Boundedness and compactness of positive integral operators on cones of homogeneous groups

Positivity 13 (2009), 497–518c© 2008 Birkhauser Verlag Basel/Switzerland1385-1292/030497-22, published online October 28, 2008DOI 10.1007/s11117-008-2217-8 Positivity

Boundedness and compactnessof positive integral operators on conesof homogeneous groups

Usman Ashraf, Muhammad Asif and Alexander Meskhi

Abstract. Necessary and sufficient conditions on a weight function v guaran-teeing the boundedness/compactness of integral operators with positive ker-nels defined on cones of homogeneous groups from Lp to Lq

v are established,where 1 < p, q < ∞ or 0 < q ≤ 1 < p < ∞. Behavior of singular numbers forthese operators is also studied.

Mathematics Subject Classification (2000). Primary 26A33, 42B25; Secondary43A15, 46B50, 47B10, 47B34.

Keywords. Operators with positive kernels, potentials, homogeneous groups,trace inequality, weights, singular numbers of kernel operators.

1. Introduction

In this paper boundedness/compactness criteria from Lp(E) to Lqv(E) are estab-

lished for the operator

Kf(x) =∫

Er(x)

k(x, y)f(y)dy, x ∈ E, (1)

with positive kernel k, where 1 < p, q < ∞ or 0 < q ≤ 1 < p < ∞, Er(x) and Eare certain cones in homogeneous groups and k satisfies the conditions which inone-dimensional case are similar to those of [20].

A full characterization of pair of weights (v, w) governing the boundednessof integral operators with positive kernels from Lp

w to Lqv, 1 0,

The work was partially supported by the INTAS grant No. 05-1000008-8157 and the GeorgianNational Foundation Grant No. GNSF/ST06/3-010.

498 U. Ashraf et al. Positivity

from Lp(R+) to Lqv(R+), 1 < p, q < ∞, 1/p < α < 1 have been obtained in

[19] and [28]. This result was generalized in [20] (see also [6], Ch. 2) for integraloperators with positive kernels involving fractional integrals.

Two-weight Lpw − Lq

v criteria for the Hardy operators have been derived byB. Muckenhaupt [22] for p = q, J. Bradley [3], V. Kokilashvili [14] and V. Maz’ya[18] for p ≤ q. Necessary and sufficient conditions guaranteeing the boundednessof Rα from Lp

w(R+) to Lqv(R+), where 1 1 were derived in

[17], [31] (see also [16,26,32]).The two-weight problem for higher-dimensional Hardy-type operators defined

on cones in Rn involving Oinarov [25] kernels was studied in [11,34] (see also [29],

for Hardy-type transforms on star-shaped regions).In the present paper we present also two-sided estimates of Schatten-von

Neumann norms for the operator with positive kernel

Kuf(x) = u(x)∫

Er(x)

k(x, y)f(y)dy, x ∈ E, (2)

where u is a measurable function on E.Constants (often different constants in the same series of inequalities) will

generally be denoted by c.

2. Preliminaries

A homogeneous group G is a simply connected nilpotent Lie group G on whichLie algebra g is given one-parameter group of transformations δt = exp(A log t),t > 0, where A is a diagonalized linear operator on G with positive eigenvalues.For G the mappings exp o δt o exp−1, t > 0, are automorphisms on G, which willbe denoted by δt. The number Q = trA is called homogeneous dimension of G.The symbol e will stand for the neutral element in G.

It is possible to equip G with a homogeneous norm r : G → [0,∞) which iscontinuous function on G and smooth on G\e satisfying the following conditions:(i) r(x) = r(x−1) for every x ∈ G;

(ii) r(δtx) = t · r(x) for every x ∈ G and t > 0;

(iii) r(x) = 0 if and only if x = e;

(iv) there exists co ≥ 1 such that

r(xy) ≤ co(r(x) + r(y)), x, y ∈ G.

A ball in G, centered at x and with radius ρ , is defined as

B(x, ρ) = y ∈ G; r(xy−1) < ρ.

It can be observed that δρB(e, 1) = B(e, ρ).

Vol. 13 (2009) Boundedness and compactness 499

Let us fix a Haar measure | · | in G so that |B(e, 1)| = 1. Then |δtE| = tQ|E|,in particular, |B(x, s)| = sQ for x ∈ G, s > 0.

Examples of homogeneous groups are: Euclidean n-dimensional spaces, Heisen-berg group, upper triangular groups etc (see [8] for the definition and basic prop-erties of homogeneous groups).

Let S be the unit sphere in G i.e. S = x ∈ G : r(x) = 1, and let A be ameasurable subset of S with positive measure. We denote by E a measurable conein G defined by

E := x ∈ G : x = δsx, 0 < s < ∞, x ∈ A.

We denote

Et := y ∈ E : r(y) < t.

Now we define the kernel operator by (1), where k(x, y) is non-negative functionon (x, y) ∈ E × E : r(y) < r(x).

In the sequel we shall also use the notation:

Sx := Er(x)/2c0 , Fx := Er(x)\Sx,

where the constant c0 is from the triangle inequality for r.

Let Ω be a measurable subset of G. A locally integrable almost everywherepositive function w on Ω is called a weight. Denote by Lp

w(Ω) (0 < p < ∞) theweighted Lebesgue space, which is the space of all measurable functions f : Ω → C

with finite norm (quasi-norm if 0 < p < 1)

‖f‖Lpw(Ω) =

⎛⎝∫

Ω

|f(x)|pw(x)dx

⎞⎠

1/p

.

Proposition A. Let G be a homogeneous group. There is a (unique) Radon measureσ on S such that for all u ∈ L1(G),

∫

G

u(x)dx =

∞∫

0

∫

S

u(δsy)sQ−1dσ(y)ds.

For the details see e.g. [8], p. 14.

Theorem A. ([6] (Ch. 1)). Let (X,µ) be a measure space and let φ : X → R be aµ-measureable non-negative function. Suppose that 1 < p ≤ q < ∞ and v and ware weight functions on X. Then the operator

(Hφf) (x) =∫

y∈X: φ(y)<φ(x)

f(y)dµ

is bounded from Lpw(X) to Lq

v(X) if and only if

supt>0

⎛⎜⎝

∫

x:φ(x)>t

v(x)dµ

⎞⎟⎠

1/q⎛⎜⎝

∫

x:φ(x)<t

w1−p′(x)dµ

⎞⎟⎠

1/p′

< ∞, p′ = p/(p − 1).

Taking X = E, dµ = dx, v = u, w = 1, φ(x) = r(x) in the previous statementand observing that

supt>0

⎛⎜⎝∫

E\Et

u(x)dx

⎞⎟⎠

1/q

tQ/p′ ≈ supj∈Z

⎛⎜⎝

∫

E2j+1\E2j

u(x)dx

⎞⎟⎠

1/q

2jQ/p′,

we have the next statement.

Lemma 2.1. Let 1 < p ≤ q < ∞. Then the operator

(Hf) (x) =∫

Er(x)

f(y)dy

is bounded from Lp(E) to Lqu(E) if and only if

A := supj∈Z

Aj := supj∈Z

⎛⎜⎝

∫

E2j+1\E2j

u(x)dx

⎞⎟⎠

1/q

2jQ/p′< ∞.

Lemma 2.2 ([12] (Ch. XI)). Let (X,µ) and (Y, ν) be σ-finite measure spaces andlet 1 < p, q < ∞. Suppose that for positive function a : X × Y → R, we have∥∥∥‖a(x, y)‖

Lp′ν (Y )

∥∥∥Lq

µ(X)< ∞.

Then the operator

Af(x) =∫

Y

a(x, y)f(y)dν(y)

is compact from Lpν(Y ) to Lq

µ(X).

We shall need also the following statement (see [18] for 1 ≤ q 1. Then the inequality⎛⎝

∞∫

0

v(x)

⎛⎝

x∫

0

f(t)dt

⎞⎠

q

dx

⎞⎠

1/q

≤ c

⎛⎝

∞∫

0

(f(x))pw(x)dx

⎞⎠

1/p

, f ≥ 0,

holds if and only if⎛⎜⎜⎝

∞∫

0

⎡⎢⎣⎛⎝

∞∫

t

v(x)dx

⎞⎠⎛⎝

t∫

0

w1−p′(x)dx

⎞⎠

q−1⎤⎥⎦

p/(p−q)

w1−p′(t)dt

⎞⎟⎟⎠

(p−q)/(pq)

< ∞.

The next lemma is known as the Ando’s theorem (see [1] and [15], Sections5.3 and 5.4) which in our case is formulated for the set E.

Lemma 2.4. Let 0 < q < ∞, 1 < p < ∞ and q < p. Suppose that v and w areHaar-measurable almost everywhere positive functions on E. If the operator

AEf(x) =∫

E

a(x, y)f(y)dy, x ∈ E,

is bounded from Lpw(E) to Lq

v(E), then AE is also compact.

Let H be a separable Hilbert space and let σ∞(H) be the class of all compactoperators T : H → H, which form an ideal in the normed algebra B of all boundedlinear operators on H. To construct a Schatten-von Neumann ideal σp(H) (0 <p ≤ ∞) in σ∞(H), the sequence of singular numbers sj(T ) ≡ λj(|T |) is used,where the eigenvalues λj(|T |) (|T | ≡ (T ∗T )1/2) are non-negative and are repeatedaccording to their multiplicities and arranged in decreasing order. A Schatten-vonNeumann quasinorm (norm if 1 ≤ p ≤ ∞) is defined as follows:

‖T‖σp(H) ≡⎛⎝∑

j

spj (T )

⎞⎠

1/p

, 0 < p < ∞,

with the usual modification if p = ∞. Thus we have ‖T‖σ∞(H) = ‖T‖ and ‖T‖σ2(H)

is the Hilbert-Schmidt norm given by the formula

‖T‖σ2(H) =(∫ ∫

|a(x, y)|2dxdy

)1/2

for the integral operator

Af(x) =∫

a(x, y)f(y)dy.

We refer, for example, to [4,13] for more information concerningSchatten-von Neumann ideals.

Estimates of Schatten-von Neumann ideal norms for the Hardy-type trans-forms and the Riemann-Liouville operators were established in [5,7,23,24]. Thesame problem for operators with positive kernels involving one-sided potentialswas studied in [21].

Let ρ be a weight function on E. We denote by lp(L2

ρ(E))

the set of allmeasurable functions g : E → R for which

‖g‖lp(L2ρ(E)) =

⎛⎜⎜⎝∑n∈Z

⎛⎜⎝

∫

E2n+1\E2n

|g(x)|2ρ(x)dx

⎞⎟⎠

p/2⎞⎟⎟⎠

1/p

< ∞.

The next statement deals with interpolation result which is a consequence ofmore general statements from [33] (see also [2]).

Proposition B. Let 1 ≤ p0, p1 ≤ ∞, 0 ≤ θ ≤ 1, 1p = 1−θ

p0+ θ

p1. Suppose that

ρ is a weight function on E. If A is a bounded operator from lpi(L2

ρ(E))

intoσpi

(L2

ρ(E)), where i = 0, 1, then it is also bounded from lp

(L2

ρ(E))

to σp

(L2

ρ(E)).

Moreover,

‖A‖lp(L2ρ(E))→σp(L2(E)) ≤ ‖T‖1−θ

lp0 (L2ρ(E))→σp0 (L2(E))‖T‖θ

lp1 (L2ρ(E))→σp1 (L2(E)).

The following statement is obvious for p = ∞; when 1 ≤ p < ∞ it followsfrom Lemma 2.11.12 of [27].

Proposition C. Let 1 ≤ p ≤ ∞ and let fk, gk be orthonormal systems in aHilbert space H. If T ∈ σp(H), then

‖T‖σp(H) ≥(∑

n

|〈Tfn, gn〉|p)1/p

.

3. Boundedness and compactness criteria

In this section we establish boundedness/compactness criteria for K acting fromLp(E) to Lq

v(E), 1 < p ≤ q < ∞. First we need some definitions regarding thekernel k.

Definition 3.1. Let k be a positive function on (x, y) ∈ E × E : r(y) < r(x) andlet 1 < λ < ∞. We say that k ∈ Vλ, if there exist positive constants c1, c2 and c3

such that

(i) k(x, y) ≤ c1k(x, δ1/(2co)x) (3)

for all x, y ∈ E with r(y) < r(x)/(2co);

k(x, y) ≥ c2k(x, δ1/(2co)x) (3′)

for all x, y ∈ E with r(x)/(2co) < r(y) < r(x);

(ii)∫

Fx

kλ′(x, y)dy ≤ c3r

Q(x)kλ′(x, δ1/(2co)x), λ′ = λ/(λ − 1), (4)

for all x ∈ E.

Example. Let G = Rn, r(xy−1) = |x − y|, δtx = tx, x, y ∈ R

n. If k(x, y) = |x −y|λ−n, then k ∈ Vλ, where n/λ < α ≤ n. Indeed, it is easy to see that if y ∈ Sx, then|x| ≤ |x−y|+|y| ≤ |x−y|+|x|/2. Hence |x|/2 = k(x, x/2) ≤ k(x, y). Consequently,(3) holds. Further, it is easy to see that (3′) is also satisfied. Moreover, we have

∫

Fx

|x − y|(α−n)λ′dy =

∞∫

0

∣∣∣y ∈ Fx : |x − y|(α−n)λ′> ε

∣∣∣ dε

≤|x|(α−n)λ′∫

0

(· · · ) +

∞∫

|x|(α−n)λ′

(· · · ) := I1 + I2.

For I1 we have

I1 =

|x|(α−n)λ′∫

0

|B(0, |x|)| dα = c|x|(α−n)λ′+n,

while for I2 we observe that

I2 ≤∞∫

|x|(α−n)λ′

∣∣∣y : |y| < |x|, |x − y| ≤ ε1/(α−n)λ′∣∣∣ dε

≤ c

∞∫

|x|(α−n)λ′

εn/(α−n)λ′dε = cα,n

(|x|(α−n)λ′)1+n/(α−n)λ′

= cα,n|x|n+(α−n)λ′.

Example. It is easy to see that if the following two conditions are satisfied for k:

(i) k(δtx, δτ y) ≤ c1k(δtx, δsz)

for all t, τ, s, x, y, z with 0 < τ < s < t; x, y, z ∈ A;

(ii)t∫

t/(2co)

kλ′(δtx, δτ y)τQ−1dτ ≤ c2t

Q · kλ′((δtx, δt/(2co)x), t > 0, x ∈ A,

then k ∈ Vλ.

Example. Let k(x, y) = k(r(x), r(y)) be a radial kernel. It is easy to check that ifthere exist positive constants c1 and c2 such that(i) k(s, l) ≤ c1k(s, t), 0 < l < t < s,

(ii)t∫

t/(2c0)

kλ′(t, s)sQ−1ds ≤ c2t

Qkλ′(t, t/(2c0)), 1 < λ < ∞, then k ∈ Vλ.

Theorem 3.2. Let 1 < p ≤ q < ∞ and let v be a weight on E. Suppose that k ∈ Vp.Then K is bounded from Lp(E) to Lq

v(E) if and only if

B ≡ supj∈Z

B(j) ≡ supj∈Z

⎛⎜⎝

∫

E2j+1\E2j

v(x)kq(x, δ1/(2co)x)dx

⎞⎟⎠

1/q

(2j)Q/p′

< ∞.

Proof. Sufficiency. Let f ≥ 0 on E. We have

‖Kf‖qLq

v(E)≤c

⎡⎣∫

E

v(x)

⎛⎝∫

Sx

k(x, y)f(y)dy

⎞⎠

q

dx +∫

E

v(x)

⎛⎝∫

Fx

k(x, y)f(y)dy

⎞⎠

q

dx

⎤⎦

= c(I1 + I2).

By condition (3) and Lemma 2.1 we find that

I1 ≤ c

∫

E

v(x)kq(x, δ1/(2co)x)

⎛⎜⎝∫

Er(x)

f(y)dy

⎞⎟⎠

q

dx ≤ c

⎛⎝∫

E

fp(x)dx

⎞⎠

q/p

,

while Holder’s inequality and condition (4) yield

I2 ≤∫

E

v(x)

⎛⎝∫

Fx

kp′(x, y)dy

⎞⎠

q/p′ ⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p

dx

≤ c

∫

E

v(x)kq(x, δ1/(2co))rQq/p′(x)

⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p

dx

≤ c∑j∈Z

∫

E2j+1\E2j

v(x)kq(x, δ1/(2co))rQq/p′(x)

⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p

dx

≤ c∑j∈Z

⎛⎜⎝

∫

E2j+1\E2j

v(x)kq(x, δ1/(2co)x)rQq/p′(x)dx

⎞⎟⎠

×

⎛⎜⎝

∫

E2j+1\E2j−1/co

fp(y)dy

⎞⎟⎠

q/p

≤ cBq‖f‖qLp(E).

Sufficiency has been proved.

Necessity. To prove necessity we take the functions fj(x) = χE2j+1 (x). Thensimple calculations show that ‖fj‖Lp(E) = c2jQ/p and

‖Kfj(x)‖qLq

v(E)≥

∫

E2j+1\E2j

v(x)

⎛⎝∫

Fx

fj(y)k(x, y)dy

⎞⎠

q

dx

≥ c

⎛⎜⎝

∫

E2j+1\E2j


⎞⎟⎠ 2jQq.

Finally, due to the boundedness of K we have the desired result. Theorem 3.3. Let 1 < p ≤ q < ∞. Suppose that k ∈ Vp. Then K is compact fromLp(E) to Lq

v(E) if and only if(i) B < ∞; (5)

(ii) limj→−∞

B(j) = limj→+∞

B(j) = 0. (6)

Proof. Sufficiency. Let 0 < a < b < ∞ and represent Kf as follows:

Kf = χEaK(fχEa

) + χEb\EaK(fχEb

)+ χE\Eb

K(fχEb/2c0) + χE\Eb

K(fχE\Eb/2co)

≡ P1f + P2f + P3f + P4f.

For P2 we have

P2f(x) =∫

E

k∗(x, y)f(y)dy,

where k∗(x, y) = χEb\Ea(x)χEr(x)(y)k(x, y). Further, observe that

S :=∫

E

⎛⎝∫

E

(k∗(x, y))p′dy

⎞⎠

q/p′

v(x)dx =∫

Eb\Ea

⎛⎜⎝∫

Er(x)

kp′(x, y)dy

⎞⎟⎠

q/p′

v(x)dx

≤ c

∫

Eb\Ea

⎛⎝∫

Sx

kp′(x, y)dy

⎞⎠

q/p′

v(x)dx + c

∫

Eb\Ea

⎛⎝∫

Fx

kp′(x, y)dy

⎞⎠

q/p′

v(x)dx

:= S1 + S2.

Taking into account the condition k ∈ Vp we have

Si ≤ c

∫

Eb\Ea

kq(x, δ1/(2co)x)rqQ/p′(x)v(x)dx

≤ cbq/p′∫

Eb\Ea

kq(x, δ1/(2co)x)v(x)dx < ∞, i = 1, 2.

Finally we have S < ∞ and consequently, by Lemma 2.2 we conclude that P2 iscompact for every a and b. Analogously we can obtain that P3 is compact.

Further, by the arguments used in the proof of Theorem 3.2 we have

‖P1‖ ≤ cB(a) ≡ c supt≤a

⎛⎜⎝∫

Ea\Et


⎞⎟⎠

1/q

tQ/p′;

‖P4‖ ≤ cB(b) ≡ c supt≥b

⎛⎜⎝∫

E\Et

v(x)kq(x, δ1/(2cox)dx

⎞⎟⎠

1/q

(tQ − bQ

)1/p′.

Therefore

‖K − P2 − P3‖ ≤ ‖P1‖ + ‖P4‖ ≤ c(B(a) + B(b)

).

It remains to show that condition (6) implies

lima→0

B(a) = limb→∞

B(b) = 0.

Let a > 0. Then a ∈ [2m, 2m+1) for some integer m. Consequently B(a) ≤ B(2m+1).Further, for 0 < t < 2m+1, there exists j ∈ Z, j ≤ m, such that t ∈ [2j , 2j+1).Hence

tqQ/p′∫

E2m+1\Et

v(x)kq(x, δ1/(2co)x))dx ≤

≤ 2(j+1)qQ/p′m∑

k=j

∫

E2k+1\E2k


≤ cBq(k)2jqQ/p′∞∑

k=j

2−kqQ/p′= cBq(k).

Taking into account this estimate we shall find that

B(2m) ≤ supk≤m

Bq(k) ≡ S(m).

If a → 0, then m → −∞. Consequently S(m) → 0, which implies that B(2m) → 0as a → 0. Finally we have that B(a) → 0 as a → 0.


Now we take arbitrary b > 0. Then b ∈ [2m, 2m+1) for some m ∈ Z. HenceB(b) ≤ B(2m). Further, for t > 2m, there exists k ≥ m, k ∈ Z such that t ∈[2k, 2k+1). For such a t we have

⎛⎜⎝∫

E\Et


⎞⎟⎠(tQ − 2mQ

)q/p′≤

≤

⎛⎜⎝

∫

E\E2k


⎞⎟⎠(2(k+1)Q − 2mQ

)q/p′

≤ c2kqQ/p′∞∑

j=k

⎛⎜⎝

∫

E2j+1\E2j

v(x)k(x, δ1/(2co)x)q(x)dx

⎞⎟⎠

≤ cBq(k)2kqQ/p′∞∑

j=k

2−jqQ/p′ ≤ cBq(k).

Consequently, B(2m) ≤ supk≥m

B(k). From the last inequality we conclude that the

condition limk→+∞

B(k) = 0 implies limb→∞

B(b) = 0.

Necessity. Let K be compact. Since (5) is obvious we have to prove (6). Letj ∈ Z and put fj(y) = χE2j+1\E2j−1/c0

(y)2−jQ/p. Then for φ ∈ Lp′(E), we have

∣∣∣∣∣∣∫

E

fj(y)φ(y)dy

∣∣∣∣∣∣ ≤

⎛⎜⎝

∫

E2j+1\E2j−1/c0

|fj(y)|pdy

⎞⎟⎠

1/p⎛⎜⎝

∫

E2j+1\E2j−1/c0

|φ(y)|p′dy

⎞⎟⎠

1/p′

=

⎛⎜⎝

∫

E2j+1\E2j−1/c0

|φ(y)|p′dy

⎞⎟⎠

1p′

→ 0

as j → −∞ or j → +∞. On the other hand,

‖Kfj‖Lqv(E) ≥

⎛⎜⎝

∫

E2j+1\E2j

(Kfj(x))qv(x)dx

⎞⎟⎠

1/q

≥ c

⎡⎢⎣

∫

E2j+1\E2j

⎛⎝∫

Fx

k(x, y)fj(y)dy

⎞⎠

q

v(x)dx

⎤⎥⎦

1/q

≥ c

⎡⎢⎣

∫

E2j+1\E2j

kq(x, δ1/(2co)x)

⎛⎝∫

Fx

fj(y)dy

⎞⎠

q

v(x)dx

⎤⎥⎦

1/q

≥ c

⎛⎜⎝

∫

E2j+1\E2j

kq(x, δ1/(2co)x)v(x)dx

⎞⎟⎠

1/q

2jQ/p′= cB(j).

As a compact operator maps a weakly convergent sequence into a strongly con-vergent one, we conclude that (6) holds.

Now we consider the case when q < p.

Theorem 3.4. Let 0 < q 1. Suppose that k ∈ Vp. Then thefollowing conditions are equivalent:

(i) K is bounded from Lp(E) to Lqv(E);

(ii) K is compact from Lp(E) to Lqv(E);

(iii)

D ≡

⎛⎜⎜⎝∫

E

⎛⎜⎝

∫

E\Er(x)

kq(y, δ1/(2co)y)v(y)dy

⎞⎟⎠

p/(p−q)

r(x)Qp(q−1)/(p−q)dx

⎞⎟⎟⎠

(p−q)/pq

< ∞.

Proof. First we show that (iii) ⇒ (i). Suppose that f ≥ 0. Keeping the notationof sufficiency of the proof of Theorem 3.2, using Proposition A and condition (3)we have

I1 ≤ c

∫

E

v(x)kq(x, δ1/(2c0)x)

⎛⎝∫

Sx

f(y)dy

⎞⎠

q

dx

= c

∞∫

0

V (t)

⎛⎜⎝

t/(2c0)∫

0

F (τ)dτ

⎞⎟⎠

q

dt,

where

V (t) := tQ−1

∫

A

v(δtx)kq(δtx, δt/(2c0)x)dσ(x),

F (t) := tQ−1

∫

A

F (δtx)dσ(x).


Notice that

D =

⎛⎜⎝

∞∫

0

⎛⎝

∞∫

t

V (τ)dτ

⎞⎠

p/(p−q)

tQp(q−1)/(p−q)+Q−1dt

⎞⎟⎠

(p−q)/(pq)

= c

⎛⎜⎝

∞∫

0

⎛⎝

∞∫

t

V (τ)dτ

⎞⎠

p/(p−q)

×⎛⎝

t∫

0

τ (Q−1)(1−p)(1−p′)dτ

⎞⎠

p(q−1)/(p−q)

t(Q−1)(1−p)(1−p′)dt

⎞⎟⎠

(p−q)/(pq)

.

Consequently, due to Lemma 2.3, Holder’s inequality and Proposition A we findthat

I1 ≤ c

⎛⎝

∞∫

0

t(Q−1)(1−p)F p(t)dt

⎞⎠

q/p

= c

⎛⎝

∞∫

0

t(Q−1)(1−p)

⎛⎝∫

A

f(δtx)dσ(x)

⎞⎠

p

t(Q−1)pdt

⎞⎠

q/p

≤ c

⎛⎝

∞∫

0

tQ−1

⎛⎝∫

A

fp(δtx)dσ(x)

⎞⎠ dt

⎞⎠

q/p

= c

⎛⎝∫

E

fp(x)dx

⎞⎠

q/p

.

Further, applying Holder’s inequality twice and condition (4) we find that

I2 ≤∫

E

v(x)

⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p⎛⎝∫

Fx

kp′(x, y)dy

⎞⎠

q/p′

dx

≤ c

∫

E

v(x)

⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p

rQq/p′(x)kq(x, δ1/(2c0)x)dx

= c∑j∈Z

∫

E2j+1\E2j

v(x)

⎛⎝∫

Fx

fp(y)dy

⎞⎠

q/p

r(x)Qq/p′kq(x, δ1/(2c0)x)dx


≤ c∑j∈Z

⎛⎜⎝

∫

E2j+1\E2j−1/co

fp(y)dy

⎞⎟⎠

q/p ∫

E2j+1\E2j

v(x)rQq/p′(x)kq(x, δ1/(2co)x)dx

≤ c

⎛⎜⎝∑

j∈Z

∫

E2j+1\E2j−1/co

fp(y)dy

⎞⎟⎠

q/p

×

⎛⎜⎜⎝∑j∈Z

⎛⎜⎝

∫

E2j+1\E2j


⎞⎟⎠

p/(p−q)⎞⎟⎟⎠

(p−q)/p

≤ c‖f‖qLp(E)

⎛⎜⎜⎝∑j∈Z

⎛⎜⎝

∫

E2j+1\E2j


⎞⎟⎠

p/(p−q)⎞⎟⎟⎠

(p−q)/p

≡ c‖f‖qLp(E)(D)q.

Further,

(D)pq/(p−q) ≤ c∑j∈Z

2Qq(p−1)j/(p−q)

⎛⎜⎝

∫

E2j+1\E2j


⎞⎟⎠

p/(p−q)

≤ c∑j∈Z

∫

E2j \E2j−1

r(y)Qp(q−1)/(p−q)

×

⎛⎜⎝

∫

E\Er(y)

v(x)kq(x, δ1/(2c0)x)dx

⎞⎟⎠

p/(p−q)

dy

≤ cDpq/(p−q).

Now we show that (i)⇒(iii). Let vn(x) = v(x)χEn\E1/n(x), where n is an integer

with n ≥ 2. Let

fn(x) =

⎛⎜⎝

∫

E\Er(x)

vn(z)kq(z, δ1/(2c0)z)dz

⎞⎟⎠

1/(p−q)

r(x)Q(q−1)/(p−q).

It is easy to see that

‖f‖Lp(E) =

⎛⎜⎜⎝∫

E

⎛⎜⎝

∫

E\Er(x)

kq(y, δ1/(2co)y)vn(y)dy

⎞⎟⎠

p/(p−q)

×r(x)Qp(q−1)/(p−q)dx)1/p

< ∞.

On the other hand, by condition (3′) and integration by parts we find that

‖Kf‖Lqv(E) ≥

⎛⎝∫

E

vn(x)

⎛⎝∫

Fx

fn(y)k(x, y)dy

⎞⎠

q

dx

⎞⎠

1/q

≥ c

⎛⎝∫

E

vn(x)kq(x, δ1/(2c0)x)

⎛⎝∫

Fx

fn(y)dy

⎞⎠

q

dx

⎞⎠

1/q

= c

⎛⎜⎜⎝∫

E

vn(x)kq(x, δ1/(2co)x)

⎛⎜⎜⎝∫

Fx

⎛⎜⎝

∫

E\Er(y)

vn(z)kq(z, δ1/(2co)z)dz

⎞⎟⎠1/(p−q)

×r(y)Q(q−1)/(p−q)dy)q

dx)1/q

≥ c

⎛⎜⎜⎝∫

E


⎛⎜⎝

∫

E\Er(x)


⎞⎟⎠

q/(p−q)

×⎛⎝∫

Fx

r(y)Q(q−1)/(p−q)dy

⎞⎠

q

dx

⎞⎠

1/q

≥ c

⎛⎜⎜⎝∫

E


⎛⎜⎝

∫

E\Er(x)


⎞⎟⎠

q/(p−q)

×r(x)Qq(p−1)/(p−q)dx)1/q

= c

⎛⎝

∞∫

0

tQ−1

⎛⎝∫

A

vn(δtx)kq(δtx, δt/(2co)x)dσ(x)

⎞⎠⎛⎝

∞∫

t

τQ−1

×⎛⎝∫

A

vn(δτ z)kq(δτ z, δτ/(2co)z)dσ(z)

⎞⎠dτ

⎞⎠

q/(p−q)

tQq(p−1)/(p−q)dt

⎞⎟⎠

1/q

= c

⎛⎜⎝

∞∫

0

d

⎛⎝

∞∫

t

τQ−1

⎛⎝∫

A

vn(δτ x)kq(δτ x, δτ/(2co)x)dσ(x)

⎞⎠ dτ

⎞⎠

p/(p−q)

×tQq(p−1)/(p−q)dt)1/q

= c

⎛⎜⎝

∞∫

0

⎛⎝

∞∫

t

τQ−1

⎛⎝∫

A

vn(δτ x)kq(δτ x, δτ/(2co))dσ(x)

⎞⎠ dτ

⎞⎠

p/(p−q)

×tQq(p−1)/(p−q)dt)1/q

= c

⎛⎜⎝

∞∫

0

tQ−1

⎛⎝

∞∫

t

τQ−1

⎛⎝∫

A

vn(δτ x)kq(δτ x, δτ/(2co)x)dσ(x)

⎞⎠dτ

⎞⎠

p/(p−q)

×tQq(p−1)/(p−q)−Qdt)1/q

= c

⎛⎜⎜⎝∫

E

⎛⎜⎝

∫

E\Er(x)

vn(x)kq(x, δ1/(2co)x)dx

⎞⎟⎠

p/(p−q)


⎞⎟⎟⎠

1/q

.

Now the boundedness of K and Fatou’s lemma completes the proof of the impli-cation (i)⇒ (iii).

Lemma 2.4 implies that (i)⇔(ii). The theorem has been proved.

4. Schatten-von Neumann norm estimates

In this section we give necessary and sufficient conditions guaranteeing two-sidedestimates of the Schatten-von Neumann norms for the operator (2).

We denote by k0 the function r(x)Qk2(x, δ1/(2co)x).

Theorem 4.1. Let 2 ≤ p < ∞ and let k ∈ V2. Then Ku ∈ σp(L2(E)) if and only ifu ∈ lp(L2

k0(E)). Moreover, there exist positive constants b1 and b2 such that

b1‖u‖lp(L2k0

(E)) ≤ ‖Ku‖σp(L2(E)) ≤ b2‖u‖lp(L2k0

(E)).

Proof. Sufficiency. First observe that

J(x) :=∫

Er(x)

k2(x, y)dy ≤ ck0(x),

where c is a positive constant. Indeed, splitting the integral over Er(x) into twoparts and taking into account the condition k ∈ V2 we have

J(x) =∫

Sx

k2(x, y)dy +∫

Fx

k2(x, y)dy ≤ c1k0(x) + c2k0(x) ≤ c3k0(x).

Hence, the Hilbert-Schmidt formula yields

‖Ku‖σ2(L2(E))=

⎛⎜⎝

∫

E×Er(x)

(k(x, y)u(x))2 dxdy

⎞⎟⎠

1/2

=

⎛⎜⎝∫

E

u2(x)

⎛⎜⎝∫

Er(x)

k2(x, y)dy

⎞⎟⎠ dx

⎞⎟⎠

1/2

≤ c

⎛⎝∫

E

u2(x)k0(x)dx

⎞⎠

1/2

= c

⎛⎜⎝∑

n∈Z

∫

E2n+1\E2n

u2(x)k0(x)dx

⎞⎟⎠

1/2

= c‖u‖l2(L2k0

(E)).

On the other hand, by Theorem 3.2 and Proposition B we have that there is apositive constants c such that

‖Ku‖σp(L2(E)) ≤ c‖u‖lp(L2k0

(E)),

where 2 ≤ p < ∞.Necessity. Let Ku ∈ σp

(L2(E)

). We set

fn(x) = χE2n+1\E2n (x)|E2n+1 \ E2n |− 12 ;

gn(x) = u(x)χE2n+1\E3·2n−1 (x)k1/20 (x)α−1/2

n ,

where

αn =∫

E2n+1\E3·2n−1

u2(x)k0(x)dx.

Then it is easy to verify that fn and gn are orthonormal systems in L2(E).By proposition C we find that

∞ > ‖Ku‖σp(L2(E)) ≥(∑

k∈Z

|< Kufn, gn >|p)1/p

=

⎡⎢⎣∑

n∈Z

⎛⎜⎝

∫

E2n+1\E3·2n−1

k1/20 (x)(Kfn)(x)u2(x)α−1/2

n dx

⎞⎟⎠

p⎤⎥⎦

1/p

≥ c

⎡⎢⎣∑

n∈Z

⎛⎜⎝

∫

E2n+1\E3·2n−1

k1/20 (x)u2(x)α−1/2

n

⎛⎜⎝

∫

Er(x)\E2n

k(x, y)fn(y)dy

⎞⎟⎠ dx

⎞⎟⎠

p⎤⎥⎦

1/p

≥ c

⎡⎢⎣∑

n∈Z

⎛⎜⎝

∫

E2n+1\E3·2n−1

k(x, δ1/(2c0)x)u2(x)α−1/2n 2−nQ/2(r(x)Q− 2nQ)dx

⎞⎟⎠p

⎤⎥⎦

1/p

≥ c

⎡⎢⎣∑

n∈Z

⎛⎜⎝

∫

E2n+1\E3·2n−1

u2(x)k0(x)α−1/2n dx

⎞⎟⎠

p⎤⎥⎦

1/p

= c

[ ∞∑n=0

αp/2n

]1/p

.

Let us now take

f ′n(x) = χE3·2n−1\E3·2n−2 (x)|E3·2n−1\E3·2n−2 |− 1

2 ;

g′n(x) = u(x)χE3·2n−1\E2n (x)k1/2

0 (x)β−1/2n ,

where

βn =∫

E3·2n−1\E2n

u2(x)k0(x)dx

and argue as above. Then we will find that

∞ > ‖Ku‖σp(L2(E)) ≥ c

[ ∞∑n=0

βp/2n

]1/p

.

Summarizing the estimates obtained above we have⎛⎜⎜⎝∑n∈Z

⎛⎜⎝

∫

E2n+1\E2n

u2(x)k0(x)dx

⎞⎟⎠

p/2⎞⎟⎟⎠

1/p

≤(∑

n∈Z

(αn + βn)p/2

)1/p

≤ c(‖Ku‖σp(L2(0,∞)) + ‖Ku‖σp(L2(0,∞)))

≤2c‖Ku‖σp(L2(0,∞)).

5. Convolution-type operators with radial kernels

Let ϕ be a positive function on [0,∞) and let

Kf(x) =∫

Er(x)

ϕ(r(xy−1))f(y)dy, x ∈ E.

We say that ϕ belongs to Uλ, 1 < λ < ∞, if

(a) there exists a positive constants c1 and c2 such that

ϕ(r(xy−1)) ≤ c1ϕ(r(x)), y ∈ Sx,

ϕ(r(xy−1)) ≥ c2ϕ(r(x)), y ∈ Er(x);

(b) there is a positive constant c3 for which the inequality

∫

Fx

ϕλ′(r(xy−1))dy ≤ c3(r(x))Qϕλ′

(r(x)).

Example. Let ϕ(t) = tα−Q, where Q/λ < α < Q. Then it is easy to see thatϕ ∈ Uλ. Indeed, (a) is obvious due to the properties of the quasi-norm r. Let usshow (b). We have

I : =∫

Fx

ϕλ′(r(xy−1)dy =

∫

Fx

(r(xy−1)(α−Q)λ′dy

=

∞∫

0

|B(0, r(x))| ∩ y ∈ E : (r(xy−1))(α−Q)λ′> s|ds

=

(r(x)(α−Q)λ′∫

0

(· · · ) +

∞∫

(r(x)(α−Q)λ′

(· · · ) := I1 + I2.

It is clear that I1 ≤ (r(x))Qϕλ′(r(x)), while for I2 we find that

I2 ≤∞∫

(r(x)(α−Q)λ′

sQ

(α−Q)λ′ ds = c(r(x))(α−Q)λ′+Q = c(r(x))Qϕλ′(r(x)).

The statements of this section are proved just in the same way as for theoperator K( see the previous sections); therefore the proofs are omitted.

Theorem 5.1. Let 1 < p ≤ q < ∞ and let k ∈ Up. Then(i) K is bounded from Lp(E) to Lq

v(E) if and only if

B ≡ supj∈Z

B(j) ≡ supj∈Z

⎛⎜⎝

∫

E2j+1\E2j

v(x)ϕq(r(x))dx

⎞⎟⎠

1/q

(2j)Q/p′

< ∞;

(ii) K is compact from Lp(E) to Lqv(E) if and only if B < ∞ and limj→−∞ Bj =

limj→+∞ Bj = 0.

Theorem 5.2. Let 0 < q 1. Suppose that ϕ ∈ Up. Then thefollowing conditions are equivalent:(i) K is bounded from Lp(E) to Lq

v(E);(ii) K is compact from Lp(E) to Lq

v(E);(iii)

⎛⎜⎜⎝∫

E

⎛⎜⎝

∫

E\Er(x)

ϕq(r(y))v(y)dy

⎞⎟⎠

p/(p−q)


⎞⎟⎟⎠

(p−q)/(pq)

< ∞.

Let now ϕ(t) = tQϕ2(t) and let k(x) = ϕ(r(x)). Suppose that

Kuf(x) = u(x)∫

Er(x)

ϕ(r(xy−1))f(y)dy, x ∈ E,

where u is a measurable function on E. Then we have

Theorem 5.3. Then Ku ∈ σp(L2(E)) if and only if u ∈ lp(L2k(E)). Moreover, there

exist positive constants b1 and b2 such that

b1‖u‖lp(L2k(E)) ≤ ‖Ku‖σp(L2(E)) ≤ b2‖u‖lp(L2

k(E)).

Acknowledgement

The authors would like to express their gratitude to the referee for valuable remarksand suggestions.

References

[1] T. Ando, On the compactness of integral operators, Indag. Math., (N.S.) 24 (1962),235–239.

[2] J. Bergh, J. Lofstrom, Interpolation spaces: An introduction, GrundlehrenMath.Wiss. vol. 223, Springer, Berlin (1976).

[3] J.S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978),405–408.


[4] M.Sh. Birman, M. Solomyak, Estimates for the singular numbers of integral opera-tors, Uspekhi Mat. Nauk. 32(1) (1977), 17–82. English transl. Russian Math. Surveys32 (1977) (Russian).

[5] D.E. Edmunds, W.D. Evans, D.J. Harris, Two-sided estimates of the approximationnumbers of certain Voltera integral operators, Stud. Math., 124(1) (1997), 59–80.

[6] D.E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and compact integral operators,Kluwer, Dordrecht (2002).

[7] D.E. Edmunds, D. Stepanov, The measure of non-compactness and approximationnumbers of certain Voltera integral operators, Math. Ann. 298 (1994), 41–66.

[8] G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups, Princeton Univ.Press, University of Tokyo Press, Princeton, New Jersey (1982).

[9] I. Genebashvili, A. Gogatishvili, V. Kokilashvili, Solution of two-weight problems forintegral transforms with positive kernels, Georgian Math., J. 3(1) (1996), 319–342.

[10] I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight theory for integraltransforms on spaces of homogeneous type, Pitman Monographs and Surveys in Pureand Applied Mathematics 92, Longman, Harlow (1998).

[11] P. Jain, P.K. Jain, B. Gupta, Higher dimensional compactness of Hardy operatorsinvolving Oinarov-type kernels, Math. Ineq. Appl. 9(4) (2002), 739–748.

[12] L.P. Kantorovich, G.P. Akilov, Functional analysis, Pergamon, Oxford (1982).

[13] H. Konig, Eigenvalue distribution of compact operators, Birkauser, Boston (1986).

[14] V.M. Kokilashvili, On Hardy’s inequality in weighted spaces, Soobsch. Akad. Nauk.SSR, 96 (1979), 37–40 (Russian).

[15] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustilnik, P.E. Sobolevskii, Integral operatorsin spaces of summable functions, (Nauka, Moscow, 1966), Engl. Transl. NoordhoffInternational Publishing, Leiden (1976) (Russian).

[16] A. Kufner, L.E. Persson, Integral inequalities with weights, Academy of Sciences ofthe Czech Republic (2000).

[17] F.J. Martin-Reyes, E. Sawyer, Weighted inequalities for Riemann-Liouville fractionalintegrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), 727–733.

[18] V.G. Maz’ya , Sobolev spaces, Springer, Berlin (1985).

[19] A. Meskhi, Solution of some weight problems for the Riemann-Liouville and Weyloperators, Georgian Math. J. 5(6), (1998), 565–574.

[20] A. Meskhi, Criteria for the boundedness and compactness of integral transforms withpositive kernels, Proc. Edinb. Math. Soc. 44(2) (2001), 267–284.

[21] A. Meskhi, On singualar numbers for some integral operators, Revista Mat. Compl.,14 (2001), 379–393.

[22] B. Muckenhoupt, Hardy’s inequality with weights, Stud. Math., 44 (1972), 31–38.

[23] K. Nowak, Schatten ideal behavior of a generalized Hardy operator, Proc. Am. Math.Soc., 118 (1993), 479–483.

[24] J. Newman, M. Solomyak, Two-sided estimates on singular values for a class ofintegral operators on the semi-axis. Int. Equ. Operat. Theory, 20 (1994), 335–349.

[25] R. Oinarov, Two-sided estimate of certain classes of integral operators. Trans. Math.Inst. Steklova 204 (1993), 240–250 (Russian).


[26] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Math. Series,219, Longman Sci. and Tech. Harlow (1990).

[27] A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, Cambridge(1987).

[28] D.V. Prokhorov, On the boundedness of a class of integral operators, J. Lond. Math.Soc., 61(2) (2000), 617–628.

[29] G. Sinnamon, One-dimensional Hardy-inequalities in many dimensions, Proc. RoyalSoc. Edinburgh Sect, A 128A (1998), 833–848.

[30] G. Sinnamon, V. Stepanov, The weighted Hardy inequality: new proof and the casep = 1, J, Lond. Math. Soc. 54 (1996), 89–101.

[31] V. Stepanov, Two-weight estimates for the Riemann–Livouville operators, Izv. Akad.Nauk SSSR. 54 (1990), 645–656 (Russian).

[32] V. Stepanov, Weighted norm inequalities for integral operators and related topics,In: Nonlinear Analysis. Function Spaces and Applications, 5 Olympia Press, Prague(1994) 139–176.

[33] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, (1978), 2nd ed. Johann Ambrosius Barth, Heidelberg (1995).

[34] A. Wedestig, Weighted inequalities of Hardy-type and their limiting inequalities, Doc-toral Thesis, Lulea University of Technology, Department of Mathematics (2003).

Usman Ashraf and Muhammad AsifAbdus Salam School of Mathematical SciencesGC UniversityLahore, Pakistane-mail: [email protected]

[email protected]

Alexander MeskhiA. Razmadze Mathematical InstituteGeorgian Academy of Sciences1, M. Aleksidze St.0193 Tbilisi, Georgiae-mail: [email protected]

Received 26 December 2007; accepted 7 July 2008

To access this journal online:www.birkhauser.ch/pos

Boundedness and compactness of positive integral operators on cones of homogeneous groups

Documents

Transcript of Boundedness and compactness of positive integral operators on cones of homogeneous groups