## Abstract

We present a Kato-type inequality for bounded domains , *n*≥2.

## 1. Introduction

Hardy's inequality is an important tool in the study of the spectral properties of partial differential equations. This inequality states that for a function , *n*≥3The corresponding ‘first-order’ analogue of the Hardy inequality was established by Kato and plays an important role in the study of relativistic quantum mechanical systems. Specifically, Kato inequality for (Kato 1966, p. 307) states that for , *n*≥2(1.1)where .

The analogue of Hardy's inequality for a bounded domain , *n*≥2 with a Lipschitz boundary iswhere (Edmunds & Evans 2004, p. 212; see also Davies 1984, 1999; Lewis 1988 for references and details).

The purpose of this article is to establish the Kato-type inequality for a bounded domain . Since is a non-local operator, there are three possibilities to define the r.h.s. of (1.1) in the case of . One possibility is to use the r.h.s. of (1.1) but restrict ourselves only to functions with compact support inside *Ω*. Another possibility is based on the fact that (see Lieb & Loss 1997)So we can define the analogue of the r.h.s. of (1.1) for *Ω* as(1.2)The third possibility is to consider the square root of the internal Dirichlet–Laplacian operator in the domain *Ω*.

In this article, we consider the first two definitions, since they are more interesting for relativistic quantum mechanics (localization of kinetic energy). The case of a Kato-type inequality for the square root of the internal Dirichlet–Laplacian in fact follows for nice domains from Hardy's inequality sincefor operators *A*, *B*>0 (see Birman & Solomjak 1987, theorem 2, p. 232).

Let us briefly describe the content of the paper. In §2 for functions *f* such that supp *f*⊂*Ω*_{1} for some , we show that(1.3)where is the distance from *x* to ∂*Ω*_{1}, i.e. . Later, we obtain the inequality(1.4)Initially, we prove (1.4) for radial functions (proposition 3.1) and then for all (theorem 4.2). Though we give (1.4) for some restricted class of bounded domains *Ω*, we expect that theorem 4.2 is true for Lipschitz domains. But we will not discuss this in the current article.

## 2. Kato-type inequality for functions with compact support

*Let Ω*_{1} *be a convex bounded domain, such that* *for some domain* *, n*≥2. *We suppose that* *and* supp *f*⊂*Ω*_{1}. *Then for some constant* *, the inequality* *(1.3)* *holds*.

In view of the inequality , without loss of generality we may assume that *f*(*x*) is a real-valued function. Next we apply the Lieb–Yau trick (see Lieb & Yau 1988) to get inequality, which is a basic tool in the proofs of theorems 1 and 2.

*Let* , *, where* *,* . *We assume that* *,*(2.1)*and*(2.2)*for any* *and some constant M*>0. *Then*(2.3)*for any* .

On expanding brackets in the l.h.s. of (2.3), we get(2.4)Applying the Cauchy–Schwarz inequality and using (2.1), (2.2) gives(2.5)The inequality (2.3) follows from (2.4) and (2.5). ▪

*Let us suppose that K*(*x*, *y*) *and f*(*x*) *satisfy the conditions of* lemma 2.2 *and* supp *f*⊂*Ω*_{1} *for some Ω*_{1}⊂*B*. *Then*(2.6)

An application of lemma 2.2 withgiveswherePassing to the limit *ϵ*→0 completes the proof ▪

In view of corollary 2.3, it suffices to prove that(2.7)for any *x*∈*Ω*_{1} and some . The convexity of *Ω*_{1} implies that for any , there exists an (*n*−1)-dimensional plane *π*_{z} in , such that *z*∈*π*_{z} and . For any *x*∈*Ω*_{1}, we take *x*_{0}=*x*_{0}(*x*), such that . Let be the half of with boundary which does not contain *Ω*_{1}. Clearly,For any *z*∈∂*Ω*_{1}, we putwhere *B*_{s}(*z*) is a ball with centre at *z* and radius *s*. From , we conclude that . Consequently, we have(2.8)Let us choose Cartesian coordinates (*y*_{1}, …, *y*_{n}) in with centre at *x*_{0} and axes such that . Then, and . Making the change of variables in (2.8) giveswhereSince *Ω*_{1} is bounded, it follows that for some constant ,for all *x*∈*Ω*_{1}. Therefore, , and soCombining the above estimates we obtain (2.7) with

## 3. Lower estimate for the integral representation (1.2). Case of radial functions

*We suppose that* *and* supp *f*⊂[0,1). *Then for some absolute constant c*_{4}<0 *we get*(3.1)*where* *, n*≥2*, is a ball with centre at the origin and radius R*=1.

Let us briefly outline the content of this section. The proof of proposition 3.1 is preceded by proofs of some auxiliary results. In lemma 3.2, we show that integral on the l.h.s. of (3.1) is equivalent (up to multiplication by a constant) to one-dimensional integral (3.3). In order to estimate (3.3) from below, we apply the Lieb–Yau trick (lemma 2.2) with test functionfor *ω*∈(0,1/4) and then integrate in *ω* both sides of the obtained inequality. Lemmas 3.3 and 3.4 are needed to get a lower estimate for the term on the r.h.s. of (2.3). At the end of this section we piece together all the lemmas to establish proposition 3.1.

*Under the conditions of* *proposition 3.1**, for some constant* *, we have*(3.2)*where*(3.3)

Let us change the coordinates *x*, *y* in the integral in (3.2) to spherical coordinates , , whereWe choose the direction of the axes in *y*-space, such that the direction of axis *ϕ*_{1}=*π*/2 coincides with the vector * x*, i.e. the angle between

*x*and

*y*is equal to

*ϕ*

_{1}, and soRecall that the absolute value of the Jacobian of this change of variables is equal toIt follows thatwhereDenote by

*J*(

*k*) the Euler-type integral(3.4)Then usingwe obtain(3.5)From (3.4) and the elementary inequality for , we find thatSinceit follows thatorAn application of the elementary inequalityimplies thatHence,andSubstituting these estimates into (3.5) we obtainTaking we arrive at (3.2). ▪

*Let ϕ*(.) *be a positive increasing function and*(3.6)*Then for any r*∈(0,1)(3.7)*where μ*=(1−*r*)^{−1}*, n*≥2.

Step 1. We havewhereSince *h*(*s*) is increasing, then *I*<0 for *s*<*r*, and soThus,(3.8)

Step 2. Let us make the change of the variables(3.9)in the integral on the r.h.s. of (3.8). Elementary calculations give(3.10)(3.11)(3.12)and(3.13)Consequently, using (3.6) and (3.9)–(3.13) we get(3.14)Substituting *μ*=(1−*r*)^{−1} into (3.14) and making the change we arrive at(3.15)Combining (3.8) and (3.15) completes the proof. ▪

*There exist absolute constants c*_{7}>0 *and κ*>0 *such that for any* 0<*ω*<1/4 *and μ*>1(3.16)*where*(3.17)

Substituting (3.17) into the l.h.s of (3.16) and usingwe get(3.18)withand(3.19)

Let us estimate *A*(*μ*) and *B*(*μ*). Since *μ*^{−1}≤1 and for *u*<1, it follows that(3.20)For any *R*∈(1,+∞) we putLet *γ*_{R} be oriented anticlockwise and segments *γ*_{R}^{1}, *γ*_{R}^{2} oriented from left to right. Due to the fact that for any *ε*<1an application of Cauchy's theorem gives(3.21)Combining (3.20) and (3.21) we havewhereUsing the elementary inequalitywe get(3.22)Making the change of the variables *t*=*z*^{−1} we getand so(3.23)Moreover,(3.24)Applying (3.23) and (3.24) to Taylor's expansion of *ψ*(*ω*)we see that *ψ*(*ω*)*ω*^{−2} increases for *ω*>0, and sofor . Therefore, by (3.22) we have(3.25)where(3.26)

We proceed with *B*(*μ*). According to (3.19)Sincefor all and all *u*>0, it follows that(3.27)where(3.28)Combining (3.18) with (3.25) and (3.27) we havefor all *μ*>1 and . Taking *κ*=100 and using (3.26) and (3.28) we obtain (3.16) with ▪

*One has**for all μ*≥1 *and some absolute constant c*_{10}<0.

After elementary calculations we get(3.29)whereSinceand so for *μ*≥*e*, it follows thatfor *μ*≥*e*. On the other hand, since we getfor . Note that, by (3.29), . Takingwe complete the proof. ▪

Using the left inequality in (3.2) and the fact thatwe find(3.30)An application of lemma 2.2 with *m*=1, *B*=[0,1],for any positive function *h*(.) gives(3.31)whereLet *h*(.) and *ϕ*(.) be defined by (3.6) and (3.17), respectively. An application of lemmas 3.3 and 3.4 yields(3.32)where, as before, *μ*=(1−*r*)^{−1}. Combining (3.30)–(3.32) we obtain(3.33)Integrating both sides of (3.33) in *ω* and using lemma 3.5 we haveWe putwhere(3.34)Thus,(3.35)Let us substitute (3.34) into (3.35) and change the variables on the r.h.s. of (3.35) to Cartesian coordinates. Then (3.1) follows. ▪

## 4. Lower estimate for the integral representation (1.2). General case

Here we generalize inequality (3.1) to the case of non-radial functions. Furthermore, we obtain the analogue of (3.1) for certain class of domains *Ω*.

*Let* *, n*≥2 *such that* supp *f*⊂*B*_{1}(0). *Then*(4.1)*where c*_{4}<0 *is the absolute constant from* *proposition 3.1*.

In view of the inequality , without loss of generality, we may assume that *f*(*x*) is real-valued.

For any *e*∈*S*^{n} (*S*^{n} is the unit sphere in ), we puti.e. *T*^{e} is rotation in around *e* through angle *π*. Obviously,(4.2)for all .

Making the change of the variables , and using (4.2) and we have(4.3)According to the Cauchy–Schwarz inequality,and sowhereUsing this and integrating (4.3) over all *e*∈*S*^{n} gives(4.4)Note that *ψ*(*x*) depends only on |*x*|. Hence, we can apply proposition 3.1. It follows thatSincefor any *e*∈*S*^{n}, it follows that(4.5)Combining (4.4) and (4.5) we complete the proof. ▪

*Let Ω be a domain in* *, n*≥2. *We assume that there exist diffeomorphism**and some constant* *, such that for all* (4.6)*Then for some constant* *and any* *we have**where* .

Step 1. As before, we put for any domain *D*. Given any *x*∈*Ω* we put and take *u*_{0}, *x*_{1} are such thati.e. *u*_{0}, *x*_{1} deliver minima to corresponding functionals. Applying the right inequality in (4.6) we getwhere *x*_{0}=*ϕ*(*u*_{0}). By choice of *x*_{1} we have , and so(4.7)Along these lines using the left inequality in (4.6) we get(4.8)

Step 2. Making the change of variables *x*=*ϕ*(*u*), *y*=*ϕ*(*v*), using (4.6) and letting we get(4.9)An application of lemma 4.1 yieldsUsing (4.7) we find (recall ) that(4.10)for some . We apply (4.8) and (4.10), and then again make the change of variables *x*=*ϕ*(*u*) and use (4.6) to get(4.11)Combining (4.9) and (4.11), and letting we complete the proof. ▪

## Acknowledgments

We are grateful for financial support through EPSRC grant RCMT090. We also thank Prof. W. D. Evans for valuable discussions.

## Footnotes

- Received March 8, 2005.
- Accepted October 31, 2005.

- © 2006 The Royal Society