c Pleiades Publishing, Ltd., 2008. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2008, Vol. 260, pp. 25–36.  Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 32–43.

Function Spaces of Lizorkin–Triebel Type on an Irregular Domain O. V. Besov a Received November 2007

Abstract—On an irregular domain G ⊂ Rn of a certain type, we introduce function spaces of fractional smoothness s > 0 that are similar to the Lizorkin–Triebel spaces. We prove embedding theorems that show how these spaces are related to the Sobolev and Lebesgue spaces Wpm (G) and Lp (G). DOI: 10.1134/S0081543808010033

In 1938 S.L. Sobolev proved his well-known embedding theorem Wpm (G) ⊂ Lq (G), m−

m ∈ N,

1 < p < q < ∞,

n n + ≥ 0, p q

(1) (2)

for domains G ⊂ Rn satisfying the cone condition (see [1]). Relation (2) (which determines the maximum possible value of q in theorem (1)) is also a necessary condition for the embedding. Sobolev’s result has been extended to domains of a more general form: domains of the classes J n−1 and Ip, 1 − 1 (V.G. Maz’ya, 1960, 1975, see [2]), John domains (Yu.G. Reshetnyak [3, 4]), and n

p

n

domains with the flexible cone condition (O.V. Besov, 1983, see [5]). Definition 1 [6]. For σ ≥ 1 a domain G ⊂ Rn is said to satisfy the flexible σ-cone condition if, for some T > 0 and 0 < κ0 ≤ 1, for each x ∈ G there exists a piecewise smooth path γ = γx : [0, T ] → G,

γ(0) = x,

|γ  | ≤ 1 a.e.,

(3)

such that dist(γ(t), Rn \ G) ≥ κ0 tσ

for 0 < t ≤ T.

(4)

If σ = 1, this condition is also called the flexible cone condition. The domains that do not satisfy the flexible cone condition will be called irregular. Note that a domain with the flexible σ-cone condition may have the form of an exterior powerlaw peak with exponent σ in a neighborhood of some boundary point. In this paper we consider only domains satisfying the flexible σ-cone condition with σ ≥ 1 (mainly with σ > 1). It was shown in [6] that embedding (1) on a domain with the flexible σ-cone condition holds if m−

σ(n − 1) + 1 n + ≥ 0. p q

(5)

For m = 1 this result was proved by T. Kilpel¨ainen and J. Mal´ y [7]. D.A. Labutin found that (5) is also a necessary condition for this embedding [8]. An example of a domain with the flexible σ-cone condition is given by the peak    V (σ) = x = (x1 , . . . , xn ) = (x , xn ) : |x | < 1, |x |1/σ < xn < 1 + 1 − |x |2 . a Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.

25

26

O.V. BESOV

For the peak (G = V (σ)), embedding (1) holds if  m − [σ(n − 1) + 1]

1 1 − p q

 ≥ 0;

(6)

this was shown by V.G. Maz’ya and S.V. Poborchii (see [8]). They also proved that (6) is a necessary condition for embedding (1) in this case. As can be seen from (5) and (6), the minimal smoothness required for embedding (1) of function spaces defined on a domain with the flexible σ-cone condition is equal to σ(n − 1) + 1 n − > 0, p q

(7)

and for the peak V (σ) it is equal to  [σ(n − 1) + 1]

1 1 − p q

 (8)

> 0.

The left-hand sides of (7) and (8) can be regarded as quantitative expressions of the loss of smoothness under embedding (1). s(m) Below we introduce classes of functions Lp,θ (G) of fractional smoothness s > 0 on domains G satisfying the flexible σ-cone condition. For these classes the embeddings s(m)

Wpm (G) ⊂ Lq,θ (G),

s(m)

Lp,θ (G) ⊂ Lq (G)

hold with the same loss of smoothness (7) and (8). In the case of σ = 1, these spaces coincide with the well-known Lizorkin–Triebel spaces (see [5]). Let us now pass on to a detailed exposition of our results. 1. NOTATION AND DEFINITIONS Below N is the set of natural numbers; n ∈ N, n ≥ 2; Rn is the Euclidean n-space; G ⊂ Rn is a domain such that G = Rn . For t > 0, E ⊂ Rn , and y ∈ Rn , denote y + tE := {x : x = y + tz, z ∈ E},

y + E = y + 1E,

B(x, t) := {y : |y − x| < t} = x + B(0, t),

B0 = B(0, 1);

χ is the characteristic function of the ball B(0, 1) or the interval [−1, 1]; and 2ρ(x) := dist(x, Rn \G). All the sets involved are assumed to be Lebesgue measurable, and |E| denotes the Lebesgue measure of a set E. All functions are real-valued and locally integrable. For E ⊂ Rn , f : E → R, and 1 ≤ p ≤ ∞, we set ⎛ f p,E = f |Lp (E) = ⎝



⎞1/p |f (x)|p dx⎠

,

f p = f p,Rn ;

E

1/p means ess supE |f |; L(E) = L1 (E). For 1 ≤ θ < ∞ and 0 < if p = ∞, then E |f (x)|p dx ∗ T < ∞, denote by Lθ the space of measurable functions g on (0, T ) with the (finite) norm ⎛ T ⎞1/θ dt g|L∗θ = ⎝ |g(t)|θ ⎠ , t

L∗∞ = L∞ ((0, T )).

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By Wpm (G), m ∈ N, 1 ≤ p < ∞, denote the Sobolev space of functions defined on G with the (finite) norm  Dα f |Lp (G) . f |Wpm (G) = |α|≤m

For a domain G ⊂ Rn and a number δ > 0, we define   Gδ = x : x ∈ G, dist{x, Rn \ G} > δ . The number δ > 0 is assumed to be so small that Gδ = ∅. We also set τa f (x) = f (x + a)

for a ∈ Rn ,

σt f (x) = f (tx)

for t > 0.

Denote by Pm−1 the linear space of polynomials of the form



α |α|≤m−1 cα x .

Let

π = π (m−1) : L(B0 ) → L(B0 ) be some projection onto Pm−1 and πa,t = τa−1 ◦ σt−1 ◦ π ◦ σt ◦ τa , Dm−1 (t)f (x) = t−n f − πx,tf |L(B(x, t)) . For E ⊂ Rn we set  Dm−1 (t, E)f (x) =

Dm−1 (t)f (x) if B(x, t) ⊂ E, if B(x, t) ⊂ E.

0

Remark 1.1. The local approximation f −πx,tf |L(B(x, t)) of f by its projection to the polynomials Pm−1 is equivalent to its best local approximation in L(B(x, t)) by the polynomials Pm−1 . Therefore, everywhere below Dm−1 (t)f (x) can be replaced with Em−1 (t)f (x) := t−n

inf

P ∈Pm−1

f − P |L(B(x, t)) .

For x ∈ G denote f [m] (x) := ρ(x)−m Dm−1 (ρ(x))f (x). If f is m times continuously differentiable, then f [m] (x) ≈



|Dα f (x)|.

|α|=m

Suppose that a domain G ⊂ Rn satisfies the flexible σ-cone condition with σ ≥ 1. Then, for some T, κ0 ∈ (0, 1], each point x ∈ G can be assigned a piecewise smooth path γ = γx : [0, T ] → G for which properties (3) and (4) hold and, moreover,  T  y − γx (t) dt 1 ≤ χ . sup κ0 ρ(γx (t)) ρ(γx (t)) κ0 x,y∈G

(1.1)

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Condition (1.1) indicates that the “truncated flexible cone”   γ(t) + B(γ(t), κ0 ρ(γ(t))) 0≤t≤T

has a bounded “multiplicity of self-intersection.” If γx is a path satisfying (3) and (4), then it always can be corrected so that it also satisfies (1.1) (see [6] for details). Under such a correction one may need to decrease the constant κ0 in (4) to a certain value independent of x. Definition 1.1. Let G ⊂ Rn be a domain satisfying the flexible σ-cone condition with σ ≥ 1. Suppose that for some T > 0, κ0 > 0, and δ ∈ (0, T ), each point x ∈ G \ Gδ is assigned a path γx : [0, T ] → G that has properties (3), (4), and (1.1). Assume also that the function x → γx (t) is locally piecewise continuous. To be more precise, assume that G \ Gδ can be covered by half-open pairwise disjoint cubes Qj lying in G whose edges are parallel to the coordinate axes, the cover {Qj }∞ 1 is locally finite, and γx (t) is uniformly continuous on (Qj \ Gδ ) × [0, T ] for each j ∈ N. A domain G equipped with a set {γ} = {γx }x∈G\Gδ of such paths will be called a σ-domain. Definition 1.2. Let σ ≥ 1, G ⊂ Rn be a σ-domain, 1 ≤ p < ∞, 1 ≤ θ ≤ ∞, and 0 < s < m, s(m) s(m) where m ∈ N. By Lp,θ (G) = Lp,θ (G, {γ}) we will denote the normed space of functions f ∈ Lp (G, loc) with the (finite) norm     f |Ls(m) (G) = f |Lp (Gδ ) + f |ls(m) (G, {γ}), p,θ

(1.2)

p,θ

where ⎛ ⎛ T ⎞p/θ ⎞1/p   s(m) θ −sθ dt   ⎝ ⎝ Dm−1 (t, Gt )f (x) t f |l ⎠ dx⎠ p,θ (G, {γ}) = t ⎛ +⎝



⎛ ⎝

G\Gδ

G

T



0

θ (t + ρ(x))m (ρ(γ(t)))−m Dm−1 (ρ(γ(t)))f (γ(t)) (t + ρ(x))−sθ

⎞p/θ ⎞1/p dt ⎠ dx⎠ t + ρ(x)

0

(1.3) with 0 < δ < T and γ = γx . If m − 1 ≤ s < m, then we will write s instead of s(m) in (1.2) and (1.3). 2. THE MAIN RESULTS Everywhere below G is a σ-domain in Rn with σ ≥ 1. Theorem 2.1. Let G ⊂ Rn be a 1-domain (σ = 1), s > 0, and 1 < p, θ < ∞. Then the space s(m) Lp,θ (G, {γ}) coincides with the Lizorkin–Triebel space Lsp,θ (G) up to equivalence of the norms. Theorem 2.2. Let G be a σ-domain, σ ≥ 1, 0 < s < m ∈ N, 1 < p < q < ∞, 1 ≤ θ < p, and m−s−

σ(n − 1) + 1 n + ≥ 0. p q

(2.1)

Then s(m)

Wpm (G) ⊂ Lq,θ (G, {γ}).

(2.2)

Let us equip the domain G = V (σ) with a set of paths {γx } constructed as follows. Let 0 < δ < T < 1 and x ∈ V (σ) \ V (σ)δ . For xn ≤ 1, γx is a part of the polygonal chain composed of PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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the segments [(x , xn ), (0 , xn )] and [(0 , xn ), (0 , 2)]. For xn > 1, γx is a segment of the straight line passing through x and (0, 1). Theorem 2.3. Let σ > 1, the σ-domain G = V (σ) be equipped with the above set of paths {γ}, 1 ≤ θ < p < q < ∞, and   1 1 − ≥ 0. m − s − [σ(n − 1) + 1] p q Then s(m)

Wpm (V (σ)) ⊂ Lq,θ (V (σ), {γ}).

(2.3)

Theorem 2.4. Let σ > 1, the σ-domain G = V (σ) be equipped with the above set of paths {γ}, 1 < p < q < ∞, 1 ≤ θ ≤ ∞, and  s − [σ(n − 1) + 1]

1 1 − p q

 ≥ 0.

Then s(m)

Lp,θ (V (σ), {γ}) ⊂ Lq (V (σ)).

(2.4)

Theorems 2.2–2.4 show that the loss of smoothness under embeddings (2.2), (2.3), and (2.4) is the same as in (7) and (8). Examples demonstrate that this loss of smoothness cannot be reduced under embeddings (2.2), (2.3), and (2.4). 3. INTEGRAL REPRESENTATION OF FUNCTIONS AND POINTWISE ESTIMATES Let G ⊂ Rn be a domain, x ∈ G, and Γ = (Γ1 , . . . , Γn ) : [0, T ] → G be a piecewise smooth curve such that Γ(t) = x for 0 ≤ t ≤ ε, where 0 < ε < T , and |Γ (t)| ≤ 1 a.e. Let r : [0, T ] → [0, ∞) be a piecewise smooth function such that r > 0 for t > 0,

r(+0) = 0,

B(Γ(t), 2rt) ⊂ G r1 (t) =

1 r(t) , 4m t

|r  (t)| ≤ c0

a.e. on [0, T ],

for t ∈ [0, T ],

Γ0 (t) = Γ(t) − Γ(0) = Γ(t) − x.

For a locally integrable function f on G and t > 0, denote by ft (x) a special averaging of f at x of the form (see [9]) ft (x) =

 m m−1 

 aj ck (r1 (t))f x + jr(t)v + Γ0 (t)(1 + kr1 (t)) η(v) dv,

j=1 k=0

1   , aj = (−1)j−1 m where η ∈ C0∞ B 0, 4m j , and m−1  k=0

ck (t) = 1,

m−1 

ck (t)(1 + kt)l = 0 for l = 1, . . . , m − 1,

k=0

which implies that ck (t) = Qm−1,k (t)t−m+1 , where Qm−1,k is a polynomial of degree m − 1. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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(3.1)

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The following equality was proved in [9]1 for α = 0:   α ∂ aj (f − Pm−1 ) x + jr(t)v + Γ0 (t)(1 + kr1 (t)) Dx ft (x) = ∂t j,k  n    r (t) (−jr(t))−|α| Dα η(v) × ck (r1 (t))r1 (t)(−jr(t))−|α| Dα η(v) − ck (r1 (t)) r(t) i=1

+

r  (t) r(t)

−|α|

(−jr(t))

D (vi Di η(v)) − α





0  Γ0 i (t)(1 + kr1 (t)) + Γi (t)kr1 (t)

−|α|−1

(−jr(t))

 D Di η(v) dv, α

(3.2) where Pm−1 is an arbitrary polynomial of degree m − 1. Equality (3.2) with an arbitrary |α| > 0 can be obtained from the case of α = 0 by differentiating all the terms; one should only take into account that differentiating the integrand with respect to x is equivalent to differentiating f − Pm−1 with respect to v and introducing the factor (jr(t))−|α| ; then, integrating by parts, one can transfer the derivative with respect to v to η(v), vi Di η(v), or Di η(v). It follows from (3.2) that T D fε (x) = − α

∂ α D ft (x) dt + Dα fT (x), ∂t

0 < ε ≤ T,

(3.3)

ε

with

|D fT (x)| ≤ C

|f (Γ(T ) + y)| dy,

α

(3.4)

|y|
     ∂ α  D ft (x) ≤ C tm−1 |(f − Pm−1 )(x + y)|r(t)−m−n−|α| χ 2 y − Γ(t) dy   ∂t r(t) ≤ Ctm−1 r(t)−m−|α| Dm−1 (r(t))f (Γ(t)),

(3.5)

where C is independent of f , x, and Γ. Equality (3.3) also holds for ε = 0 with f0 = f provided that x is a Lebesgue point of D α f and limt→0 r1 (t) > 0. It follows from (3.5) that if B(Γ(t), 2r(t)) ⊂ G, then      ∂ α  D ft (x) ≤ Ctm−1 r(t)−m−n−|α| χ y − Γ(t) Dm−1 (r(t))f (y) dy. (3.6)   ∂t r(t) 4. PROOF OF THEOREM 2.1 For a regular domain G, i.e., for a domain G satisfying the flexible cone condition, the norm in the Lizorkin–Triebel space Lsp,θ (G) is defined by means of integrals of averaged absolute values of differences of functions (see [5, § 29; 10]) and has the form ⎧ T ⎫1/θ   ⎨ ⎬        dt θ −s m  f |Ls (G) = f |Lp (Gδ ) +  (4.1) p,θ ⎩ t δ (t, G)f (·) t ⎭  ,   0

1 The reasoning in [9, Section 3] needs to be corrected. We will assume that r (t) = 1

p

1 r(t) 4m t

and require in addition that limt→0 r1 (t) > 0. Moreover, after eliminating misprints in exponents, one obtains D(t) = btm(m−1)/2 and Dk (t) = Qm−1,k (t)t(m−1)(m−2)/2 . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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where

⎧ ⎪ −1 ⎪ | |B ⎨ 0

m

δ (t, E)f (x) =

⎪ ⎪ ⎩

|Δm (ty)f (x)| dy

31

for B(x, mt) ⊂ E,

|y|<1

for B(x, mt) ⊂ E.

0

It was shown in [10] that the norm (4.1) is equivalent to the norm ⎛ ⎛ T ⎞p/θ ⎞1/p θ −sθ dt ⎠ dx⎠ . f |Lp (Gδ ) + ⎝ ⎝ Dm−1 (t, Gt )f (x) t t 0

G

Therefore, to prove Theorem 2.1, it suffices to estimate the last term in (1.3): ⎛ ⎝



⎛ T   ⎝ (t + ρ(x))m ρ(γ(t))−m Dm−1 (ρ(γ(t)))f (γ(t)) θ (t + ρ(x))−sθ

G\Gδ

⎞p/θ ⎞1/p dt ⎠ dx⎠ t + ρ(x)

0

≤ C f |Lsp,θ (G) . (4.2) Let us apply estimate (7) from [10]: Dm−1 (t, G)f (x) ≤ Ct

−n

t δm (v, B(x, t))f 1

dv . v

(4.3)

0

For x ∈ G \ Gδ and 0 < t ≤ T , denote F (t, x) := (t + ρ(x))m−s ρ(γ(t))−m Dm−1 (ρ(γ(t)))f (γ(t))      t t F (t, x) + 1 − χ F (t, x) = : F1 (t, x) + F2 (t, x). =χ ρ(x) ρ(x) For 0 < t ≤ ρ(x) we have 12 ρ(x) ≤ ρ(γ(t)) ≤ 32 ρ(x). Therefore, F1 (t, x) ≤ C1 ρ(x)−s Dm−1 (ρ(γ(t)))f (γ(t))   2 ρ(γ(t)) f (γ(t) + y) dy Dm−1 ≤ C2 ρ(x)−s ρ(γ(t))−n 3 −s−n



|y|<ρ(γ(t))/2

≤ C3 ρ(x)



Dm−1

|y|<3ρ(x) −s−n

 2 ρ(γ(t)) f (x + y) dy 3



≤ C4 ρ(x)

Dm−1 (ρ(x))f (x + y) dy

|y|<3ρ(x) −s−n



≤ C5 ρ(x)

−n

ρ(x)

|y|<3ρ(x)

−s−n

≤ C6 ρ(x)



ρ(x)   m δ (v, B(x, ρ(x)))f (· + y) dv dy 1 v 0

ρ(x) dv v s ϕ(v, x + y) dy, v

|y|<3ρ(x) 0

where ϕ(v, x) = v −s δm (v, G)f (x). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Applying the H¨older inequality with respect to v, we obtain ϕ(·, x + y)|L∗θ dy. F1 (t, x) ≤ C7 ρ(x)−n |y|≤3ρ(x)

Therefore, for Φ(x) = ϕ(·, x)|L∗θ , we have ρ(x)−1/θ F1 (·, x)|Lθ ([0, T ]) ≤ C8 (M Φ)(x), where M is the Hardy–Littlewood maximal operator, which is bounded in Lp for 1 < p < ∞. Then ρ−1/θ F1 |Lp (Lθ ) ≤ C9 ϕ|Lp (L∗θ ) .

(4.4)

For t ≥ ρ(x) we have c0 t ≤ ρ(γ(t)) ≤ ct and, using (4.3), obtain F2 (t, x) ≤ C10 t

−s−n

ct ct  dv  m −n δ (v, B(γ(t), ρ(γ(t))))f  ≤ C10 t 1 v 0

0

|y|
" v #s t

ϕ(v, x + y) dy

dv v

= C10 (Aϕ)(t, x). The operator A for c = c1 = 1 is bounded in Lp (L∗θ ) (this fact was proved in [10]). Thus, F2 |Lp (L∗θ ) ≤ C11 ϕ|Lp (L∗θ ) .

(4.5)

Inequalities (4.4) and (4.5) yield (4.2) by virtue of (4.1). 5. PROOF OF THEOREM 2.2 Lemma 5.1. There exist constants C1 and C2 such that        n −n f − πx,t f L B x + y, 2 t  dy t Dm−1 (t)f (x) = f − πx,t f |L(B(x, t)) ≤ C1 t   3 

≤ C2 |y|
Dm−1



|y|
2 t f (x + y) dy. 3

(5.1)

  Proof. Note that B x, 6t ⊂ B(x, t) ∩ B x + y, 23 t for |y| < 2t . Without loss of generality, we  will assume that the projection polynomial πx,t f is constructed by the restriction of f to B x, 6t . Let us first prove the second inequality in (5.1). For |y| < 2t               2   f − πx,t f L B x + y, 2 t  = f − π x+y, 32 t f − πx,t f − πx+y, 32 t f L B x + y, t     3 3               2 2      ≤ f − πx+y, 2 t f L B x + y, t  + πx,t f − πx+y, 2 t f L B x + y, t   3 3 3 3   2 n t f (x + y) + C3 N, (5.2) ≤ t Dm−1 3 where C3 is independent of x, y, and t,       t  ,   N = πx,t f − πx+y, 2 t f L B x, 3 6  PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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and the last inequality in (5.2) follows from the fact that any two norms on Pm−1 are  equivalent. Since the projection polynomial πx,tg is constructed by the restriction of g to B x, 6t , for |y| < 2t we have              t  2       ≤ C t f − π N ≤ C4 f − πx+y, 2 t f L B x, 2 f L B x + y, 4 x+y, t   3 3 6  3   2 t f (x + y). ≤ C4 tn Dm−1 3 This and (5.2) imply the second inequality in (5.1). Let us prove the first inequality in (5.1), where we set x = 0 without loss of generality; i.e., we want to prove the inequality (g = f − π0,t f ) |g(y)| dy ≤ C1 t

−n



|g(z)| dz = C1 t

−n



|y|
B(0,t)



2y χ t

   3(y − z) χ dy |g(z)| dz. 2t

It suffices to show that t−n



χ

2y t

   "z # 3(y − z) χ dy ≥ Cχ 2t t

for some C > 0 independent of t. It is easy to see that this estimate is equivalent to its particular case with t = 1, and in this case it is geometrically obvious. This completes the proof of the lemma. To prove Theorem 2.2, we need the estimate   f |Ls(m) (G) ≤ C f |Wpm(G) . q,θ

(5.3)

Let us first estimate the last term on the right-hand side of (1.3) with q instead of p. By Lemma 5.1, to this end it suffices to estimate F |Lq (G \ Gδ ) , where T F (x)θ =

⎡ ⎣(t + ρ(x))m−s ρ(γ(t))−n

⎤θ



f [m] (γ(t) + z) dz ⎦

|z|<ρ(γ(t))/2

0

dt . t + ρ(x)

 α α α n Let g = M |α|=m |D f |0 , where |D f |0 is the extension of |D f | by zero to R \ G and M is the Hardy–Littlewood maximal operator (which is bounded in Lp for p > 1). Then f [m] ≤ Cg on G. If θ > 1, we apply the H¨older inequality with respect to z with the exponents θ and θ  : 

T F (x) ≤ C θ

(m−s)θ−1

(t + ρ(x)) 0

−n





ρ(γ(t))

θ g(γ(t) + z) dz dt = : C(A(gθ ))(x).

|z|<ρ(γ(t))/2

The operator A was studied in [6] (see [6, (2.13)]). Lemma 4.3 and Corollary 4.2 (with a = b = 0) from [6] imply that under condition (2.1) the operator A is a bounded operator from Lp( to Lq( for p( = pθ and q( = θq , 1 < p( < q( < ∞. Therefore, F |Lq (G \ Gδ ) ≤ C g|Lp ≤ C1



Dα f |Lp (G) .

(5.4)

|α|=m

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Now let us estimate the first term on the right-hand side of (1.3) with q instead of p. By Lemma 5.1, it suffices to estimate H|Lq (G) , where T H(x)θ =

⎡





⎣t−n−s

Dm−1

|y|
0

⎤θ



2 dt t, G 2 t f (x + y) dy ⎦ ≤C 3 3 t

T

⎡

⎤θ



⎣tm−s

g(x + y) dy ⎦

|y|
0

dt t





α with g = M |α|=m |D f |0 (M is the maximal operator). If θ > 1, we apply the H¨older inequality with respect to y with the exponents θ and θ  :

T H(x) ≤ C θ

t



(m−s)θ−1 −n

(g(x + y)) dy dt ≤ C1 θ

t

|y|
0

(g(x + y))θ

|y|
dy |y|n−(m−s)θ

.

Then, by the Hardy–Littlewood inequality (see, e.g., [5, Section 2.20]), we obtain ⎛ H|Lq (G) = ⎝

⎞θ/q



θ qθ

θ

H(x)

dx⎠

G

for m − s −

n p

+

n q

⎛ ≤ C2 ⎝

⎞θ/p



θ pθ

g(x)

dx⎠

θ       α  ≤ C3  |D f | Lp (G)  

(5.5)

|α|=m

G

≥ 0.

  s(m) Inequalities (5.4) and (5.5) give an estimate for the seminorm f |lq,θ (G) in terms of    α    |α|=m |D f | Lp (G) . To complete the proof of the theorem, it suffices to note that the estimate f |Lq (Gδ ) ≤ Cδ f |Wpm (G)

for m −

n n + ≥0 p q

follows from classical theorems. 6. PROOF OF THEOREM 2.3 As above in the proof of Theorem 2.2, the main estimate reduces to an estimate of the norm F |Lq (V (σ) \ V (σ)δ ) , which in turn reduces to the estimate A(g )|Lq( ≤ C g |Lp( , θ

θ

p where p( = , θ

q q( = , θ

) g=M



* |D f |0 . α

|α|=m

Here the operator A has the same form as in the proof of Theorem 2.2 but with   1 1 − . m − s ≥ [σ(n − 1) + 1] p q The (Lp(, Lq()-boundedness of such an operator was proved in [11] (see Theorem 5.3 and Section 6 there). Therefore, estimate (5.4) holds with G replaced by V (σ), and this is the main estimate in   s(m) the proof. Estimates for the other terms of the norm f |Lp,θ (V (σ)) can be obtained in essentially the same way as in the proof of Theorem 2.2. Remark. We do not present the proof of Theorem 2.4 here, because we hope to prove and publish a more general result in the short run. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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7. NECESSARY CONDITIONS FOR THE EMBEDDINGS The examples presented below for σ ≥ 1 provide some necessary conditions on the parameters for the embeddings in Theorems 2.2–2.4. Consider a specific domain G ⊂ Rn satisfying the flexible σ-cone condition (“mushrooms”). Let +∞ j G = j=0 G , where   G0 = (−1, 1)n−1 × (−1, 0) = x : −1 < xi < 1 for i = 1, . . . , n − 1; −1 < xn < 0 . Take sequences {rj } and {τj } (j ∈ N) such that 0 < rj < 1, 0 < τj < 1, rj ↓ 0, and τj ↓ 0 (as j → ∞). Define Gj with j ∈ N as     Gj = τj e1 + x ∈ Rn−1 : |x | < rj × (3rj , 4rj ) ∪ x ∈ Rn−1 : |x | < rjσ × (−1, 4rj ), where e1 = (1, 0, . . . , 0). Let us specify the sequences {rj } and {τj } so that Gj ∩ Gj+1 = ∅ for j ∈ N. On the domain G thus defined, consider the following sequence of functions fj , j ∈ N: fj = 0 on G0 and on G\Gj , and   xn − rj for x ∈ Gj , fj (x) = η rj where η ∈ C0∞ ((0, 1)) and η ≡ 0. We equip G \ Gδ with paths γ = γx as follows: for x ∈ Gj with xn > 3rj , the corresponding path lies on a two-segment polygonal chain, with the first segment joining x to the line x = τj e1 and the second segment lying on this line. It is easy to show that −m+ p1 +

fj |Wpm (G) ∼ rj

σ(n−1) p

,

−s+ n q

fj |Lsq,θ (G) ≥ c0 rj

as j → ∞. Comparing these two relations as j → ∞, we see that condition (2.1) is necessary for embedding (2.2). Let us find a necessary condition for the embedding from Theorem 2.3. Consider the function   xn − ε , η ∈ C0∞ ((0, 1)), η ≡ 0, 0 < ε < 1. fε (x) = η ε Then, for ε → 0, we have −m+ p1 + σ(n−1) p

fε |Wpm (V (σ)) ∼ ε

−s+ σ(n−1)+1 q

fε |Lsq,θ (V (σ)) ≥ c0 ε

,

.

Comparing these two relations as ε → 0, we see that the condition   1 1 − ≥0 m − s − [σ(n − 1) + 1] p q is necessary for embedding (2.3). Let us find a necessary condition for the embedding from Theorem 2.4. Consider the function  "x # 1 for t ≤ 1, n ∞ , ζ ∈ C (R), ζ(t) = fε (x) = ζ ε 0 for t ≥ 2. Then −s+ σ(n−1)+1 p

fε |Lsp,θ (V (σ)) ∼ ε

,

fε |Lq (V (σ)) ∼ ε

σ(n−1)+1 q

as ε ↓ 0. Comparing these two relations as ε → 0, we see that the condition   1 1 − ≥0 s − [σ(n − 1) + 1] p q is necessary for embedding (2.4). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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O.V. BESOV

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project nos. 05-01-01050 and 06-01-04006), INTAS (project no. 05-1000008-8157), the program “Contemporary Problems of Theoretical Mathematics” of the Russian Academy of Sciences, and a grant of the President of the Russian Federation (project no. NSh-2841.2006.1). REFERENCES 1. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 1991). 2. V. G. Maz’ya, Sobolev Spaces (Leningr. Gos. Univ., Leningrad, 1985; Springer, Berlin, 1985). 3. Yu. G. Reshetnyak, “Integral Representations of Differentiable Functions in Domains with Nonsmooth Boundary,” Sib. Mat. Zh. 21 (6), 108–116 (1980) [Sib. Math. J. 21, 833–839 (1981)]. 4. V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives, and Quasiconformal Mappings (Nauka, Moscow, 1983); Engl. transl.: Quasiconformal Mappings and Sobolev Spaces (Kluwer, Dordrecht, 1990). 5. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996) [in Russian]. 6. O. V. Besov, “Sobolev’s Embedding Theorem for a Domain with Irregular Boundary,” Mat. Sb. 192 (3), 3–26 (2001) [Sb. Math. 192, 323–346 (2001)]. 7. T. Kilpel¨ ainen and J. Mal´ y, “Sobolev Inequalities on Sets with Irregular Boundaries,” Z. Anal. Anwend. 19 (2), 369–380 (2000). 8. D. A. Labutin, “Sharpness of Sobolev Inequalities for a Class of Irregular Domains,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 232, 218–222 (2001) [Proc. Steklov Inst. Math. 232, 211–215 (2001)]. 9. O. V. Besov, “Spaces of Functions of Fractional Smoothness on an Irregular Domain,” Mat. Zametki 74 (2), 163–183 (2003) [Math. Notes 74, 157–176 (2003)]. 10. O. V. Besov, “Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains,” Mat. Zametki 74 (3), 340–349 (2003) [Math. Notes 74, 326–334 (2003)]. 11. O. V. Besov, “On the Compactness of Embeddings of Weighted Sobolev Spaces on a Domain with Irregular Boundary,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 232, 72–93 (2001) [Proc. Steklov Inst. Math. 232, 66–87 (2001)].

Translated by K. Besov

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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2008