Basics of the interpolation of spaces

Basics of the interpolation of spaces Jorge Eliecer Hernández Hernández August 7, 2017

1 Introduction The following is a brief compilation of the main basic notions in the General Theory of Interpolation of Spaces. It contains some basic concepts and theorems that serve as an introduction to such an interesting area of Mathematics.

2 Basic and Notations Let (X; ; ) be a measure space. A function f : X ! R (or C) is said to be simple if the imagin set contains a nite different number of values, each in a subset of X with nite measure. We will use the following notation: 1. With SF (X; ) we will denote the set of simple functions de ned on X: 2. If a > 0 , f 2 SF (X; ) , f (x) = cj for all x 2 Aj ; we de ne 1a 0 Z n a X 1=a 1=a kf k1=a = jf (x)j d =@ jcj j A j=1

and in the case a = 0; then

kf k1 = sup jf (x)j : x2X

3. We de ne and

n BX (a) = f 2 SF (X; ) : kf k1=a + BX (a) = ff 2 BX (a) : f (x)

o 1 0g :

Note that if f 2 BX (a), then we can write

f = g a h for all x such that f (x) 6= 0; 1=a

+ wheret f (x) 6= 0; g 2 BX (1) is de ned by g(x) = jf (x)j , and h 2 BX (0) is de ned by h(x) = (f (x)= jf (x)j : In fact, a f (x) 1=a = f (x) if x 6= 0: (g(x))a h(x) = jf (x)j jf (x)j

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+ Also we have that if y 2 R; g 2 BX (a); h 2 BX (0); then g a+iy h 2 BX (a); indeed, Z Z 1=a 1=a 1=a jhj d g a+iy d = g a+iy h Z jgj d 1:

Now we stablish and proof the classical Theorem of J. Hadamard, known as Three lines of Hadamard.

Theorem 1 Let f (z) be a function de ned on C; where z = x + iy for all x 2 (a1 ; a2 ) y continuous in [a1 ; a2 ] : Suposse that yhere exists A; c such that Aecjyj ; x 2 [a1 ; a2 ] :

jf (x + iy)j

Let M1 ; M2 positive real numbers de ned by

M1 = sup jf (a1 + iy)j < 1 y

M2 = sup jf (a2 + iy)j < 1: y

Then we have a) If x 2 [a1 ; a2 ] , then jf (x + iy)j max fM1 ; M2 g b) Let M (a) = supy jf (a + iy)j ; then M (a) M (a1 ) M (a2 )1 where a = a1 + (1

)a2 :

Next result is an extension of n variables case.

n

Theorem 2 Let f(Xi ; i )gi=1 be a collection os measures spaces and B(f1 ; :::; fn ) de ned for fr 2 BXr (ar ) = Br (ar ); liner in each variable. De ne M (a1 ; ::; an ) = sup fjB(f1 ; ::; fn j : fr 2 Br (ar )g :

Then we have that the set of Rn ; where M is nite is convex and log M is a convex function.

3 Some introductory Notes The following result is known as Riesz-Thorin Theorem, and establish the conditions for the interpolation of a linear operator between Banach spaces.

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Theorem 3 Let (E; ) ; (F; ) measures spaces. Let T a linear operator de ned in the space SF (E; ) such that T (f ) is a measurable function in SF (F; ); and for 1 p1 ; p2 ; q1 ; q2 we have kT f kq1 M1 kf kp1 kT f kq2 M2 kf kp2 for all f 2 SF (E; ): I p < 1 then T extends by continuity to a linear operator, say T : Lp (E; ) ! Lq (F; ), with 1 1 1 1 = + y = + for all 2 (0; 1) p p1 p2 q q1 q2 and , kT k M1 M21 :

This result say to us that if a linear operator satisfy its continuity for simple functions between the espaces Lp1 (E; ) and Lq1 (F; ); and between the spaces Lp2 (E; ) and Lq2 (F; ); then also it will be continuous between the spaces Lp (E; ) y Lq (F; ) ; where p is a linear combination of p1 ; p2 and q is linear combination of q1 ; q2 : This is what this is all about the interpolation theory, to nd continuous extensions of linear operators. An example is described by the Theorem of Titmarsh-Hausdorff-Young.

Theorem 4 Let 1

2; then the Fourier transform Z 1 b F(f )(u) = f (u) = f (x)e ihu;xi dx (2 )n=2 Rn is a continuous linear operator from Lp (Rn ) in Lp (Rn ) with norm p

kFk

(2 )

(1=p 1=2)n

:

Another example.

Theorem 5 Let p > 1 and q; r such that 1 1 1 = + 1: r q p If f 2 Lp and g 2 Lq then the convolution Z h(x) = f (x y)g(y)dy Rn

there exists in a.e, h 2 Lr ; and

khkr

kf kp kgkq :

(Chapter head:)Interpolación de Operators. Basic Concepts 3

4 Basic Notions De nition 1 Let A0 ; A1 be Banach spaces inmersos in a Hausdorff topological vector i0

i1

space U ; that is, the inclusions A0 ,! U and A1 ,! U are continuous. Then A0 ; A1 are said to be compatibles spaces, and A = (A0 ; A1 ) is an interpolation couple. Basically we look at the spaces A0 \ A1 and A0 + A1 : Theorem 6 Let A = (A0 ; A1 ) be an interpolation couples. Then the space (A) = A0 \ A1 ; kak

where kak

is a Banach space.

(A)

(A)

= max kakA0 ; kakA1

Similarly we have. Theorem 7 Let A = (A0 ; A1 ) be an interpolation couple. Then the space de ned by (A) = A0 + A1 ; kak

where is a Banach space.

kak

(A)

(A)

= inf kakA0 + kakA1 : a = a0 + a1 ;

De nition 2 Let A = (A0 ; A1 ) be an interpolation couple. A Banach space A is said to be intermediate respect to A if i

i

(A) ,! A ,! (A): The following proposition tells us that given a linear operator acting between the spaces of two interpolation pairs, and whose constraint to each component space is continuous, then the operator remains continuous between the intersecting and sum spaces. Proposition 8 Let T : A0 + A1 ! B0 + B1 be a linear operator such that TA0 = T jA0 : A0 ! B0

are continuous, then are continuous.

TA1 = T jA1 : A1 ! B1 T : (A) ! (B) T : (A) ! (B) 4

De nition 3 Let A = (A0 ; A1 ) and B = (B0 ; B1 ) couples. If T satisfy the conditions of above theorem then T is said to be an admissible operator between the couples A; B:

Proposition 9 Let A = (A0 ; A1 ) and B = (B0 ; B1 ) couples. If T : (A) ! (B) is continuous then the restrictions TA0 y TA1 are continuous, in consequence T is an admissible operator. Next we concrete the de nition of interpolation space. De nition 4 Let A an intermediate space respect to the copulae A = (A0 ; A1 ): If for every admissible operator T : A ! A we have TA : A ! A is continuous, and A is an interpolation space between A0 and A1 . In a general form, if A is an intermediate space respect to A and B is an intermediate space respect to B; and for every admissible operator T : A ! B we have T : A ! B is continuous, then A; B are interpolation spaces respect to the couples A; B: . [1]

Carothers N.L. A Short Course on Banach Space Theory. Department of Mathematics and Statistics. Bowling Green University. 2002 [2] Diestel J. Sequences ans Series in Banach spaces. Springer-Verlag New york. 1983 [3] Dunford N.,Schwartz J. Linear Operators, I. Interscience Publishers, Inc.,New York. 1957 [4] Johnson W.B., Lindenstrauss, L. Handbook of the Geometry of Banach Spaces. Elsevier. Vol. 1. 2001 [5] Megginson Robert., An introduction to Banach space Theory. Springer-Verlag New York. 1998 [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematikund ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977). [7] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Ergebnisse der Mathematikund ihrer Grenzgebiete 97, Springer-Verlag, Berlin (1979). [8] Hewitt E, Stromberg K., Real and Abstract Analysis. Springer-Verlag New York. 1975. [9] Mukherjea,P.,Potoven, K. Real and Functional Analysis. 17. Nro. 4 (1971) 587. [10] Rudin W., Real and Complex Analysis. McGraw-Hill. Second edition. 1974.

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