Potential Analysis 16: 347–372, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

347

Wolff’s Inequality for Radially Nonincreasing Kernels and Applications to Trace Inequalities CARME CASCANTE1, JOAQUIN M. ORTEGA1, and IGOR E. VERBITSKY2,

1 Departament de Matemàtica Aplicada i Anàlisi, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain (e-mail: {cascante,ortega}@mat.ub.es) 2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: [email protected])

(Received: 3 October 2000; accepted: 15 November 2000) Abstract. We extend Th. Wolff’s inequality to a general class of radially decreasing convolution kernels. As an application we obtain characterizations of nonnegative Borel measures µ on Rn such that the trace inequality
TK f
Lq (dµ) ≤ C
f
Lp (dx) holds for every f in Lp (dx). Mathematics Subject Classifications (2000): 42B35, 31B15. Key words: potentials, Wolff’s inequality, trace inequality.

1. Introduction In this paper, we obtain a generalization of Th. Wolff’s inequality for potentials with arbitrary nonnegative radially nonincreasing kernels. We start with the classical case of Riesz kernels, Iα (x) = c(α, n)|x|α−n , 0 < α < n, on Rn , where c(α, n) is a normalization constant, which sometimes will be dropped for the sake of convenience. Let f be a nonnegative Borel measurable function, and let µ be a nonnegative Borel measure on Rn . The Riesz potential Iα (f dµ) is defined by
Iα (f dµ)(x) = c(α, n)

Rn

f (y) dµ(y), |x − y|n−α

x ∈ Rn .

If f ≡ 1, then the Riesz potential Iα (dµ) of the measure µ will be denoted by Iα µ.  First two authors partially supported by DGICYT Grant PB98-1242-C02-01, and CIRIT Grant 1998SGR00052.  Third author partially supported by NSF Grant DMS-0070623.

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There are two nonlinear potentials which play an important role in potential theory related to homogeneous Sobolev spaces, namely, the Havin–Maz’ya potential:

Vα,p µ(x) = Iα [(Iα µ)p −1 ](x), and the Hedberg–Wolff potential: 
+∞  µ(B(x, r)) p −1 dr , Wαp µ(x) = r n−αp r 0 where 1/p + 1/p = 1, and B(x, r) is a ball of radius r centered at x ∈ Rn . The energy of a nonnegative Borel measure µ in Rn is defined by
p Vαp µ(x) dµ(x). Eαp (µ) =
Iα µ
Lp (dx) = Rn

Since Wαp µ(x)
C1 Vαp µ(x), for every x ∈ Rn , it follows that
Wαp µ(x) dµ(x)
C1 Eαp (µ). Rn

A converse to the preceding inequality is Wolff’s inequality (see [9]):
Wαp µ(x) dµ(x), Eαp (µ)
C2 Rn

where the constants Ci , i = 1, 2, above depend only on α, p, and n. A simple proof of Wolff’s inequality due to Adams, which can be found, e.g., in [4], is based on a theorem of Muckenhoupt and Wheeden [14]. This theorem p is the fractional shows that Eαp (µ)
Mα µ
p , where Mα µ(x) = supr>0 µ(B(x,r)) r n−αp maximal function. Several other proofs of Wolff’s inequality are known, although the original approach [9], which is based on direct calculations involving a dyadic analogue of the Riesz potential, has its own advantages. It can be used in a more general framework, and is also suitable for applications to nonlinear equations (see [10]). Some generalizations of Wolff’s inequality are known (see [3] and [5]) where it is assumed that the kernel K : (0, +∞) → R+ is a nonnegative nonincreasing lower semicontinuous function such that K(t)t n satisfies a Dini condition, that is, such that, for every r > 0,
r dt K(t)t n
CK(r)r n−1 . t 0 In this case, the generalized Wolff potential is given by
+∞ K(t)p µ(B(x, t))p −1 t n−1 dt. WK,p (µ)(x) = 0

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

We set

349




TK (f dµ)(x) =

Rn

K(|x − y|)f (y) dµ(y).

Then an analogue of Wolff’s theorem says that
p WK,p (µ)(x) dµ(x).
TK (dµ)
p Rn

One of our main results in Section 2 shows that there is an extension of Wolff’s inequality which holds for arbitrary nonnegative nonincreasing lower semicontinuous kernels, without any kind of Dini’s type condition imposed on K, provided that in the definition of the generalized Wolff potential we replace the function K(t) by
1 t K(t) = n K(r)r n−1 dr. t 0 This modified kernel has been used before in [11] and [3]. The main technical tools in the proof of our result are a discretization technique, which goes back to Fefferman and Stein [8], and Wolff [9], and an extension of the inequality of Muckenhoupt and Wheeden mentioned above due to Kerman and Sawyer [11]. In the last section of the paper, we discuss applications of modified Wolff’s inequalities to the study of trace inequalities for generalized potential operators. In the case K = Iα , and 0 < q, p < +∞, the classes of positive Borel measures µ such that the trace inequality
Iα f
Lq (µ)
C
f
p ,

∀f ∈ Lp (dx),

(1.1)

holds have been widely studied. The case where 1 < p
q was considered in [1, 11, 12], where different characterizations of (1.1) were established, initially in terms of capacities of arbitrary compact sets, and later in equivalent noncapacitary forms. (See [4], and also [18] for a survey of these and more recent results.) In [2], D. Adams extended the capacitary characterization to general radial nonincreasing kernels K by showing that the trace inequality
TK f
Lq (µ)
C
f
p ,

∀f ∈ Lp (dx),

(1.2)

holds if and only if there exists a constant M > 0 such that, for any compact set E ⊂ Rn , µ(E)
MCK,p (E)q/p . Here CK,p is the capacity associated with K. In the case 1 < p < q < +∞, and for K = Iα , it is well known that it is enough to verify this capacitary condition only on balls E = B(x, r). In other words, (1.1) holds if and only if sup x∈Rn ,r>0

q

µ(B(x, r))r (αp−n) p < ∞,

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where 0 < α < n/p; it is not difficult to see that (1.1) is valid only in the trivial case µ = 0 if α  n/p. Our first result in Section 3 shows that a similar statement is true in the general situation. The capacitary condition for balls is equivalent to the condition 
+∞ (p−1) pq p n−1 K(t) t dt < ∞. (1.3) sup µ(B(x, r)) x∈Rn ,r>0

r

Thus, we will prove that (1.3) characterizes the trace inequality (1.2) for radially nonincreasing kernels in case 1
p < q. For q < p, [12] and [13] obtained a capacitary condition which characterizes the trace inequality (1.1) in this case. In [6] (see also [18] for the case q < 1), a noncapacitary characterization has been established which shows that (1.1) holds q(p−1) if and only if Wαp (µ) is in L p−q (dµ). An analogous characterization for inequality (1.2) was obtained in [5] under the assumption that K(t)t n−1 satisfies a Dini condition. We will show in the present paper that this Dini condition can be removed by considering the generalized Wolff potential mentioned above. In other words, (1.2) holds if and only if WK,p µ ∈ L

q(p−1) p−q

(1.4)

(dµ).

Our last result demonstrates that the trace inequality in the case q < p is equivalent to trace inequalities for certain discrete versions of the potential operator. If D = {Q} is the family of all dyadic cubes in Rn , and x ∈ Rn , we define
 K(rQ )χQ (x) f dy, TKD f (x) = Q∈D

Q

and prove that the uniform discrete trace inequality
|TKD+z f (x)|q dµ(x)
C
f
qp , ∀f ∈ Lp (dx), sup z∈Rn

Rn

is equivalent to the trace inequality for the kernel K. A final remark on our notation: we will adopt the usual convention of using the same letter for various absolute constants whose values may change in each occurrence, and we will write A  B if there exists an absolute constant such that A
MB. We will say that two quantities A and B are equivalent if both A  B and B  A, and in that case we will write A B. 2. Wolff’s Theorem for Radially Nonincreasing Kernels We begin this section with some definitions. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. We will assume that K(t)t n−1 is integrable in a neighborhood of 0.

351

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

For a measurable nonnegative function f on Rn , and x ∈ Rn , we define
TK f (x) = K(|x − y|)f (y) dy, Rn

and similarly, if µ is a positive Borel measure on Rn ,
K(|x − y|) dµ(y). TK µ(x) = Rn

If µ is a positive locally finite Borel measure on Rn , 1 < p < +∞, and 1/p + 1/p = 1, the energy of µ related to K and p is given by
(TK µ)p (x) dx. EKp (µ) = Rn



In [11], Kerman and Sawyer related this energy to the Lp -norm of a fractional maximal function. This fractional maximal function is defined by MK µ(x) = sup K(r)µ(B(x, r)), r>0

where K(r) =

1 rn




r

K(t)t n−1 dt, 0

for r > 0. Observe that in case K(t) = 1/t n−α is the Riesz kernel, it follows that MK coincides up to a constant with the usual fractional maximal function Mα µ(x) = sup r>0

µ(B(x, r)) . r n−α

We can now state the Kerman–Sawyer inequality. THEOREM 2.1 ([11]). Let 1 < q < +∞, and let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Then there exists a constant C > 0 such that, for any positive locally finite measure µ on Rn , 1
MK µ
q

TK µ
q
C
MK µ
q . C In fact, we will also need a pointwise inequality used in the proof of the above theorem which states that under the same assumptions, there exists C > 0 such that, for any x ∈ Rn , MK µ(x)
CM(TK µ)(x),

(2.1)

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where M(TK µ) denotes the Hardy–Littlewood maximal function of TK µ. In particular, for any x ∈ Rn , r > 0, K(r)µ(B(x, r))
CM(TK µ)(x).

(2.2)

It is shown in [3], that although K does not necessarily satisfy a doubling condition, the function K always does. More precisely, the following lemma holds (see Lemma 4.2.1 in [3]). LEMMA 2.2 ([3]). K is a positive continuous function such that: (i) For any r > 0, K(r)/n
K(r). (ii) K is nonincreasing and satisfies the doubling condition, i.e., for any r > 0, K(2r)
K(r)
2n K(2r). We want to introduce an appropiate Wolff-type potential for an arbitrary kernel K. The following lemmas suggest two natural candidates to consider. The first one is Lemma 5.2.1 in [3]. LEMMA 2.3 ([3]). Let K : (0, +∞) → R+ be a nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0, 1 < q < +∞, and let µ be a positive locally finite Borel measure on Rn . There exists a constant C > 0 such that

+∞ K(t)q µ(B(x, t))q−1 t n−1 dt dµ(x).
MK µ
qq
C Rn

0

In particular, the above and Theorem 2.1 gives that p

EKp (µ)
MK µ
p

+∞
C K(t)p µ(B(x, t))p −1 t n−1 dt dµ(x). Rn

(2.3)

0

The second lemma is a modification of Lemma 5.2.2 in [3]. LEMMA 2.4. Under the assumptions of Lemma 2.3, there exists a constant C > 0 such that

+∞ q K(t)K(t)q−1 µ(B(x, t))q−1 t n−1 dt dµ(x).
MK µ
q  C Rn

0

In particular p

EKp (µ)
MK µ
p

+∞  C K(t)K(t)p −1 µ(B(x, t))p −1 t n−1 dt dµ(x). Rn

0

(2.4)

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

353

Proof. Since the Hardy–Littlewood maximal function M is of strong-type (q, q), and M(TK µ)  TK µ a.e. (with respect to dx), we have
MK µ
qq
M(TK µ)
qq
= M(TK µ)(x)M(TK µ)q−1 (x) dx n
R
q−1  TK µ(x)M(TK µ) (x) dx = Rn

TK (M(TK µ)q−1 )(y) dµ(y). Rn

We next obtain a lower estimate for TK M(TK µ)q−1 . Using polar coordinates, we deduce
q−1 K(|y − x|)M(TK µ)q−1 (x) dx K ∗ M(TK µ) (y) = Rn

+∞ K(t) M(TK µ)q−1 (x) dσ (x), = |y−x|=t

0

where dσ is the surface measure on the unit sphere in Rn . Using (2.1), we have that for any t > 0, M(TK µ)(x)  CK(2t)µ(B(x, 2t)), and the fact that K has the doubling property, together with the fact that if |y − x| = t, B(y, t) ⊂ B(x, 2t), gives that for any t > 0, |y − x| = t, M(TK µ)(x)  CK(t)µ(B(y, t)). Consequently,
TK M(TK µ)

q−1




+∞

(y)  C

K(t)q−1 µ(B(y, t))q−1 dσ (x) dt

K(t)


|y−x|=t

0 +∞

 C

K(t)K(t)q−1 µ(B(y, t))q−1 t n−1 dt.

0

Thus
TK µ
qq  C


Rn




+∞

K(t)K(t)q−1 µ(B(y, t))q−1 t n−1 dt dµ(y).

✷

0

Now it is natural to consider the following two modifications of the Wolff potential. If µ is a positive Borel measure in Rn and 1 < p < +∞, we set
+∞ µ(B(x, t))p −1 K(t)K(t)p −1 t n−1 dt, (2.5) WKp µ(x) = 0
+∞ W Kp µ(x) = µ(B(x, t))p −1 K(t)p t n−1 dt. (2.6) 0

Lemma 2.2 gives that for any x ∈ Rn , WKp µ(x)
nW Kp µ(x).

(2.7)

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CARME CASCANTE ET AL.

Thus, by Lemmas 2.3 and 2.4,

WKp µ(x) dµ(x)  EK,p (µ)  Rn

W Kp µ(x) dµ(x).

Rn

We will see that the integrals involving the above two Wolff-type potentials satisfy a converse inequality as well:

W Kp µ(x) dµ(x)
C WKp µ(x) dµ(x). (2.8) Rn

Rn

Altogether, this will give the proof of Wolff’s theorem in this general setting. We recall that in case K(t) = 1/t n−α , 0 < α < n, is the Riesz kernel, it follows that K K, and the potentials defined above coincide up to constants with the Hedberg–Wolff potential [9]. More generally, if K(t)t n satisfies a Dini condition, that is, if for any r > 0,
1 r K(r) = n K(t)t n−1 dt
CK(r), r 0 then W Kp µ(x)
CWKp µ(x). The proof of (2.8) will be deduced from a discrete estimate that involves dyadic versions of the two Wolff-type potentials previously defined. For technical reasons, we introduce a third dyadic potential. If D = {Q} is the family of all dyadic cubes in Rn , we define  d n µ(x) = K(rQ )K(rQ )p −1 rQ µ(Q)p −1 χQ (x), WKp Q∈D



d Kp µ(x) = W





n K(crQ )K(rQ )p −1 rQ µ(Q)p −1 χQ (x),

Q∈D

and



d

W Kp µ(x) =





n K(rQ )p rQ µ(Q)p −1 χQ (x),

Q∈D

where we denote by rQ the side length of Q in D, and c = c(n) > 1 is a fixed constant which depends only on n, such that if x ∈ Q, then Q ⊂ B(x, 2c rQ ). Sometimes it will be more convenient to deal with dyadic “shifted” versions of the above potentials defined by  n K(rQ )K(rQ )p −1 rQ µ(Q)p −1 χQ (x), WKd,zp µ(x) = Q∈Dz d,z Kp µ(x) W

=







n K(crQ )K(rQ )p −1 rQ µ(Q)p −1 χQ (x),

Q∈Dz

and d,z

W Kp µ(x) =

 Q∈Dz





n K(rQ )p rQ µ(Q)p −1 χQ (x),

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

where Dz now denotes the shifted dyadic lattice Dz = D + z = {Q + z}Q ∈D . The fact that K and K are nonincreasing, and that K satisfies the doubling condition, easily gives that there exists C > 0 such that, for any x ∈ Rn , d,z Kp µ(x)
CWKp µ(x), W

(2.9)

and d,z

W Kp µ(x)
CW Kp µ(x).

(2.10)

Indeed, for any x ∈ Rn , and l ∈ Z, there exists a unique cube Ql ∈ Dz , such that x ∈ Ql and |Ql | = 2ln . Hence Ql ⊂ B(x, 2c rQl ), and we deduce  d,z Kp µ(x) = K(crQl )K(rQl )p −1 |Ql |µ(Ql )p −1 W l∈Z










K(c2l )K(2l )p −1 2ln µ(B(x, c2l−1 ))p −1

l∈Z





l∈Z


C

c2l





K(c2l )K(2l )p −1 2ln µ(B(x, c2l−1 ))p −1

c2l−1


l∈Z

c2l

K(t)K

c2l−1

 t p −1 c



t n µ(B(x, t))p −1

dt t

dt
CWKp µ(x), t

where in the last inequality we have used the fact that K satisfies a doubling condid Kp in order tion. This gives (2.9). We recall that we have introduced the potential W to obtain an estimate of this type. The proof of (2.10) is done in a similar way. We can now prove the following proposition. PROPOSITION 2.5. Let µ be a nonnegative Borel measure on Rn , and let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. (i) There exists a constant C > 0 such that, for any x ∈ Rn , d µ(x)
CW Kp µ(x). WKp

(ii) There exists a constant C1 > 0 such that

d d W Kp µ(x) dµ(x)
C1 WKp µ(x) dµ(x). Rn

Rn

d d Kp µ by W µ. The same is true if we replace WKp Proof. Statement (i) follows from Lemma 2.2. We now prove (ii). We observe that if Q ∈ D, and rQ = 2k for some k ∈ Z, then

µ(Q)  1  n )r µ(Q ) = K(r K(2l )2ln µ(Q)K(rQ ). Q Q n kn rQ 2 l
k Q ⊂Q

(2.11)

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CARME CASCANTE ET AL.

The proof of statement (ii) is based on some elementary estimates. The first one is analogous to Lemma 6.3.2 in [3] and Lemma 4.1 in [19]. We include a proof for the sake of completeness. LEMMA 2.6. Let s  1. There exists a constant C(s) such that for any (λQ )Q∈D , λQ ∈ R+ , we have: 

s λQ


C(s)

Q∈D



λQ

 

s−1 λQ

(2.12)

.

Q ⊂Q

Q∈D

Proof. We first prove the lemma for s = m ∈ Z+ using induction. The estimate (2.12) is obvious for m = 1. We now assume that (2.12) holds for s = m. Then       m−1    m+1 λQ
C(m) λQ λQ λQ Q∈D

Q ∈D m

Q∈D

=



λQ

 

Q ∈D

+



+

λQ

Q ⊂Q

λ

Q ∈D

 

Q

Q ⊂Q

λQ

 

Q ⊂Q

m−1 λ

Q

.

Q ⊂Q

Next, the second term in the last sum equals to   m−1    λQ λQ λQ , Q∈D

Q ⊂Q

Q ⊂Q

since Q ⊂ Q, we deduce that the above is bounded by   m−1 m       λQ λQ λQ = λQ λQ , Q∈D

Q ⊂Q

Q ⊂Q

Q∈D

Q ⊂Q

which gives the lemma for s nonnegative. In general, let s = m + ε, 0 < ε < 1. Using the integer case estimate for m, we obtain  m+1   s Q∈D λQ λQ
C(m)   1−ε Q∈D Q∈D λQ  m  Q ⊂Q λQ = C(m) λQ   1−ε Q∈D Q ∈D λQ  m  Q ⊂Q λQ
C(m) λQ   1−ε Q∈D Q ⊂Q λQ

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

= C(m)



λQ

= C(m)

Q∈D

m+ε−1 λQ

Q ⊂Q

Q∈D



 

λQ

 

s−1 λQ

✷

.

Q ⊂Q

LEMMA 2.7. Let s > 1. There exists a constant C > 0 such that, for any (λQ )Q∈D , λQ ∈ R+ , s s−1   1   1  λQ |Q|
C λQ λQ . (2.13) |Q| |Q| Q∈D Q∈D Q ⊂Q

Q ⊂Q

Proof. By Lemma 2.6 we have: s   s−1  1   1  λQ |Q|
C λ λ Q Q |Q| |Q|s−1 Q∈D Q∈D Q ⊂Q Q ⊂Q Q ⊂Q     s−1  1 = C λQ λQ . |Q|s−1 Q

Q ⊂Q

Q ⊂Q

However, the above expression on the right-hand side is bounded by   s−1  1 λQ s−1 λQ . C(s) |Q | Q

✷

Q ⊂Q

We now complete the proof of statement (ii) of Proposition 2.5. We use (2.11), and n µ(Q)  0. Then by (2.12), we obtain apply (2.13) with λQ = K(crQ )rQ
d W Kp µ(x) dµ(x) Rn  p K(rQ )µ(Q) |Q| = Q∈D

p  1  n K(rQ )rQ µ(Q ) |Q|

|Q| Q∈D Q ⊂Q  p −1  1  n n K(rQ )rQ µ(Q) K(rQ )rQ µ(Q )
C |Q| Q∈D Q ⊂Q
 d
C K(rQ )K(rQ )p −1 |Q|µ(Q)p
C WKp µ(x) dµ(x). Q∈D

Rn

d Kp is completely analogous and boils down to This gives (ii). The proof for W  p ✷ replacing rQ in Q∈D (K(rQ )µ(Q)) |Q| by crQ .

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CARME CASCANTE ET AL.

Our next goal is to show that statement (ii) of Proposition 2.5 extends to the continuous versions of Wolff’s potentials. PROPOSITION 2.8. Let µ be a nonnegative Borel measure on Rn , and let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let 1 < p < +∞. Then

W Kp µ(x) dµ(x)
C WKp µ(x) dµ(x). Rn

Rn

Proof. The proof is based on an idea of Fefferman and Stein [8] (see also [15] d,z and [6]) based on the averaging of W Kp over the shifts of the dyadic lattice D. For j  0 let j

W Kp µ(x) =




2j





µ(B(x, r))p −1 K(r)p r n

0

dr . r

If x ∈ Bj = B(0, 2j ), then
C j d,z W Kp µ(x)
W Kp µ(x) dz. |Bj +3 | Bj+3

(2.14)

Assuming for a moment that (2.14) holds, we will finish the proof. We use Fubini’s theorem to obtain


C j d,z W Kp µ(x) dµ(x)
W Kp µ(x) dz dµ(x) |Bj +3 | Bj Bj+3 Bj

C d,z W Kp µ(x) dµ(x) dz = |Bj +3 | Bj+3 Bj

C d,z W Kp µ(x) dµ(x) dz.
|Bj +3 | Bj+3 Rn d,z Kp , and we obtain Next, we apply statement (ii) of Proposition 2.5 to W


C j d,z Kp W Kp µ(x) dµ(x)
µ(x) dµ(x) dz. W n |B | j +3 Bj Bj+3 R

We deduce, using (2.9), that the above is bounded by


C WKp µ(x) dµ(x) dz
C WKp µ(x) dµ(x). |Bj +3 | Bj+3 Rn Rn Lebesgue’s monotone convergence theorem finally gives:

W Kp µ(x) dµ(x)
C WKp µ(x) dµ(x). Rn

Rn

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

It remains to check (2.14). We proceed as in [6], and consider for k ∈ Z, x ∈ Bk = B(0, 2k ), and l
k, ,l the set of points z ∈ Bk+3 such that there exists Q ∈ D, l(Q) = 2l+2 , and I = B(x, 2l ) ⊂ Q + z. It is proved in [15] that |,l |  C|Bk+3 | 2kn .

(2.15)

The fact that K satisfies the doubling condition gives:
2j dr j W Kp µ(x) = µ(B(x, r))p −1 K(r)p r n r 0 
2l dr µ(B(x, r))p −1 K(r)p r n = l−1 r l
j 2  µ(B(x, 2l ))p −1 K(2l )p 2ln .
C l
j

Applying (2.15) to l
j and x ∈ Bj , we deduce
 C l p −1
µ(Q + z)p −1 χQ+z (x) dz µ(B(x, 2 )) |,l | ,l l(Q+z)=2l+1
 C µ(Q + z)p −1 χQ+z (x) dz.
|Bj +3 | Bj+3 l+1 l(Q+z)=2

Hence, if x ∈ Bj , j W Kp µ(x)

C
|Bj +3 |


C |Bj +3 |












n µ(Q + z)p −1 K(rQ )p rQ χQ+z (x) dz

Bj+3 l
j l(Q+z)=2l+1




d,z

W Kp µ(x) dz.

✷

Bj+3

Finally, the Kerman–Sawyer inequality (2.1), together with Lemmas 2.3, 2.4, and Proposition 2.8, yields the following version of Wolff’s theorem. THEOREM 2.9. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let 1 < p < +∞, and let µ be a positive locally finite Borel measure on Rn . Then

WKp µ(x) dµ(x) W Kp µ(x) dµ(x). EKp (µ) Rn

Rn

3. Applications to Trace Inequalities for Radially Nonincreasing Functions Let µ be a positive locally finite Borel measure on Rn , let 0 < q, p < +∞, and let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous

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function such that limt →+∞ K(t) = 0. Our first application of Wolff’s theorem for arbitrary nonincreasing radial kernels is a characterization in the case p < q of the class of measures µ such that the trace inequality
TK f
Lq (dµ)
C
f
Lp (dx)

(3.1)

holds for all Borel measurable nonnegative functions f . We recall that, for a nondecreasing and nonnegative K, and 1 < p < +∞, the generalized capacity of a set E ⊂ Rn is given by CK,p (E) = inf{
f
pp : f  0, TK f  1 on E}. By definition, CK,p is translation invariant, so in what follows we will write CK,p (B(r)) = CK,p (B(x, r)). When 1 < p
q < +∞, Adams (see [1] and [4]), obtained a capacitary characterization of the nonnegative measures satisfying the trace inequality (3.1). THEOREM 3.1 ([2]). Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such +∞that lim t →+∞ K(t) = 0, 1 < p
q < +∞. Assume also that for any r > 0, r K(t)p t n−1 dt < +∞, and that K(t)p t n−1 is not integrable in any neighborhood of 0. Let µ be a nonnegative Borel measure on Rn . Then the following properties of µ are equivalent: (i) There is a constant A1 > 0 such that 1/q 

q (TK f ) (x) dµ(x)
A1 Rn

1/p p

f (x) dx

,

Rn

for any f  0. (ii) There is a constant A2 > 0 such that, for any compact set E ⊂ Rn ,


TK µ|E
p
A2 µ(E)1/q . (iii) There is a constant A3 > 0 such that for any compact set E ⊂ Rn , µ(E)1/q
A3 CK,p (E)1/p . Moreover, the least imbedding constant A1 in (i) is equivalent to sup { µ(E)1/q CK,p (E)−1/p : E compact }. When 1 < p < q < +∞, and K is the Riesz kernel Iα , or the Bessel kernel, condition (iii) can be replaced by a simpler one (see [4]) in which the family of all compact sets is replaced by the familly of all (closed) balls, that is, condition (iii) q is equivalent to µ(B(x, r))
r (n−αp) p . Our first application of Wolff’s inequality states that this is still true for general radially decreasing convolution kernels.

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

THEOREM 3.2. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0, and 1 < p < q < +∞. +∞ Assume also that for any r > 0, r K(t)p t n−1 dt < +∞, and that K(t)p t n−1 is not integrable in any neighborhood of 0. Let µ be a nonnegative Borel measure on Rn . Then the following properties of µ are equivalent: (i) There is a constant A1 > 0 such that 1/q 

(TK f )q (x) dµ(x)
A1 Rn

1/p f (x)p dx

,

Rn

f  0.

(ii) There is a constant A2 > 0 such that for any x ∈ Rn , r > 0, µ(B(x, r))
A2 CK,p (B(r))q/p . Proof. REMARK 3.3. In [3, Theorem 4.4.1], a two-sided estimate for the capacity CK,p of a ball was obtained: 
+∞ 1−p p n−1 K(t) t dt . CK,p (B(x, r)) r

See also [10] for a weighted analogue of this relation. Thus, condition (ii) can be rewritten equivalently as 
+∞ (1−p) pq p n−1 K(t) t dt . µ(B(x, r))
A2 CK,p (B(x, r)) r

The fact that (i) implies (ii) is obvious. In order to prove the sufficiency of this characterization, we will show that (ii) implies (iii) in Theorem 3.1. Let E be a compact set in Rn . By Theorem 2.9, we have:
p W Kp µ|E (x) dµ|E (x)
TK µ|E
p Rn

+∞ µ|E (B(x, t))p −1 K(t)p t n−1 dt dµ|E (x) = n 0 R

R µ|E (B(x, t))p −1 K(t)p t n−1 dt dµ|E (x) + = Rn 0

+∞ µ|E (B(x, t))p −1 K(t)p t n−1 dt dµ|E (x), + Rn

R

where R > 0 is to be chosen later. Without any assumption on µ, we obtain that
+∞ µ|E (B(x, t))p −1 K(t)p t n−1 dt R
+∞ p −1 K(t)p t n−1 dt  µ(E)p −1 CK,p (B(R))1/(1−p),
µ(E) R

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and consequently,

+∞ µ|E (B(x, t))p −1 K(t)p t n−1 dt dµ|E (x) Rn

R




µ(E)p CK,p (B(R))1/(1−p). The hypothesis on µ gives that

R µ|E (B(x, t))p −1 K(t)p t n−1 dt dµ|E (x) n R 0
R q CK,p (B(t))(p −1) p K(t)p t n−1 dt.
Cµ(E) 0

Now, if we choose R > 0 so that CK,p (B(R)) = µ(E)p/q , we deduce that
R q p p /q + µ(E) CK,p (B(t))(p −1) p K(t)p t n−1 dt,
TK µ|E
p  µ(E) 0

and in order to finish we just have to show that in the above sum, the second summand is also bounded by µ(E)p /q , or equivalently:
R q CK,p (B(t))(p −1) p K(t)p t n−1 dt  µ(E)p /q −1 . 0

However, this is an immediate consequence of the fact that the function

( p −1) q



q



CK,p (B(t)) q p is a primitive of CK,p (B(t))(p −1) p K(t)p t n−1 , together with the choice of R. Observe that this argument breaks down when p = q. ✷ Our second application of Wolff’s theorem for radially nonincreasing kernels is a characterization of the trace inequality in the case q < p. This characterization will extend the one obtained in [6] and [5], and it will be given in terms of the Wolff-type potential W Kp µ introduced in the preceding section. THEOREM 3.4. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let µ be a locally finite positive Borel measure on Rn , and 1
q < p < +∞. Then the following conditions are equivalent: (i) ( Rn (TK f )q (x) dµ(x))1/q
C( Rn f (x)p dx)1/p , f  0. d,z (ii) supz∈Rn Rn W Kp µ(x)q(p−1)/(p−q) dµ(x) < +∞. (ii) Rn W Kp µ(x)q(p−1)/(p−q) dµ(x) < +∞. Moreover, the least imbedding constant C in (i) is equivalent to 1/p


W Kp µ
Lq(p−1)/(p−q) (dµ) .

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

Proof. The proof generalizes the one in Theorem 3.1 in [5], where it is shown the theorem for nonincreasing functions K such that K(t)t n−1 satisfies a Dini condition, see also [6]. For the sake of completeness, we just sketch it very briefly. We begin by showing that (i) ⇒ (ii). Duality gives that (i) is equivalent to p

p


TK (g dµ)
Lp (dσ )
C
g
Lq (dµ) ,

(3.2)



for any g ∈ Lq (dµ), g  0. Theorem 2.9 applied to the positive measure ν = g dµ, together with (2.10), easily gives: p


TK (g dµ)||Lp (dσ ) p
 
dm z C W Kp (g dµ)(x)g(x) dµ(x) = C g dµ K(rQ )p |Q|. Rn

Q∈D+z

Q



The fact that (i) holds, gives then that for any g ∈ Lq (dµ), p  
p g dµ K(rQ )p |Q|
C
g
Lq (dµ) . Q∈D+z

Q









Let cQ = K(rQ )p |Q|µ(Q)p . If ψ ∈ Lq /p (dµ), ψ  0, we consider  1/p
 d z 1/p 1 (x) = sup ψ(y) dµ(y) . g(x) = Mµ ψ x∈Q µ(Q) Q The above estimate implies  Q ψ(x) dµ(x) p

g
Lq (dµ)
C
ψ
Lq /p (dµ). cQ µ(Q) Q∈D+z Duality again yields  cQ dz χQ ∈ Lq(p−1)/(p−q)(dµ), W Kp µ = µ(Q) Q∈D+z which is condition (ii). Next, we show that (ii) ⇒ (i). Duality and Theorem 2.9 gives that (i) is satisfied if for any g ∈ Lq (dµ), g  0,

p W Kp (g dµ)(x)g(x) dµ(x) TK (g dµ)p (x) dσ (x)
C
g
Lq (dµ) . Rn

Rn

Now, we consider the translated dyadic Hardy–Littlewood maximal function with respect to µ given by
1 H Ldz sup |h(y)| dµ(y). Mµ h(x) = x∈Q+z Q∈D µ(Q + z) Q+z

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We have:

dz

dz

W Kp (g dµ)(x)
MµH Ldz g(x)p −1 W Kp µ(x). Hölder’s inequality with exponent r = q /(p − 1), gives
dz W K (g dµ)(x)g(x) dµ(x) Rn


C


Rn

MµH Ldz g(x)q

1/q 
dx Rn

r 1/r  dz W K µ(x)g(x) dµ(x) .



Using now the boundedness of MµH Ldz from Lq (dµ) to Lq (dµ) (see [15]), and Hölder’s inequality with λ = q /r > 1, we see that the above integral is bounded by 1/(r λ ) 
dz p r λ W Kp µ(x) dµ(x) . C
g
Lq (dµ) Rn

Since r λ = q(p − 1)/(p − q), and we are assuming that (ii) holds, the preceding estimate and (2.14) easily give that
j p W Kp (g dµ)(x)g(x) dµ(x)
C
g
Lq (dµ) . Bj

Applying Lebesgue’s monotone convergence theorem, we obtain (i). We next prove (ii) ⇒ (iii). Hölder’s inequality with exponent q(p − 1)/(p − q), together with (2.14) gives that
q(p−1)/(p−q)  j W K µ(x) dµ(x) Bk

q(p−1)/(p−q)  dz C W Kp µ(x) dσ (z) dµ(x).
|Bj +3 | Bj Bj+3 Fubini’s theorem and (ii) easily give that
q(p−1)/(p−q)  j W K µ(x) dµ(x)
C, Bk

and letting j → +∞, we obtain (iii). The implication (iii) ⇒ (ii) follows from the pointwise estimate (2.10). ✷

4. Discrete Trace Inequalities for Radially Nonincreasing Functions In this section we will show that the trace inequality is equivalent to uniform trace inequalities for certain discrete versions of the potential operator. We begin with

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

365

some definitions. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0 and D = {Q} is the family of all dyadic cubes in Rn . We consider the kernel KD given by  K(rQ )χQ (x)χQ (y), KD (x, y) = Q∈D n

for x, y in R . We define the operator TKD by
 K(rQ )χQ (x) f dσ (y), TKD f (x) = Q

Q∈D

where f is a nonnegative Borel measurable function, or more generally, if µ is a positive locally finite Borel measure on Rn ,  K(rQ )µ(Q)χQ (x). TKD µ(x) = Q∈D

If 1 < p < +∞, the energy of µ is given by p

 p  K(rQ )µ(Q)χQ (x) dx. TKD µ(x) dx = EKD p (µ) = Rn

Rn

Q∈D

We also introduce a dyadic fractional maximal function MKd µ given by MKd µ(x) = supx∈Q K(rQ )µ(Q). Our first result can be viewed as a discrete version of the Kerman–Sawyer theorem for radially nonincreasing kernels; see [11]. THEOREM 4.1. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0, 1 < p < +∞, and µ a locally finite positive Borel measure on Rn . The following quantities are equivalent: p
 K(rQ )µ(Q)χQ (x) dx, (a) EKD p (µ) = Rn




Q∈D

p

(b) Rn

MKd µ(x) dx.

Proof. We introduce some more definitions. Let 1 < s < +∞, 3 = (λQ )Q∈D , λQ ∈ R+ , and let σ be a positive measure. We define s
 λQ χQ (x) dσ (x), A1 (3) = |Q|σ Rn Q∈D s−1   1  λQ λQ , A2 (3) = |Q|σ Q∈D Q ⊂Q  s
1  sup λQ dσ (x). A3 (3) = Rn x∈Q |Q|σ Q ⊂Q

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The proof of the theorem will be a consequence of the following proposition. PROPOSITION 4.2. Let σ be a positive locally finite Borel measure on Rn . (a) Let 1 < s
2. Then there exists a constant C > 0 which depends only on s such that, for any 3 = (λQ )Q∈D , λQ ∈ R+ , A1 (3)
CA2 (3). (b) Let 1 < s < +∞. Then there exists a constant C > 0 which depends only on s such that, for any 3 = (λQ )Q∈D , λQ ∈ R+ , 1 A1 (3)
A3 (3)
CA1 (3). C Proof. Let 1
s
2. By Lemma 2.6, s
 λQ χQ (x) dσ (x) A1 (3) = |Q|σ Rn Q∈D s−1  λQ
  λQ χ (x) dσ (x).
C(s) Q | |Q| |Q σ σ Q Q∈D Q ⊂Q

Hölder’s inequality with exponent 1/(s − 1)  1, gives that s−1
  λQ 1 χQ (x) dσ (x) |Q|σ Q |Q |σ Q ⊂Q s−1  s−1 
 1  λQ 1 χQ (x) dσ (x) = λQ .
|Q|σ Q |Q |σ |Q|σ Q ⊂Q

Q ⊂Q

Consequently, A1 (3)
C(s)

 Q∈D

 λQ

s−1 1  λQ = C(s)A2 (λ), |Q|σ Q ⊂Q

and that proves (a). Next, let 1 < s < +∞. Then  s
1  sup λQ dσ (x) A3 (3) = Rn x∈Q |Q|σ Q ⊂Q s 
λQ H Ld Mσ χQ (x) dσ (x)
CA1 (3),
|Q|σ Rn Q∈D where we have used the fact that the dyadic Hardy–Littlewood maximal function MσH Ld is of strong type (s, s). This proves the estimate A3 (3)
CA1 (3).

(4.1)

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

367

We will obtain the remaining estimate A1 (3)
CA3 (3) using an induction argument. If 1 < s
2, then part (a) gives s
 λQ A1 (3) = χQ (x) dσ (x) |Q|σ Rn Q∈D s−1   1  λQ λQ
C |Q|σ Q∈D Q ⊂Q  s−1
 1  λQ = C χQ (x) λQ dσ (x) |Q|σ Rn Q∈D |Q|σ Q ⊂Q  s−1
 λQ 1  χQ (x) sup λQ dσ (x)
C |Q| |Q| σ x∈Q σ Rn Q∈D Q ⊂Q


CA1 (3)

1/s

A3 (3)

1/s

,

where in the last estimate we have used Hölder’s inequality with exponent s. Thus if 1 < s
2 we have obtained that A1 (3)
CA3 (3). Now if k > 1, and we assume that the above inequality holds for any k − 1 < r
k, we must show that it also holds for k < r
k + 1. Using again Lemma 2.6 we have s
 λQ χQ (x) dσ (x) A1 (3) = |Q| σ Rn Q∈D s−1  λQ
  λQ χQ (x) dσ (x).
C |Q|σ Q |Q |σ Q∈D Q ⊂Q

Applying the induction hypothesis for k − 1 < s − 1
k to the measure χQ σ , and to the set (λQ )Q , where λQ = 0 outside Q, we obtain s−1
  λQ χQ (x) dσ (x) |Q |σ Q Q ⊂Q  s−1
1  sup λQ dσ (x),
C Q x∈Q ⊂Q |Q | Q ⊂Q

and  s−1  λQ
1  sup λQ dσ (x) A1 (3)
C |Q|σ Q x∈Q ⊂Q |Q | Q∈D Q ⊂Q  s−1
 λQ 1  χQ (x) sup λQ dσ (x)
C Rn Q∈D |Q|σ x∈Q ⊂Q |Q | Q ⊂Q

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s 1/s  λQ
C χQ (x) dσ (x) × |Q| σ Rn Q∈D 1/s s 
 1  λQ dσ (x) sup × x∈Q |Q | Rn Q ⊂Q


CA1 (3)

1/s

A3 (3)

1/s

.

Hence, A1 (3)
CA3 (3).

✷

n In order to complete the proof of Theorem 4.1, we set λQ = K(rQ )rQ µ(Q), apply statement (b) of Proposition 4.2 with s = p , and recall that in Section 2 we proved that 1  1  n λQ = K(rQ )rQ ✷ µ(Q ) µ(Q)K(rQ ). |Q| |Q| Q ⊂Q

Q ⊂Q

Our next result is a discrete version of Wolff’s theorem for radially nonincreasing kernels. THEOREM 4.3. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let 1 < p < +∞, and let µ a locally finite positive Borel measure on Rn . The following quantities are equivalent: p
 K(rQ )µ(Q)χQ (x) dx, (a) EKD p (µ) = Rn






(b)


Q∈D

Rn

d WKp µ(x)p dx =

Rn





n K(rQ )K(rQ )p −1 µ(Q)p rQ ,

Q∈D

p

(c)



d WKp µ(x) dx =







n K(rQ )p µ(Q)p rQ .

Q∈D

Proof. If 3 = (λQ )Q∈D ⊂ R+ , we set s  1  λQ |Q|σ . B2 (3) = |Q|σ Q∈D Q ⊂Q

The proof of the theorem will be a consequence of part (b) of the following proposition. PROPOSITION 4.4. Let 1 < s < +∞, and let σ be a positive locally finite Borel measure on Rn .

369

WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

(a) If 1 < s
2, then A1 (3) A2 (3). (b) Suppose that σ is a reverse doubling measure (σ ∈ RD), that is, there exists τ > 1 such that |2Q|σ  τ |Q|σ , for every cube Q. Then there exist constants Ci > 0, i = 1, . . . , 4, depending only on s, such that, for any 3 = (λQ )Q∈D , λQ ∈ R + , A1 (3)
C1 A2 (3)
C2 B2 (3)
C3 A4 (3)
C4 A1 (3).

(4.2)

Proof. We first show that if 1 < s < +∞, then there exists C > 0 such that

A2 (3)
CA1 (3)1/s A3 (3)1/s .

(4.3)

Indeed, A2 (3) =



 λQ

Q ⊂Q

Q∈D

where µQ = ( |Q|1 σ  Q∈D

s−1  1  λQ = λQ µQ , |Q|σ Q∈D



Q ⊂Q

λQ )s−1 . Then

  λQ µQ χQ (x) dσ (x) |Q|σ Rn Q∈D  
 λQ   χQ (x) sup µQ χQ (x) dσ (x).
|Q|σ x∈Q Rn Q∈D


λQ µQ =

Hölder’s inequality gives that the above is bounded by s 1/s 
1/s  s  λQ χQ (x) dσ (x) sup µQ χQ (x) dσ (x) |Q|σ Rn Rn x∈Q Q∈D 
  s 1/s λQ = χQ (x) dσ (x) × |Q| σ Rn Q∈D s 1/s  
1  sup λQ χQ (x) dσ (x) . × Rn x∈Q |Q|σ




Q ⊂Q

Hence 

λQ µQ
A1 (3)1/s A3 (3)1/s



Q∈D

and consequently A2 (3)
A1 (3)1/s A3 (3)1/s , which gives (4.3).

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CARME CASCANTE ET AL.

By part (b) of Proposition 4.4, it follows that A3 (3)
CA1 (3). This estimate, together with (4.3), yields that, for 1 < s < +∞, there exists a constant C > 0 such that A2 (3)
CA1 (3).

(4.4)

Using part (a) of Proposition 4.4 and (4.1), we deduce that, for 1 < s
2, A1 (3) A2 (3). Hence, we obtain part (a). In order to prove (b), we will check the chain of inequalities A2 (3)
C1 A1 (3)
C2 A3 (3)
C3 B2 (3)
C4 A2 (3). The first inequality is just (4.4), and the second one is statement (b) of Proposition 4.2. Next, we show that A3 (3)
CB2 (3). Indeed,  s
1  sup λQ dσ (x) A3 (3) = Rn x∈Q |Q|σ Q ⊂Q  s
 1  χQ (x) λQ dσ (x)
|Q|σ Rn Q∈D Q ⊂Q s  1  λQ |Q|σ = B2 (3). = |Q|σ Q∈D Q ⊂Q

The fact that σ ∈ RD is used as in the proof of Lemma 2.7, to deduce that A2 (3)
CA3 (3). ✷ Similarly to the proof of Theorem 3.4, we can use the discrete version of Wolff’s theorem (Theorem 4.3) to obtain the following characterization of discrete trace inequalities. THEOREM 4.5. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let µ be a locally finite positive Borel measure on Rn , 1 < q < p < +∞, and consider KD (x, y) =  Q∈D K(rQ )χQ (x)χQ (y). Then the following statements are equivalent: (a) There exists a constant C > 0 such that, for any f  0, 1/q 
q 
C
f
Lp (dx). TKD f (x) dµ(x) Rn

d

(b) W K µ =

 Q∈D





K(rQ )p |Q|µ(Q)p −1 χQ ∈ Lq(p−1)/(p−q)(dµ).

In [16] and [17], it is shown that if K satisfies an extra hypothesis, namely, that there exists ε > 0 such that for any pair of balls B and B with B ⊂ 2B,  ε rB K(rB )
C , K(rB ) rB

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WOLFF’S INEQUALITY FOR RADIALLY NONINCREASING KERNELS

then




q  TK f (x) dµ(x) Rn




sup

z∈Rn

Rn




= sup

z∈Rn

Rn



1/q

q TKD+z f (x) dµ(x)

  Q∈D+z

1/q q


f (y) dy

K(rQ )χQ (x)

1/q dµ(x)

,

Q

with constants of equivalence independent of f and µ. In [19], some results of this type are proved for general kernels K with the discrete operator TKD replaced by a pointwise smaller discrete operator. As a consequence of Theorems 3.4 and 4.5 we obtain that, for general radial nonincreasing kernels K, the trace inequality is equivalent to a uniform shifted dyadic trace inequality. THEOREM 4.6. Let K : (0, +∞) → R+ be a nonnegative nonincreasing lower semicontinuous function such that limt →+∞ K(t) = 0. Let µ be a locally finite positive Borel measure on Rn , and let 1 < q < p < +∞. Then the following are statements equivalent: (a) There exists a constant C > 0 such that, for any f  0, 1/q 
q 
C
f
p . TK f (x) dµ(x) Rn

(b) There exists a constant C > 0 such that for any f  0, 1/q 
q 
C
f
p . TKD+z f (x) dµ(x) sup z∈Rn

Rn

References 1. 2. 3. 4. 5. 6.

Adams, D.R.: ‘Traces of potentials arising from translation invariant operators’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1971), 203–217. Adams, D.R.: ‘Weighted nonlinear potential theory’, Trans. Amer. Math. Soc. 297 (1986), 73– 94. Aikawa, H. and Essén, M.: Potential Theory – Selected Topics, Lecture Notes in Math. 1633, Springer, 1996. Adams, D.R. and Hedberg, L.I.: Function Spaces and Potential Theory, Springer-Verlag, New York, 1996. Cascante, C. and Ortega, J.M.: ‘Norm inequalities for potential-type operators in homogeneous spaces’, to appear in Math. Nachr. Cascante, C., Ortega, J.M., and Verbitsky, I.E.: ‘Trace inequalities of Sobolev type in the upper triangle case’, Proc. London Math. Soc. 80 (2000), 391–414.

372 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

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