S. V. Astashkin and P. Sunehag- The Real Interpolation Method on Couples of Intersections

8/3/2019 S. V. Astashkin and P. Sunehag- The Real Interpolation Method on Couples of Intersections

http://slidepdf.com/reader/full/s-v-astashkin-and-p-sunehag-the-real-interpolation-method-on-couples-of 1/4

218 0016–2663/06/4003–0218 c2006 Springer Science+Business Media, Inc.

Functional Analysis and Its Applications, Vol. 40, No. 3, pp. 218–221, 2006

Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 66–69, 2006

Original Russian Text Copyright c by S. V. Astashkin and P. Sunehag

BRIEF COMMUNICATIONS

The Real Interpolation Method on Couples of Intersections

S. V. Astashkin and P. Sunehag

Received April 20, 2005

Abstract. Suppose that (X 0, X 1) is a Banach couple, X 0∩X 1 is dense in X 0 and X 1 , (X 0, X 1)θ,q(0 < θ < 1, 1 q < ∞) are the spaces of the real interpolation method, ψ ∈ (X 0 ∩X 1)∗ , ψ = 0, isa linear functional, N = Ker ψ, and N i stands for N with the norm inherited from X i (i = 0, 1).The following theorem is proved: the norms of the spaces (N 0, N 1)θ,q and (X 0, X 1)θ,q are equivalenton N if and only if θ ∈ (0, α) ∪ (β ∞, α0) ∪ (β 0, α∞) ∪ (β, 1), where α, β , α0 , β 0 , α∞ , and β ∞ arethe dilation indices of the function k(t) = K (t, ψ; X ∗

0, X ∗

1 ).

Key words: interpolation space, interpolation of subspaces, interpolation of intersections, real

interpolation method, K -functional, dilation index of a function, weighted L p-space.

1. Introduction. Interpolation of subspaces is still one of the most significant problems ininterpolation theory [1]. It has been stated as early as in the monograph [2] by Lions and Magenes.Various aspects of this problem have been treated in [3]–[12]. (The authors are very far from claimingthe completeness of this citation list.) An important special case of this general problem is theproblem of interpolation of intersections, which can be stated for the real method of interpolationas follows. (The definitions will be given below.) Let (X 0, X 1) be a Banach couple, i.e., a pairof Banach spaces linearly and continuously embedded in a Hausdorff topological vector space T .Every linear subspace N of T generates a normed (in general, non-Banach) couple (X 0∩N, X 1∩N ),where the norm on X i ∩ N is the restriction of the norm of the space X i (i = 0, 1). The problemis to find conditions on the triple (X 0, X 1, N ) and the parameters θ ∈ (0, 1) and q ∈ [1,∞] of the

real interpolation method under which the natural formula(X 0 ∩N, X 1 ∩N )θ,q = (X 0, X 1)θ,q ∩N (1)

is valid (in the sense that the spaces coincide as sets and the norms are equivalent).We consider the case in which N is the kernel of a linear functional ψ ∈ (X 0 ∩ X 1)∗. Then

formula (1) just means the equivalence of norms of the spaces N θ,q = (X 0 ∩ N, X 1 ∩ N )θ,q andX θ,q = (X 0, X 1)θ,q on the subspace N . In [13], [14] and independently in [15], four dilation indicesrelated to the functional ψ viewed as an element of the sum X ∗0 +X ∗1 of dual spaces were introduced,and the problem was solved under some additional condition on these indices. In this note, weannounce results in which this additional condition is dropped owing to the introduction of yetanother two similar indices. Thus we obtain a definitive, complete solution of this problem.

An important special case of this problem was treated by Krugljak, Maligranda, and Persson

[16], who, under some additional assumptions, found conditions on θ ∈ (0, 1), p ∈ [1,∞), and theweight functions w0(x) and w1(x) ensuring that

(L p(w0) ∩N, L p(w1) ∩N )θ,p = (L p(w0), L p(w1))θ,p ∩N, (2)

where L p(w) is the weighted L p-space on (0,∞) with the usual norm and N is the space of allfunctions f : (0,∞) → R satisfying

∞

0 f (x) dx = 0. In the same paper, a similar problem was posedfor the more general couple (L p0(w0), L p1(w1)). Using Theorems 1 and 2 in the present paper, onecan readily solve this problem; at the end of the paper, we state one result in this direction. (See[14] for details.)



219

2. Definitions and notation. For a normed couple (X 0, X 1) and t > 0, the Peetre K -func-tional is defined by the formula

K (t, x; X 0, X 1) = inf x=x0+x1,xi∈Xi

(x0X0+ tx1X1

), x ∈ X 0 + X 1.

If 0 < θ < 1 and 1 q < ∞, then the space X θ,q = (X 0, X 1)θ,q of the real interpolation methodconsists of all x ∈ X 0 + X 1 such that

xXθ,q = ∞0

(t−θK (t, x : X 0, X 1))q dtt1/q

< ∞.

One can give another definition of this space (with an equivalent norm) in terms of the J -func-tional

J (t, x; X 0, X 1) = max(xX0, txX1

), x ∈ X 0 ∩X 1.

Specifically, X θ,q consists of all x ∈ X 0+X 1 representable in the form x =

k∈Z 2θkxk (convergencein X 0 + X 1) with the norm

inf

k∈Z

(J (2k, xk; X 0, X 1))q1/q

, (3)

where the infimum is taken over all such representations.

If X 0∩X 1 is dense in X 0 and X 1 and ψ ∈ (X 0∩X 1)∗

, then we can consider the Banach couple(X ∗0 , X ∗1 ) of dual spaces, and ψ ∈ X ∗0 + X ∗1 [1, §3.7]. In what follows, an important role will beplayed by the K -functional k(t) = K (t, ψ; X ∗0 , X ∗1 ) and the functions

M (t) = sups>0

k(ts)

k(s), M 0(t) = sup

0<smin(1,1/t)

k(ts)

k(s), M ∞(t) = sup

smax(1,1/t)

k(ts)

k(s).

They are submultiplicative for t > 0, and therefore, the following numbers, called the dilation

indices of k(t), are well defined:

α = limt→0

log2 M (t)

log2 t, α0 = lim

t→0

log2 M 0(t)

log2 t, α∞ = lim

t→0

log2 M ∞(t)

log2 t,

β = limt→∞

log2 M (t)

log2 t , β 0 = limt→∞

log2 M 0(t)

log2 t , β ∞ = limt→∞

log2 M ∞(t)

log2 t .

It is readily seen that 0 α α0 β 0 β 1 and 0 α α∞ β ∞ β 1.

3. Main results. Suppose that (X 0, X 1) is a Banach couple such that X 0 ∩X 1 is dense in X 0and X 1 . Let ψ ∈ (X 0 ∩ X 1)∗ be a nonzero functional, let N = Ker ψ, and let N i be the normedspace N with norm inherited from X i (i = 0, 1).

Theorem 1. The norms of the interpolation spaces N θ,q = (N 0, N 1)θ,q and X θ,q = (X 0, X 1)θ,qare equivalent on N if and only if

θ ∈ (0, α) ∪ (β ∞, α0) ∪ (β 0, α∞) ∪ (β, 1). (4)

Moreover, if θ ∈ (0, α)∪(β 0, α∞)∪(β, 1), then N θ,q = (N 0, N 1)θ,q is dense in X θ,q = (X 0, X 1)θ,q ;if θ ∈ (β ∞, α0), then N θ,q is dense in some subspace of X θ,q of codimension 1.

Remark 1. The inequalities α0 β 0 and α∞ β ∞ imply that at most one of the intervals(β 0, α∞) and (β ∞, α0) can be nonempty.

Using the following definition in [15], we can state Theorem 1 in a different way. Keeping thepreceding notation and conditions, by X θ,q,ψ we denote the set of all x ∈ X 0 + X 1 representable inthe form

x =k∈Z

2θkxk, xk ∈ N (convergence in X 0 + X 1) (5)

with the norm (3), where the infimum is taken over all representations (5).



221

[13] S. V. Astashkin, Funkts. Anal. Prilozhen., 39:2 (2005), 61–64; English transl.: Funct. Anal.Appl., 39:2 (2005), 131–134.

[14] S. V. Astashkin, Algebra i Analiz, 17:2 (2005), 33–69; English transl.: St. Petersburg Math.J., 17:2 (2006), 239–266.

[15] P. Sunehag, J. Approx. Theory, 130 (2004), 78–98.[16] N. Krugljak, L. Maligranda, and L.-E. Persson, Ark. Mat., 37 (1999), 323–344.

Samara State University

e-mail: [email protected] ICT Australia, Statistical Machine Learning Programe-mail: [email protected]

Translated by S. V. Astashkin and P. Sunehag

S. V. Astashkin and P. Sunehag- The Real Interpolation Method on Couples of Intersections

Documents

Transcript of S. V. Astashkin and P. Sunehag- The Real Interpolation Method on Couples of Intersections