Separable anisotropic differential operators in weighted abstract spaces and applications

J. Math. Anal. Appl. 338 (2008) 970–983 www.elsevier.com/locate/jmaa

Separable anisotropic differential operators in weighted abstract spaces and applications ✩ Ravi P. Agarwal a,∗ , Donal O’Regan b , Veli B. Shakhmurov c a Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA b Department of Mathematics, National University of Ireland, Galway, Ireland c Istanbul University, Engineering Faculty, Department of Electrical–Electronics Engineering, Avcilar, 34320 Istanbul, Turkey

Received 9 February 2007 Available online 8 June 2007 Submitted by S. Heikkilä

Abstract In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev– l (Ω; E , E) = W l (Ω; E) ∩ L Lions type spaces Wp,γ p,γ (Ω; E0 ) are discussed when E0 , E are two Banach spaces and E0 is 0 p,γ continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E0 . These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established. © 2007 Elsevier Inc. All rights reserved. Keywords: Banach space-valued functions; Operator-valued multipliers; UMD spaces; R-bounded sets; Boundary value problems; Differential-operator equations; Interpolation of Banach spaces; Positive operators

1. Introduction Fourier multipliers in vector-valued function spaces have been studied in [3,16,18,29] and operator-valued Fourier multipliers have been investigated in [4,10,11,14,30,31]. The exposition of the theory of Lp -multipliers of the Fourier transformation, and some related references, can be found in [29, §2.2.1–§2.2.4]. Boundary value problems (BVP’s) for differential-operator equations (DOE’s) have been discussed in [1,2,13,21,32]. A comprehensive introduction to DOE’s and historical references may be found in [13] and [32]. In this paper the operator-valued multiplier theorem in E-valued weighted space Lp,γ (R n ; E) is shown, here γ = γ (x) is a weighted function from Ap , p ∈ (1, ∞). Also l (Ω; E , E) = W l (Ω; E) ∩ L we consider the E-valued anisotropic class of function spaces Wp,γ 0 p,γ (Ω; E0 ). The p,γ most regular class of interpolation space Eα , between E0 and E is found such that the mixed differential operator D α l (Ω; E , E) to L is bounded from Wp,γ 0 p,γ (Ω; Eα ). This result generalizes and improves the results [5, §9], [28, §1.7] ✩

This work was supported by project UDP-543/17062005 of Istanbul University.

* Corresponding author.

E-mail addresses: [email protected] (R.P. Agarwal), [email protected] (D. O’Regan), [email protected] (V.B. Shakhmurov). 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.05.078

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for the scalar Sobolev space, the result [15] for one-dimensional vector function spaces and the results [22–27] for Hilbert-space valued classes. Finally, we consider the differential-operator equation   Lu = aα D α u + Au + Aα (x)D α u = f (1) |α:.l|=1

|α:.l|<1

(R n ; E),

where aα are complex numbers, A and Aα (x) are possible unbounded operators in a Banach space in Lp,γ E, α = (α1 , α2 , . . . , αn ), l = (l1 , l2 , . . . , ln ) are n-tuples of nonnegative integer numbers such that |α : l| =

n  αk k=1

lk

.

For l1 = l2 = · · · = ln = 2m from (1) we obtain the following elliptic DOE n 

ak (x)Dk2m u(x) + A(x)u(x) +



Aα (x)D α u(x) = f (x).

|α|<2m

k=1

We say that the problem (1) is Lp,γ (R n ; E)-separable, if for all f ∈ Lp,γ (R n ; E) there exists a unique solution u ∈ of the problem (1) satisfying this problem almost everywhere on R n and there exists a positive constant C independent on f , such that one has the coercive estimate l (R n ; E(A), E) Wp,γ n   lk  D u k



Lp,γ R n ;E



+ AuLp,γ R n ;E   Cf Lp,γ R n ;E  .

k=1

The above estimate implies that if f ∈ Lp,γ (R n ; E) and u is the solution of the BVP’s (1) then all terms of Eq. (1) belong to Lp,γ (R n ; E) (i.e., all terms are separable in Lp,γ (R n ; E)). The above estimate implies that the problem (1) is separable in Lp,γ (R n ; E) and the inverse of the realization l (R n ; E(A), E). operator generated by (1) is bounded from Lp,γ (R n ; E) to Wp,γ The paper is organized as follows. In Section 2 we collect the necessary tools from Banach space theory and some background material is given. In Section 3 the multiplier theorems are proved. In Section 4 using these multiplier theorems, embedding theorems in E-valued weighted Sobolev type spaces are shown. In Section 5 the separability properties for Eq. (1) are established. Finally, in Section 6 by applying this result the maximal regularity of infinite systems of anisotropic partial equations is proved. 2. Notations and background Let E be a Banach space and γ = γ (x), x = (x1 , x2 , . . . , xn ) be a positive measurable function on a measurable subset Ω ⊂ R n . Let Lp,γ (Ω; E) denote the space of strongly measurable E-valued functions that are defined on Ω with the norm  1 p p    f (x) E γ (x) dx , 1  p < ∞. f Lp,γ = f Lp,γ (Ω;E) = For γ (x) ≡ 1, the space Lp,γ (Ω; E) will be denoted by Lp = Lp (Ω; E). The Banach space E is said to be ζ -convex space (see e.g. [6]) if there exists on E × E a symmetric real-valued function ζ (u, v) which is convex with respect to each of the variables, and satisfies the conditions ζ (0, 0) > 0,

ζ (u, v)  u + v,

for u = v = 1.

In the literature the ζ -convex Banach space E is often called a UMD-space and written as E ∈ UMD. It is shown [6] that a Hilbert operator  f (y) (Hf )(x) = lim dy ε→0 x −y |x−y|>ε

is bounded in the space Lp (R, E), p ∈ (1, ∞) for those and only those spaces E, which possess the property of UMD spaces. Note UMD spaces include for example Lp , lp spaces and Lorentz spaces Lpq , p, q ∈ (1, ∞).

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Definition 1. Let γ be a weight function. A Banach space E is called a γ -UMD space if E-valued martingale difference sequences are unconditional in Lp,γ (R n ; E) for p ∈ (1, ∞) i.e., there exists a positive constant Cp such that for any martingale {fk , k ∈ N0 } (see [9, §5]), any choice of signs {εk , k ∈ N} ∈ {−1, 1} and N ∈ N   N      εk (fk − fk−1 )  Cp fN Lp,γ (Ω,Σ,μ,E) . f0 +   k=1

Lp,γ (Ω,Σ,μ,E)

We say that the Banach space E has property hp,γ if the Hilbert operator acts boundedly on Lp,γ (R n ; E) for p ∈ (1, ∞). Let Σ be a sub-σ -field of subsets of Σ that contains Ω and let Lp,γ (Ω; Σ, μ, E) denote a μ measurable E-valued weighted space. Let C be a set of complex numbers and

Sϕ = ξ ; ξ ∈ C, |arg ξ |  ϕ ∪ {0}, 0  ϕ < π. A linear operator A is said to be positive in a Banach space E, with bound M if D(A) is dense on E and    −1 (A + ξ I )−1   M 1 + |ξ | B(E) with ξ ∈ Sϕ , ϕ ∈ [0, π), where M is a positive constant and I is an identity operator in E, where L(E) is the space of bounded linear operators acting in E. Sometimes instead of A + ξ I we will write A + ξ and denote it by Aξ . It is known [29, §1.15.1] there exist fractional powers Aθ of the positive operator A. Let E(Aθ ) denote the space D(Aθ ) with the graph norm defined as  p  1  uE(Aθ ) = up + Aθ u p , 1  p < ∞, −∞ < θ < ∞. Let E1 and E2 be two Banach spaces. Now (E1 , E2 )θ,p , 0 < θ < 1, 1  p  ∞ will denote an interpolation space obtained from {E1 , E2 } by the K-method [29, §1.3.1]. We denote by D(R n ; E) the space of E-valued C ∞ -functions with compact support, equipped with the usual inductive limit topology and S = S(R n ; E) will denote the E-valued Schwartz space of rapidly decreasing, smooth functions. For E = C we simply write D(R n ) and S(R n ), respectively. Now D  (R n ; E) = L(D(R n ), E) denotes the space of E-valued distributions and S  (E) = S  (R n ; E) is a space of linear continued mapping from S(R n ) into E. The Fourier transform for u ∈ S  (R n ; E) is defined by     F (u)(ϕ) = u F (ϕ) , ϕ ∈ S R n . Let γ be such that S(R n ; E1 ) is dense in Lp,γ (R n ; E1 ) (see e.g. Theorem A0 ). A function Ψ ∈ C (l) (R n ; L(E1 , E2 )) is called a multiplier from Lp,γ (R n ; E1 ) to Lq,γ (R n ; E2 ) if there exists a positive constant C such that  −1  F Ψ (ξ )F u  CuLp,γ (R n ;E1 ) L (R n ;E ) q,γ

2

for all u ∈ S(R n ; E1 ). q,γ We denote the set of all multipliers from Lp,γ (R n ; E1 ) to Lq,γ (R n ; E2 ) by Mp,γ (E1 , E2 ). For E1 = E2 = E q,γ q,γ q,γ we denote Mp,γ (E1 , E2 ) by Mp,γ (E). Let M(h) = {Ψh ∈ Mp,γ (E1 , E2 ), h ∈ H } be a collection of multipliers in q,γ Mp,γ (E1 , E2 ). A family of sets M(h) ⊂ B(E1 , E2 ) depending on h ∈ H is called a uniform collection of multipliers with respect to h if there exists a positive constant C independent on h ∈ H , such that   −1  n  F Ψh F u  n   Cu L R ,E Lq,γ R ,E2

p,γ

1

u ∈ S(R n ; E

for all h ∈ H and 1 ). A set K ⊂ B(E1 , E2 ) is called R-bounded (see e.g. [10,11,31]) if there is a positive constant C such that for all T1 , T2 , . . . , Tm ∈ K and u1, u2 , . . . , um ∈ E1 , m ∈ N,   1  1  m m         rj (y)Tj uj  dy  C  rj (y)uj  dy,      0

j =1

E2

0

j =1

E1

where {rj } is a sequence of independent symmetric [−1, 1]-valued random variables on [0, 1] and N denotes the set of natural numbers. The smallest such constant C is called the R-bound of K and is denoted by R(K).

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A family of sets K(h) ⊂ B(E1 , E2 ) depending on the parameter h ∈ H is called uniformly R-bounded with respect to h if there is a positive constant C such that for all T1 , T2 , . . . , Tm ∈ K(h) and u1, u2 , . . . , um ∈ E1 , m ∈ N,   1  1  m m         rj (y)Tj (h)uj  dy  C  rj (y)uj  dy,      0

j =1

E2

0

j =1

E1

where the constant C is independent of the parameter h (i.e., suph∈H R(K(h)) < ∞). Let

β β Un = β = (β1 , β2 , . . . , βn ), |β|  n , ξ β = ξ1 1 ξ2 2 . . . ξnβn . Definition 2. The Banach space E is said to be a space satisfying a multiplier condition with respect to p ∈ (1, ∞) and a weight function γ , when for every Ψ ∈ C (n) (R n /0; B(E)) if the set

β β ξ Dξ Ψ (ξ ): ξ ∈ R n /0, β ∈ Un p,γ

is R-bounded, then Ψ ∈ Mp,γ (E). Remark 0. The result in [30] implies that the space lp , p ∈ (1, ∞) satisfies the multiplier condition with respect to p and the weight functions

βk N n   α αj k γ= |xj | , αj k  0, N ∈ N, βk ∈ R. γ = |x| , −1 < α < p − 1, 1+ k=1

j =1

q

q

Let Wh = {Ψh ∈ Mp (E1 , E2 ), h ∈ H } be a collection of multipliers in Mp (E1 , E2 ). We say that Wh is a uniform collection of multipliers if there exists a constant M > 0 independent on h ∈ H such that   −1 F Ψh F u  n   MuL (R n ;E ) p 1 L R ;E q

2

u ∈ S(R n ; E

for all h ∈ H and 1 ). A Banach space E has property (α), (see e.g. [11]) if there exists a constant α such that    N  N           αij εi εj xij  dy  α  εi εj xij           i,j =1

L2 Ω×Ω ;E

i,j =1

L2 (Ω×Ω ;E)

for all N ∈ N, xi,j ∈ E, αij ∈ {0, 1}, i, j = 1, 2, . . . , N, and all choices of independent, symmetric, {−1, 1}-valued  on probability spaces Ω, Ω  . For example the spaces L (Ω), 1  random variables ε1 , ε2 , . . . , εN , ε1 , ε2 , . . . , εN p p < ∞ have property (α). A space E is said to have local unconditional structure (l.u.st.) (see [20]) if there exists a positive constant C such that for any finite-dimensional subspace E0 of E there exists a finite-dimensional subspace F0 with an unconditional basis such that the natural embedding I0 : E0 ⊂ E factors as I0 = I1 I2 with I1 : F0 ⊂ E, I2 : E0 ⊂ F0 and I1 I2   C. Remark 1. If E is a UMD space with property (α) then these spaces satisfy the multiplier condition with respect to γ ∈ Ap and p ∈ (1, ∞) (see Theorem A1 ). It is well known (see [14,16]) that any Hilbert space satisfies the multiplier condition. There are, however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example UMD spaces (see [10,11,31]). Definition 3. A positive operator A is said to be R-positive in the Banach space E if there exists ϕ ∈ [0, π) such that the set

  LA = 1 + |ξ | (A + ξ I )−1 : ξ ∈ Sϕ is R-bounded.

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Note that in a Hilbert space every norm bounded set is R-bounded. Therefore, in a Hilbert space all positive operators are R-positive. If A is a generator of a contraction semigroup on Lq , 1  q  ∞ [17], A has bounded imaginary powers with (−Ait )B(E)  Ceν|t| , ν < π2 [8] or if A is generator of a semigroup with Gaussian bound [12] in E ∈ UMD then those operators are R-positive. Also σ∞ (E) will denote the space of compact operators in E. Let Ω be a domain on R n and l = (l1 , l2 , . . . , ln ). l (Ω; E , E) denotes the space that consists of functions u ∈ L Now Wp,γ 0 p,γ (Ω; E0 ) which have generalized derivatives Dklk u =

∂ lk l ∂xkk

u ∈ Lp,γ (Ω; E) with norm

uWp,γ l (Ω;E ,E) = uLp,γ (Ω;E0 ) + 0

n   lk  D u k

Lp,γ (Ω;E)

< ∞.

k=1

The weight γ is said to satisfy an Ap condition [19] i.e., γ ∈ Ap , 1 < p < ∞ if there is a positive constant C such that   p−1   1 1 1 − p−1 γ (x) dx γ (x) dx  C, |Q| |Q| Q

Q

for all compacts Q ⊂ R n . Condition 0. There exist positive constants C1 and C2 such that

C1 γ (y)  γ (x)  C2 γ (y), for x ∈ Qγ = x: x ∈ Ω, |xi − yi | < γ (y), i = 1, 2, . . . , n . Let E be a Banach space. The n-dimensional Riesz projection R0 is defined by   R0 f = F −1 χ(0, ∞)n Ff, Rk = F −1 χk Ff, f ∈ Lp,γ R n ; E , where χ(Ω) denotes the characteristic function of Ω and χk denotes the characteristic function of the halfspace

x = (x1 , x2 , . . . , xn ), xk > 0 . Let Qa,b =

n 

(ak , bk ),

k=1

n   j j +1  2 k,2 k . Dj = k=1

Let l = (l1 , l2 , . . . , ln ). Consider the following anisotropic partial differential equation (PDE)  aα D α u(x) = f (x), |α:l|1

where aα are complex numbers. The above PDE equation is said to be quasi elliptic if for all ξ ∈ R n there is a positive constant C such that    n    α   C a ξ |ξk |lk . α   |α:l|=1

k=1

Theorem A0 . Let E be a Banach space, 1  p < ∞ and let γ be a positive measurable function on an open subset Ω of R n , essentially bounded on compact subsets of Ω. Then the space D(Ω; E) is dense in Lp,γ (Ω; E). Proof. For u ∈ Lp,γ (Ω; E) and n ∈ N let un : Ω → E such that  u(x) if u(x)  n, un = 0 if u(x) > n. By the dominated convergence theorem limn→∞ u − un Lp,γ (Ω;E) = 0, hence a compactly supported function can be approximated with bounded compactly supported functions, that is with compactly supported function belonging

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to Lp (Ω; E). From the standard proof of the denseness theorem in the case of spaces without weight, it follows that if u is a compactly supported function belonging to Lp (Ω; E) then there exists a compact subset K ⊂ Ω, with supp u ⊆ K, and a sequence of functions un ∈ D(Ω; E), with supp un ⊆ K, such that limn→∞ u − un Lp (Ω;E) = 0. Now since  u − un Lp,γ (Ω;E) =

  u(x) − un (x)p γ (x) dx

1

p

 1 p  sup γ (x) u − un Lp (Ω;E) x∈K

K

we have lim u − un Lp,γ (Ω;E) = 0.

n→∞

2

We now refer the reader to [4]. 3. Multipliers in weighted Lp space Proposition A0 . Let γ ∈ Ap , p ∈ (1, ∞). A Banach space E has property hp,γ provided E is γ -UMD. Proof. Theorem A0 implies that the space C(Ω; E) is dense in Lp,γ (Ω; E). Since the linear combination of E-valued spherical harmonics polynomials is dense in C(Ω; E) then it is easy to see that this set is dense in Lp,γ (Ω; E). Therefore, by the approximation argument we can assume the martingales Φk -functions in [5, Lemma 1] to be Evalued spherical polynomials. Then replacing Lp (Ω; E) by Lp,γ (Ω; E) and by the reasoning in [5, Lemma 1] we obtain the assertion. 2 In a similar way as in [33] we obtain: Proposition A1 . Let γ ∈ Ap , p ∈ (1, ∞) satisfy Condition 0 and E be a γ -UMD space (respectively γ -UMD space with l.u.st.). Then for any choice of signs εk , k ∈ Z (respectively εk , k ∈ Zn ) the function Ψ = {εk , ξ ∈ Dk } (respectively p,γ Ψ = {εν , ξ ∈ ν }) is a Mp,γ (E) multiplier. Also the reasoning in [11] yields Theorem A1 . Let γ (x) =

n 

|xk |γk ,

0  γk < p − 1, 1 < p < ∞.

k=1

Let E1 and E2 be UMD spaces with property (α) and let Ψ ∈ C (n) (R n /0; B(E1 , E2 )). If

β sup ξ β Dξ Ψh (ξ ): ξ ∈ R n /0, β ∈ Un  Kβ < ∞, h∈H

then Ψh (ξ ) is a uniform collection of multipliers from Lp,γ (R n ; E1 ) to Lp,γ (R n ; E2 ). If n = 1 then the result remains true for all E1 , E2 ∈ UMD spaces. Proof. The proof of the theorem is based on the following steps: (1) We show that if E is a UMD space then the one-dimensional Riesz projection operator R1 is bounded in Lp,ν (R; E), ν(τ ) = |τ |ν which implies that R0 =

n  k=1

Rk .

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It is known that (see e.g. [6]) the assertion E ∈ UMD implies the boundedness of Hilbert operator H in Lp (R; E). For f ∈ Lp,ν (R; E) we have ν  p ν f (y)|τ | (Hf )(τ )|τ | p = lim (2) dy = Hfν + lim Hε1 f, ε→0 ε→0 |τ − y| |τ −y|>ε

where



Hε1 f = |τ −y|>ε

ν

− pν

fν (y)(|τ | p |y| |τ − y|

− 1)

dy,

ν

fν (y) = f (y)|y| p .

Since fν (y) ∈ Lp (R; E) then there exists a positive constant M such that Hfν Lp (R;E)  Mfν Lp (R;E) . Therefore by relation (2) we have only to prove that the operator Hε1 f has uniformly bounded norm in Lp (R; E). Reasoning as in [14, Theorem 2], i.e., using the Minkovskii integral inequality we obtain ν ν   p  1     p  1 −ν p p f (y)||τ | p |y| p − 1| f (τ η)||η| p − 1| = Hε1 f Lp (R;E)  dy dτ dη dτ |τ − y| |1 − η| R

 

R

 R

Since

ν+1 p

  f (τ η)p dτ

1

p

R

ν p

||η| − 1| dη = |1 − η|

 R

R

R

ν p

||η| − 1| |η|

ν+1 p

|1 − η|

dηf Lp (R;E) .

< 1 the last integral is finite. Thus we have from above that

Hε1 f Lp (R;E)  Cf Lp (R;E) . 1 That is the Hilbert operator is bounded in Lp,ν (R; E). It is known that R1 = 2πi (iπI − H ), where I is the identity operator. By using this relation between the Hilbert and Riesz operators we obtain that the Riesz projection operator R1 is bounded in Lp,ν (R; E). Then by the reasoning in [11, Remark 2.6] and by means of the space Lp,γ (R n ; E) instead of Lp (R n ; E) we obtain the representation equality of R0 and the boundedness of the operator R0 in Lp,γ (R n ; E). (2) We now prove the R-boundedness of a collection Φ = {F −1 χ(Qa,b )Ff : a, b ∈ R n } from Lp,γ (R n ; E1 ) to Lp,γ (R n ; E2 ). By virtue of [7, Lemma 3.17] the set



K = F −1 χ(Ca )Ff : a, b ∈ R n ,

Ca =

n 

[ak , ∞)

k=1

is R-bounded. Then the R-boundedness of Φa,b from Lp,γ (R n ; E1 ) to Lp,γ (R n ; E2 ) is proved by using step (1). (3) We establish that the Fourier multiplier of certain operator-valued functions depends on the parameter h ∈ B(h) which has constant operator values on the decomposition {Aj , j ∈ Z} of (0, ∞)n . Now Theorem A0 implies that the set F −1 D(R n ; E1 ) is dense in RLp,γ (R n ; E1 ). Using this fact step (3) follows in a similar way as in [11, Theorem 3.3], i.e., by using step (2), the unconditional Schauder decompositions of the Banach space [7, Theorem 3.4] and [11, Proposition 3.2] which is valid trivially in this case. n n n (4) Let Mh , Mh,N ∈ Lloc 1 (R , B(E1 , E2 )) be Fourier multipliers from Lp,γ (R ; E1 ) to Lp,γ (R ; E2 ). In this step loc n as in [11, Proposition 3.4] we note that if Mh,N converges to Mh in L1 (R , B(E1 , E2 )) and the corresponding operators Th,N = F −1 Mh,N F are uniformly bounded in      B Lp,γ R n ; E1 , Lp,γ R n ; E2 , then the operator Th = F −1 Mh F is uniformly bounded (with respect to h), with Th   lim inf Th,N . N →∞

(5) We note that the systems of operators {j = F −1 χ(Dj )F } are unconditional Schauder decompositions of the space R0 Lp,γ (R n ; E1 ). This step follows as in [11, Lemma 3.5].

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Now by using the steps (1)–(5), [11, Lemma 2.7] and the generalized variant of [7, Lemma 3.17] as in [11, Theorem 3.6] we obtain the assertion of Theorem A1 . 2 Remark 2. It is clear that Theorem A1 is valid for the case when multiplier functions do not depend on a parameter. 4. Embedding operators The following embedding theorems in vector-valued Sobolev spaces play important roles in boundary value problems for differential equations in Banach spaces. Theorem A2 . Suppose the following conditions are satisfied: (1) E is a Banach space that satisfies the multiplier condition with respect to p and weighted function γ (x) and A is an R-positive operator in E for ϕ with 0  ϕ < π ; (2)  α = (α1 , α2 , . . . , αn ), l = (l1 , l2 , . . . , ln ) are n-tuples of nonnegative integer numbers such that  = |α : l| = n αk k=1 lk  1, 1 < p < ∞, 0 < μ  1 − ; (3) Ω ∈ R n is a region such that there exists a bounded linear extension operator acting from Lp,γ (Ω; E) to l (Ω; E(A), E) to W l (R n ; E(A), E). Lp,γ (R n ; E) and also from Wp,γ p,γ Then an embedding      l Ω; E(A), E ⊂ Lp,γ Ω; E A1−−μ D α Wp,γ is continuous and there exists a positive constant Cμ such that  α    D u  Cμ hμ uW l (Ω;E(A),E) + h−(1−μ) uLp,γ (Ω;E) L (Ω;E(A1−−μ )) p,γ

(3)

p,γ ,t

l (Ω; E(A), E) and with h > 0. for all u ∈ Wp,γ

Proof. It suffices to prove the estimate (3). We first prove (3) for the case Ω = R n . It is easy to see that    α  D u  F −  (iξ )α A1−−μ uˆ L (R n ;E) . L (R n ;E(A1−−μ )) p,γ

Moreover, for all

p,γ

l (R n ; E(A), E) u ∈ Wp,γ

we have

uWp,γ l (R n ;E(A),E) = uLp,γ (R n ;E(A)) +

n   lk  D u k

Lp,γ (R n ;E)

k=1

  = F −  uˆ L

p,γ

   F −1 Auˆ L

+ (R n ;E(A))

p,γ (R

n ;E)

+

n   −   F (iξk )lk uˆ 

Lp,γ (R n ;E)

k=1 n 

 −   F (iξk )lk uˆ 

Lp,γ (R n ;E)

.

k=1

Thus proving the inequality (3) on Ω = R n is equivalent to proving   − F (iξ )α A1−−μ uˆ  Lq,γ (R n ,E)  n    −    F (iξk )lk uˆ   Cμ hμ F −  Auˆ  + n

Lp,γ (R n ,E)

Lp,γ (R ,E)

k=1

for a suitable positive constant Cμ . Let

n  μ lk Qh (ξ ) = h A + |ξk | + h−(1−μ) . k=1



−(1−μ) 

+h

F

−

 uˆ L

p,γ (R

 n ,E)

(4)

978

R.P. Agarwal et al. / J. Math. Anal. Appl. 338 (2008) 970–983

It is easy to see that the inequality (4) will follow immediately if we can prove that the operator-function Ψh = p,γ (iξ )α A1−−μ Q−1 h (ξ ) is a uniform collection of multipliers in Mp,γ (E) with respect to h. To see this, it is suffices to show that the sets β β

ξ D Ψh (ξ ): ξ ∈ R n /{0}, β ∈ Un are R-bounded in E and the R-bounds do not depend on h. Firstly, by using a similar technique as in [25, Lemma 3.1] we have  β  β  ξ D Ψh (ξ )  C, ξ ∈ R n /{0}, β ∈ Un , (5) B(E) uniformly with respect to h. Due to the R-positivity of operator A and the estimate (5) we obtain that the sets  

n 

−1 −1 n lk −1 n AQh (ξ ): ξ ∈ R /{0} , 1+ |ξk | + h Qh (ξ ): ξ ∈ R /{0} k=1

are R bounded uniformly with respect to h. Moreover for u1, u2 , . . . , um ∈ E, m ∈ N and ξ j = (ξ1j , ξ2j , . . . , ξnj ) ∈ R n /{0} we have  m    j    rj (y)Ψh ξ uj     j =1 Lp (0,1;E)   m   j α 1−−μ −1  j     = rj (y) ξ A Qh ξ uj    j =1

Lp (0,1;E)

 m (+μ)

−(+μ) 

n n       j α  j = rj (y) ξ |ξkj |lk + h−1 |ξkj |lk + h−1 Q−1 1+ 1+ h ξ  j =1 k=1 k=1    j 1−(+μ)   × AQ−1 uj  , h ξ  Lp (0,1;E)

where {rj } is a sequence of independent symmetric {−1, 1}-valued random variables on [0, 1]. By virtue of Kahane’s contraction principle [10, Lemma 3.5] we obtain from the above equality   m   j    rj (y)Ψh ξ uj     j =1

Lp (0,1;E)

 m  (+μ) 

n      −1  j  −1  j 1−(+μ)  lk −1 AQh ξ  2M0  rj (y) 1 + |ξkj | + h uj  Qh ξ   j =1

k=1

.

Lp (0,1;E)

Then by the above estimate, in view of (5) and by product properties of the collection of R-bounded operators (see e.g. [10, Proposition 3.4]) we get that the set {Ψh (ξ ): ξ ∈ R n /{0}} is R bounded uniformly with respect to t and h. In a similar way, by using the Kahane’s contraction principle and by product and additional properties of the collection of R-bounded operators [10, Proposition 3.4] we obtain that the sets

β β ξ D Ψh (ξ ): ξ ∈ R n /{0}, β ∈ Un are R-bounded uniformly with respect to h. Then we obtain that the operator-function Ψh (ξ ) is a uniform colq,γ lection of multipliers in Mp,γ (E). Therefore, we obtain the estimate (4). Then by using an extension operator in l (Ω; E(A), E) we obtain from (4) the estimate (3). 2 Wp, γ By a similar manner as in Theorem A2 we have: Theorem A3 . Suppose all conditions of the Theorem A2 are satisfied. Then for 0 < μ < 1 −  the embedding       l D α Wp,γ Ω; E(A), E ⊂ Lp,γ Ω; E(A), E ,p

R.P. Agarwal et al. / J. Math. Anal. Appl. 338 (2008) 970–983

979

is continuous and there exists a positive constant Cμ such that  α  D u  CuWp,γ l (Ω;E(A),E) L (Ω;(E(A),E) ) p,γ

,p

l (Ω; E(A), E). for all u ∈ Wp,γ

 Proof. Reasoning as in Theorem A2 it suffices to prove that an operator-function Ψ (ξ ) = ξ α [A + nk=1 ξklk ]−1 is a multiplier from Lp,γ (R n ; E) to Lp,γ (R n ; ((E(A), E),p )). By using the R-positivity of the operator A and the interpolation spaces [29, §1.14.5] the assertion is shown. 2 5. Separable differential operators in weighted spaces Before we consider Eq. (1) we first consider the principal equation  aα D α u + Au = f, Lu =

(6)

|α:.l|=1

where aα are complex numbers, A is a possible unbounded operator in a Banach space E. Theorem A4 . Suppose the following conditions hold: (1) E is a Banach space satisfying the multiplier condition with respect to γ = (1, ∞); (2) A is an R-positive operator in E and 

K(ξ ) =

aα (iξ1 )α1 (iξ2 )α2 . . . (iξn )αn ∈ S(ϕ),

n

γk k=1 |xk | ,

0  γk < p − 1, p ∈

n    K(ξ )  C |ξk |lk , ξ ∈ R n .

|α:.l|=1

k=1

l (R n ; E(A), E) and Then for all f ∈ Lp,γ (R n ; E) Eq. (6) has a unique solution u(x) that belongs to the space Wp,γ the coercive estimate    D α u + AuLp,γ  Cf Lp,γ (7) L p,γ

|α:.l|=1

holds. Proof. By applying the Fourier transform to Eq. (6) we obtain K(ξ ) + Au(ξ ˆ ) = fˆ(ξ ), where K(ξ ) =



(8)

aα (iξ1 )α1 (iξ2 )α2 . . . (iξn )αn .

|α:.l|=1

Since K(ξ ) ∈ S(ϕ) for all ξ ∈ R n , the operator A + K(ξ ) is invertible in E. So, we obtain that the solution of (8) can be represented in the form  −1 u(x) = F −1 A + K(ξ ) fˆ. (9) Now we have   −1  AuLp,γ = F −1 A A + K(ξ ) f ^ L , p,γ  α   −1  −1  α1 α2 αn D u  F = (iξ ) (iξ ) · · · (iξ ) A + K(ξ ) fˆL 1 2 n L p,γ

Hence, it is suffices to show that the operator-functions

p,γ

.

980

R.P. Agarwal et al. / J. Math. Anal. Appl. 338 (2008) 970–983

 −1 σ1 (ξ ) = A A + K(ξ ) ,   −1 σ2 (ξ ) = (iξ1 )α1 .(iξ2 )α2 . . . (iξn )αn A + K(ξ ) |α:.l|=1

are multipliers in Lp,γ (R n ; E). To see this, it suffices to show that the following collections β β

β β

ξ D σ1 (ξ ): ξ ∈ R n /{0}, β ∈ Un , ξ D σ2 (ξ ): ξ ∈ R n /{0}, β ∈ Un are R-bounded in E. Due to the R-positivity of A, the set

σ1 (ξ ): ξ ∈ R n /{0}, β ∈ Un is R-bounded. Moreover, reasoning as in the prove Theorem A2 and in view of condition (2) we obtain that the set {σ2 (ξ ): ξ ∈ R n /{0}, β ∈ Un } is R-bounded. Then by virtue of Kahane’s contraction principle, by product properties of the collection of R-bounded operators (see e.g. [10, Lemma 3.5, Proposition 3.4]) and also the R-positivity of the operator A we obtain

R ξ β D β σ1 (ξ ): ξ ∈ R n /{0}, β ∈ Un  C,

R ξ β D β σ2 (ξ ): ξ ∈ R n /{0}, β ∈ Un  C. (10) p,γ

The estimates (10) imply that the functions σ1 (ξ ) and σ2 (ξ ) are Mp,γ (E) multipliers. Let L0 denote the differential operator in Lp,γ (R n ; E) that is generated by problem (6), that is   n  l D(L0 ) = Wp,γ R , E(A), E , L0 u = aα D α u + Au. |α:.l|=1 l (R n ; E(A), E). We The estimate (7) implies that the operator L0 has a bounded inverse from Lp,γ (R n ; E) into Wp,γ n denote by L the differential operator in Lp,γ (R ; E) that is generated by problem (1). Namely,  n  l D(L) = Wp,γ R , E(A), E , Lu = L0 u + L1 u,

where L1 u =



Aα (x)D α u.

2

|α:.l|1

Theorem A5 . Suppose all the conditions of Theorem A4 are hold and   Aα (x)A1−|α:l|−μ ∈ L∞ R n ; B(E) for 0 < μ < 1 − |α : l|. Then the operator L + λ for sufficiently large λ > 0 is separable in Lp,γ (R n ; E). Proof. In view of the condition on Aα (x) and by the virtue of Theorem A2 , there is h > 0 such that       Aα (x)D α u A1−|α:l|−μ D α u  L1 uLp,γ  L L |α:l|<1

 hμ

 

p,γ

|α:l|<1

D α uLp,γ + AuLp,γ

p,γ



+ h−(1−μ) uLp,γ

(11)

|α:l|=1 l (R n ; E(A), E). Then from the estimates (7) and (11) for u ∈ W l (R n ; E(A), E) we obtain for u ∈ Wp,γ p,γ    μ −(1−μ) L1 uLp,γ  C h (L0 + λ)uL + h uLp,γ . p,γ

(12)

l (R n ; E(A), E) we get Since uLp,γ = λ1 (L0 + λ)u − L0 uLp,γ for u ∈ Wp,γ

 1 uLp,γ  (L0 + λ)uL + L0 uLp,γ p,γ λ      1 M   α    D u L + AuLp,γ . (L0 + λ)u L +  p,γ p,γ λ λ |α:l|=1

(13)

R.P. Agarwal et al. / J. Math. Anal. Appl. 338 (2008) 970–983 2l (R n ; E(A), E) we obtain From the estimates (12) and (13) for u ∈ Wp,γ     L1 uLp,γ  Chμ (L0 + λ)uL + CMλ−1 h−(1−μ) (L0 + λ)uL p,γ

p,γ

981

(14)

.

Then choosing h and λ such that Chμ < 1, CMh−(1−μ) < λ, from (14) for sufficiently large λ we have   L1 (L0 + λ)−1  < 1. B(L (R n ;E)) p,γ

Now we have the relations

(15)

 −1 (L + λ)−1 = (L0 + λ)−1 I + L1 (L0 + λ)−1

(L + λ) = L0 + λ + L1 ,

so by using the estimate (15) and the perturbation theory of linear operators from Theorem A4 we obtain that the l (R n ; E(A), E). This implies the assertion. 2 differential operator L + λ is invertible from Lp,γ (R n ; E) into Wp,γ 6. Maximal regular infinite systems of anisotropic equations Consider the following infinity systems of equations 

∞  

aα D α um (x) + dm (x)um (x) +

|α:l|=1

dαj (x)D α uj (x) = fm (x),

x ∈ R n , m = 1, 2, . . . , ∞.

(16)

|α:l|<1 j =1

Let D = {dm }, 

Du = {dm um }, m = 1, 2, . . . , ∞, ∞ 

1 q  q |dm um | < ∞ , x ∈ R n , 1 < q < ∞. lq (D) = u: u ∈ lq , ulq (D) = Dulq = dm > 0,

u = {um },

m=1

 Theorem A6 . Let γ (x) = nk=1 |xk |γk , 0  γk < p − 1, p, q ∈ (1, ∞) and aα , dαmj ∈ L∞ (R n ) such that for 0 < μ < 1 − |α : l|   n     α1 α2 αn    C a (iξ ) (iξ ) . . . (iξ ) |ξk |lk , ξ ∈ R n , α 1 2 n   |α:.l|=1



k=1

∞ 

   −(1−|α:l|−μ) q dαj dm <∞

for a.e. x ∈ R n ,

|α:l|<1 j,m=1

1 1 +  = 1. q q

n Then for all f (x) = {fm (x)}∞ 1 ∈ Lp,γ (R ; lq ) and for sufficiently large λ the problem (16) has a unique solution ∞ l u = {um (x)}1 that belongs to the space Wp,γ (R n , lq (D), lq ) and the coercive estimate







∞    α D um (x)q

|α:l|1 R n m=1

 

C Rn

∞    fm (x)q

1

p q

 

p

γ (x) dx 1

p q

+

∞    dm um (x)q

1

p q

p

γ (x) dx

R n m=1

p

γ (x) dx

m=1

holds for the solution of the problem (16). Proof. Let E = lq , A and Ak (x) be infinite matrices, such that   A = [dm δmj ], Aα (x) = dαj (x) , j = 1, 2, . . . , ∞.

(17)

982

R.P. Agarwal et al. / J. Math. Anal. Appl. 338 (2008) 970–983

It is clear to see that this operator A is positive in lq . Therefore, by virtue of Theorem A5 we obtain that problem (16) l (R n ; l (D), l ) for all f ∈ Lp,γ (R n ; lq ) and sufficiently large λ has a unique solution u that belongs to space Wp,γ q q and    D α u + DuLp,γ (R n :lq )  Cf Lp,γ (R n :lq ) . L (R n :l ) |α:l|1

p,γ

q

From the above estimate we obtain (17).

2

Remark 3. There are many R-positive operators in different concrete Banach spaces. Therefore, putting concrete Banach spaces instead of E, and concrete R-positive differential, pseudo differential operators, or finite, infinite matrices instead of the operator A, on the differential-operator equations (1), by virtue of Theorem A5 we can obtain different class maximal regular partial differential equations or system of equations. References [1] R.P. Agarwal, M. Bohner, V.B. Shakhmurov, Maximal regular boundary value problems in Banach-valued weighted space, Bound. Value Probl. 2005 (1) (2005) 9–42. [2] S. Agmon, L. Nirenberg, Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math. 16 (1963) 121–239. [3] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997) 5–56. [4] J. Bourgain, Some remarks on Banach spaces in which martingale differences are unconditional, Ark. Mat. 21 (1983) 163–168. [5] O.V.P. Besov, V.P. Ilin, S.M. Nikolskii, Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975. [6] D.L. Burkholder, A geometrical conditions that implies the existence of certain singular integrals of Banach space-valued functions, in: Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, Wadsworth, Belmont, CA, 1983, pp. 270–286. [7] Ph. Clement, B. de Pagter, F.A. Sukochev, H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math. 138 (2000) 135–163. [8] G. Dore, A. Venni, On the closeness of the sum of two closed operators, Math. Z. 196 (1987) 189–201. [9] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, UK, 1995. [10] R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003) 788. [11] R. Haller, H. Heck, A. Noll, Mikhlin’s theorem for operator-valued Fourier multipliers in n variables, Math. Nachr. 244 (2002) 110–130. [12] M. Hieber, J. Prüss, Heat kernels and maximal Lp –Lq -estimates for parabolic evolution equations, Comm. Partial Differential Equations 22 (1997) 1647–1669. [13] S.G. Krein, Linear Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1971. [14] P. Kree, Sur les multiplicateurs dans FL aves poids, Ann. Inst. Fourier (Grenoble) 16 (1966) 121–191. [15] J.L. Lions, J. Peetre, Sur une classe d’espaces d’interpolation, Publ. Math. Inst. Hautes Etudes Sci. 19 (1964) 5–68. [16] P.I. Lizorkin, (Lp , Lq )-Multiplicators of Fourier integrals, Dokl. Akad. Nauk SSSR 152 (4) (1963) 808–811. [17] D. Lamberton, Equations d’evalution lineaires associeees a’des semigroupes de contractions dans less espaces Lp , J. Funct. Anal. 72 (1987) 252–262. [18] T.R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (2) (1984) 739–757. [19] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972) 31–38. [20] G. Pisier, Some results on Banach spaces without local unconditional structure, Compos. Math. 37 (1978) 3–19. [21] A.Ya. Shklyar, Complete Second Order Linear Differential Equations in Hilbert Spaces, Birkhäuser Verlag, Basel, 1997. [22] V.B. Shakhmurov, Theorems about compact embedding and applications, Dokl. Akad. Nauk SSSR 241 (6) (1978) 1285–1288. [23] V.B. Shakhmurov, Embedding theorems and their applications to degenerate equations, Differ. Equ. 24 (4) (1988) 475–482. [24] V.B. Shakhmurov, Embedding theorems for abstract function-spaces and their applications, Mat. Sb. 134 (1–2) (1987) 261–276. [25] V.B. Shakhmurov, Coercive boundary-value problems for strongly degenerating abstract equations, Dokl. Akad. Nauk SSSR 290 (3) (1986) 553–556. [26] V.B. Shakhmurov, Coercive boundary value problems for regular degenerate differential-operator equations, J. Math. Anal. Appl. 292 (2004) 605–620. [27] V.B. Shakhmurov, Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces, J. Inequal. Appl. 4 (2005) 329–345. [28] S.L. Sobolev, Certain Applications of Functional Analysis to Mathematical Physics, Nauka, Moscow, 1988. [29] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

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