A STABILITY VERSION OF HÖLDER’S INEQUALITY FOR 0 < p < 1

Hölder’s Inequality J.M. Aldaz vol. 9, iss. 2, art. 60, 2008

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A STABILITY VERSION OF HÖLDER’S INEQUALITY FOR 0 < p < 1

J. M. ALDAZ

Departamento de Matemáticas y Computación Universidad de La Rioja

26004 Logroño, La Rioja, Spain.

EMail:jesus.munarrizaldaz@unirioja.es

Received: 21 October, 2007

Accepted: 23 March, 2008

Communicated by: L. Losonczi 2000 AMS Sub. Class.: 26D15.

Key words: Hölder’s inequality, Reverse triangle inequality.

Abstract: We use a refinement of Hölder’s inequality for1< p < ∞to obtain the cor- responding refinement whenr ∈ (0,1). This in turn allows us to sharpen the reverse triangle inequality on the nonnegative functions inL^r, forr∈(0,1).

Acknowledgements: The author was partially supported by Grant MTM2006-13000-C03-03 of the D.G.I. of Spain.

Hölder’s Inequality J.M. Aldaz vol. 9, iss. 2, art. 60, 2008

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BykFk_t := R

|F|^t1/t

we do not mean to imply that this quantity is finite, nor do we assume thatt >0; in fact, in this note negative exponents are unavoidable.

It is well known that Hölder’s inequality can be extended to the range0< r <1, by an argument that essentially amounts to a clever rewriting of the case1< p <∞, cf. [2, pg. 191]. We denote the conjugate exponent ofrbys :=r/(r−1), and the conjugate exponent ofp byq := p/(p−1)(of course, to go from the range (0,1) to (1,∞) and viceversa, one sets r = 1/p). Hölder’s inequality for 0 < r < 1 tells us that if h and k are nonnegative functions in L^r and L^s respectively, then R hk ≥ R

h^r1/r R k^s1/s

. This entails that given functions h, w ≥ 0 inL^r, the reverse triangle inequality kh+wk_r ≥ khk_r +kwk_r holds. Nonnegativity is of course crucial.

Here we extend to the range (0,1) the following stability version of Hölder’s inequality, which appears in [1]:

Let 1 < p < ∞ and letq = p/(p−1)be its conjugate exponent. If f ∈ L^p, g ∈L^qare nonnegative functions withkfk_p,kgk_q >0, and1< p≤2, then

(1) kfk_pkgk_q 1− 1 p

f^p/2

kf^p/2k₂ − g^q/2 kg^q/2k₂

2

2

!

+

≤ kf gk₁ ≤ kfk_pkgk_q 1− 1 q

f^p/2

kf^p/2k₂ − g^q/2 kg^q/2k₂

2

2

! ,

while if2≤p < ∞, the terms1/pand1/qexchange their positions in the preceding inequalities.

Inequality (1) essentially states that kf gk₁ ≈ kfk_pkgk_q if and only if the angle between theL²vectorsf^p/2andg^q/2is small (in this sense it is a stability result). To see that on the cone of nonnegative functions (1) extends the parallelogram identity,

Hölder’s Inequality J.M. Aldaz vol. 9, iss. 2, art. 60, 2008

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rearrange the latter, for nonzerox andy in a real Hilbert space, as follows (cf. [1, formula (2.0.2)]):

(2) (x, y) = kxkkyk 1−1 2

x

kxk− y kyk

2! .

Writing (2) as a two sided inequality, adequately replacing some of the Hilbert space norms bypandqnorms, and the terms1/2by1/pand1/q, we see that (1) indeed generalizes (2). Note also thatkf^p/2k₂ =kfk^p/2p . Save in the case wherep=q = 2, the nonnegative functions f ∈ L^p andg ∈ L^q will in principle belong to different spaces, so to compare themL²is retained in (1) as the common measuring ground; to go fromL^p andL^q intoL² we use the Mazur map, which for nonnegative functions of norm 1 inL^p is simplyf 7→f^p/2(cf. [1] for more details).

Next we extend inequality (1) to the range 0 < r < 1, keeping the role of L². Unlike the case of Hölder’s inequality for1< p <∞, here we assume thathk ∈L¹. In exchange, we do not need to suppose a priori thath ∈L^r; this will be part of the conclusion.

Theorem 1. Let0 < r < 1, and let s = s/(s−1)be its conjugate exponent. If k ∈L^s,hk ∈L¹,khk_r,kkk_s >0, and1/2≤r <1, then

(3a) khkk₁ 1−r

h^1/2k^1/2

kh^1/2k^1/2k₂ − k^s/2 kk^s/2k₂

2

2

!¹_r

+

(3b) ≤ khk_rkkk_s ≤ khkk₁ 1−(1−r)

h^1/2k^1/2

kh^1/2k^1/2k₂ − k^s/2 kk^s/2k₂

2

2

!¹_r , while if0< r≤1/2, the termsrand1−rexchange their positions in the preceding inequalities.

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Proof. Suppose1/2≤ r < 1. Setp= 1/rand useq ands to denote the conjugate exponents ofpandrrespectively. Since1< p≤2, we can apply (1) to the functions f :=h^rk^randg =k^−r, which belong toL^pandL^q respectively:R

f^p =R

hk < ∞ andR

g^q =R

k^s<∞. Now the inequalities (3) immediately follow. If0< r≤1/2, then2≤p <∞, so just interchange the terms1/pand1/qin (1).

Note that from (3b), together with the hypothesiskhk_rkkk_s>0, we get

(4) 0<1−(1−r)

h^1/2k^1/2

kh^1/2k^1/2k₂ − k^s/2 kk^s/2k₂

2

2

for allr ∈ [1/2,1)(forr ∈ (1/2,1)this already follows from

x

kxk2 − _kyk^y

2

2 2

≤ 2, which is immediate from (2) whenx, y ≥ 0). The analogous result, withr instead of1−r, holds when0< r≤1/2. Thus, (3b) can be rewritten as

(5) khk_rkkk_s 1−(1−r)

h^1/2k^1/2

kh^1/2k^1/2k₂ − k^s/2 kk^s/2k₂

2

2

!⁻¹_r

≤ khkk₁

when 1/2 ≤ r < 1, while if 0 < r ≤ 1/2, the same formula holds but with r replacing1−r.

Now we are ready to obtain a sharpening of the reverse triangle inequality for nonnegative functions.

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Theorem 2. Let0< r <1. Given nonnegative functionsh, w∈L^rwithkhk_r,kwk_r >

0, setk := (h+w)^r−1/k(h+w)^r−1k_s. Then, if1/2≤r <1, we have

(6) kh+wk_r ≥ khk_r 1−(1−r)

h^1/2k^1/2

kh^1/2k^1/2k₂ −k^s/2

2

2

!−¹_r

+kwk_r 1−(1−r)

w^1/2k^1/2

kw^1/2k^1/2k₂ −k^s/2

2

2

!−¹_r

,

while if0< r≤1/2, the same inequality holds but with1−rreplaced byr.

Proof. Suppose1/2≤r <1, and note thatkis a unit vector inL^s. Hence, so isk^s/2 inL². By the nonnegativity ofhandwwe have

(7) kh+wkr=

Z (h+w)^r−1

k(h+w)^r−1k_s(h+w) = Z

hk+ Z

wk.

Since the left hand side of the preceding equality is finite, so are both integrals on the right hand side, and now the result follows by applying (4). If0< r ≤ 1/2,we argue in the same way, but withrreplacing1−rin (4).

Let us writeθ(x, y) :=

x

kxk − _kyk^y

. To conclude, we make some comments on the size ofθ(h^1/2k^1/2, k^s/2), which also apply toθ(w^1/2k^1/2, k^s/2). On a real Hilbert space,θ(x, y)is comparable to the angle between the vectorsxandy. In particular, θ(h^1/2k^1/2, k^s/2)is zero if and only if there exists at >0such thath=tw, in which casekh+wk_r =khk_r+kwk_r. Under any other circumstance, the inequality given by (6) is strictly better that the standard reverse triangle inequality.

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On the other hand, if we ask how small 1−(1−r)

h^1/2k^1/2

kh^1/2k^1/2k₂ −k^s/2

2

2

!¹_r

can be forr ∈ [1/2,1),the obvious bound θ(h^1/2k^1/2, k^s/2) ≤ √

2 is informative whenris close to 1, but useless ifr= 1/2. The analogous remark holds for

1−r

h^1/2k^1/2

kh^1/2k^1/2k₂ −k^s/2

2

2

!¹_r

when0 < r ≤ 1/2. However, nontrivial bounds also hold near1/2, since for every r∈(0,1),kh+wk_r≤2^1/r−1(khk_r+kwk_r)(see for instance Exercise 13.25 a), [2, pg. 199]). Thus, θ(h^1/2k^1/2, k^s/2)and θ(w^1/2k^1/2, k^s/2) cannot be simultaneously large. More precisely, if1/2≤r <1, then either

θ²(h^1/2k^1/2, k^s/2)≤ 1−2^r−1 1−r or

θ²(w^1/2k^1/2, k^s/2)≤ 1−2^r−1 1−r , while if0< r≤1/2, then either

θ²(h^1/2k^1/2, k^s/2)≤ 1−2^r−1 r or

θ²(w^1/2k^1/2, k^s/2)≤ 1−2^r−1

r .

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References

[1] J.M. ALDAZ, A stability version of Hölder’s inequality, J. Math. Anal.

Applics., to appear. doi:10.1016/j.jmaa.2008.01.104. [ONLINE: http://

arxiv.org/abs/0710.2307].

[2] E. HEWITTANDK. STROMBERG, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Second printing corrected, Springer-Verlag, New York-Berlin, 1969.