf(x))" (dnfix^dnfix))'1 if det df(x) > 0, [id, otherwise.

The matrix G(x) is symmetric and characterizes the local deviation of / from a conformal mapping. The matrix G(x) defines the mapping

v-2

A{x, £) = (G(x)Z, e> 2 G(x) e, X e Q, e e HG,

satisfying the conditions

1 -1er

Cv Ko (f y

where Cv is a constant independent of f.

Corollary 4.2 [5, Corollary 4.8]. Let Q ç E, f : Q — E be a mapping with bounded distortion. If w is a W^-solution to the equation

—divH^\VhW\5Vh wj =0 in an open domain W ç E, then v = w o f is a weak solution to the equation

—divH A (x, VHv) = 0

on f-1(W) n Q.

References

1. Dairbekov, N. S. The Morphism Property for Mappings with Bounded Distortion on the Heisenberg Group, Siberian Mathematical Journal, 1999, vol. 40, no. 4, pp. 682-649. DOI: 10.1007/BF02675669.

2. Dairbekov, N. S. Mappings with Bounded Distortion of Two-Step Carnot Groups, Proceedings on Analysis and Geometry, ed. S.K. Vodopyanov, Sobolev Institute Press, Novosibirsk, 2000, pp. 122-155.

3. Reshetnyak, Yu. G. Certain Geometric Properties of Functions and Mappings with Generalized Derivatives, Siberian Mathematical Journal, 1966, vol. 7, no. 4, pp. 704-732. DOI: 10.1007/BF00973267.

4. Reshetnyak, Yu. G. Space Mappings with Bounded Distortion, Translation of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989.

5. Vodopyanov, S. K. Foundations of the Theory of Mappings with Bounded Distortion on Carnot Groups, Contemporary Mathematics, 2007, vol. 424, pp. 303-344. DOI: 10.13140/RG.2.1.3112.8480.

6. Kleiner, B., Muller, S. and Xie, X. Pansu Pullback and Exterior Differentiation for Sobolev Maps on Carnot Groups, 2021, arxiv.org/abs/2007.06694v2.

7. Rotschild, L. P. and Stein, E. M. Hypoelliptic Differential Operators and Nilpotent Groups, Acta Mathematica, 1976, vol. 137, pp. 247-320. DOI: 10.1007/BF02392419.

8. Folland, G. B. and Stein, E. M. Hardy Spaces on Homogeneous Groups, Princeton Mathematical Notes, vol. 28, Princeton University Press, Princeton, N. J., 1982.

9. Bonfiglioli, A., Lanconelli, E. and Uguzonni, F. Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, 2007. DOI: 10.1007/978-3-540-71897-0.

10. Vodop'yanov, S. K. and Ukhlov, A. D.-O. Approximately Differentiable Transformations and Change of Variables on Nilpotent Groups, Siberian Mathematical Journal, 1996, vol. 37, no. 1, pp. 62-78. DOI: 10.1007/BF02104760.

11. Vodop'yanov, S. K. P-Differentiability on Carnot Groups in Different Topologies and Related Topics, Proceedings on Analysis and Geometry, ed. S. K. Vodopyanov, Sobolev Institute Press, Novosibirsk, 2000.

12. Franchi, B., Lu, G. and Wheeden, R. L. A Relationship between Poincare-Type Inequalities and Representation Formulas In Spaces of Homogeneous Type, International Mathematics Research Notices, 1996, vol. 1, pp. 1-14. D01:10.1155/S1073792896000013.

13. Hajiasz, P. Sobolev Spaces on an Arbitrary Metric Space, Potential Analysis, 1996, vol. 5, pp. 403-415. DOI: 10.1007/BF00275475.

14. Vodop'yanov, S. K. Monotone Functions and Quasiconformal Mappings on Carnot Groups, Siberian Mathematical Journal, 1996, vol. 37, no. 6, pp. 1113-1136. DOI: 10.1007/BF02106736.

15. Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York, 1983. DOI: 10.1007/978-1-4757-1799-0.

Received September 21, 2022

Sergey G. Basalaev

Novosibirsk State University,

1 Pirogova St., 630090 Novosibirsk, Russia,

Associate Professor

E-mail: [email protected]

https://orcid.org/0000-0002-4161-9948

Владикавказский математический журнал 2023, Том 25, Выпуск 1, С. 5-19

УСРЕДНЕНИЯ КОМПАКТНЫХ ОТОБРАЖЕНИЙ ГРУППЫ ЭНГЕЛЯ

Басалаев С. Г.1

1 Новосибирский государственный университет, Россия, 630090, Новосибирск, ул. Пирогова, 1 E-mail: [email protected]

Аннотация. На групппе Энгеля, снабженной левоинвариантной субримановой метрикой, исследуются контактные отображения, принадлежащие метрическим классам Соболева. В евклидовом пространстве одним из основных методов работы с негладкими отображениями является сглаживание — свертка с гладким ядром. Дополнительная трудность работы с контактными отображениями групп Кар-но состоит в том, что сглаживание контактного отображения, как правило, не контактно. Тем не менее, в рассматриваемом нами случае величину отклонения от контактности оказывается возможным оценить в достаточной мере, чтобы получить полезные результаты. Мы получаем оценки на сходимость (или в некоторых случаях расходимость) компонент дифференциала сглаженного отображения к соответствующим компонентам дифференциала Пансю контактного отображения. В качестве приложения этого результата к квазиконформному анализу приведены альтернативные доказательства сходимости усредненных горизонтальных внешних форм и перестановочности переноса внешней формы дифференциалом Пансю с внешним дифференциалом в слабом смысле. Эти результаты, в свою очередь, позволяют получить такие базовые свойства отображений с ограниченным искажением, как непрерывность по Гельдеру, дифференцируемость в смысле Пансю почти всюду, N-свойство Лузина.

Ключевые слова: группа Карно, группа Энгеля, квазиконформные отображения, ограниченное искажение.

AMS Subject Classification: 30C65, 58C25.

Образец цитирования: Basalaev S. G. Mollifications of Contact Mappings of Engel Group // Влади-кавк. мат. журн.—2023.—Т. 25, № 1.—C. 5-19 (in English). DOI: 10.46698/n0927-3994-6949-u.