Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 984259,12 pages http://dx.doi.org/10.1155/2013/984259

Research Article

Sobolev Embeddings for Generalized Riesz Potentials of Functions in Morrey Spaces L( ' over Nondoubling Measure Spaces

Yoshihiro Sawano1 and Tetsu Shimomura2

1 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

2 Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

Correspondence should be addressed to Yoshihiro Sawano; yoshihiro-sawano@celery.ocn.ne.jp Received 16 December 2012; Accepted 12 February 2013 Academic Editor: Alfonso Montes-Rodriguez

Copyright © 2013 Y. Sawano and T. Shimomura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces L(1,<P)(G) over nondoubling measure spaces.

1. Introduction

In this paper, we show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent. In [1], in the setting of the Lebesgue measure, for 0 < a < n, recall that Adams considered and proved the boundedness of the fractional integral operator Ia given by

4/ M := f I ^L dy.

Jr" \x - v\

The operator Ia is also called the fractional integral operator or the Riesz potential. We denote by B(z, r) the ball {x e R" : \x- z\ < r} with center z and of radius r > 0, and by \B(z, r)\ its Lebesgue measure, that is, \B(z, r)\ = wnrn, where wn is the volume of the unit ball in R". Let G be abounded open subset of R". We denote its diameter by dG;

dG = sup {\x - y\ : x, y e G}.

For u e L1(G), we define the integral mean over B(z,r)

*B(z,r)

u (x) dx := id/ M f u (x) dx. (3)

J B(z,r) \B(Z,r)\ JGnB(z,r)

Let 1 < p < >x>. If f is a positive function on the interval (0, rn) satisfying the doubling condition (see (23)), then we define the Morrey space to be the family of all f e

L^oc(G) for which there is a positive constant C such that

- \f(x)\pdx

J B(z,r)

' B(z,r) (4)

< Cp f (r) whenever z e G, 0 < r < dG.

The norm of f e L(p'v^(G) is defined by the infimum of the constants C satisfying the inequality above. When <p(r) = r-x (r > 0), L(p'f) (G) is denoted by LpJi(G). A direct consequence of this notation is that

LptXp (G)DLn/X (G)

for 0 < X < n and p e [1,n/X). Some prefer to use the notation

V1PM-)

= sup W(r)\\f\\l/(B(Z>

w (r) = (wnrn) 1/pf(r) llp

references [2-5].

Much about the case p > 1 is known. Recall that the Adams theorem about the boundedness of fractional integral operators [1, Theorem 3.1] asserts that

IML scii/ii,

provided the parameters p, q, X satisfy 0 < X<n, 1 < p < q < rn,

See also research papers [2-4, 6-16] and a survey [5].

Meanwhile, only a few results are known for the case p = L Trudinger [17, Theorem 1] proved that if f e L u (G) = L1(G) then exp(a\I1 f\) e L1 (G) for some constant a > 0; this implies that the operator I1 is bounded from L 1>1(G) to exp(L1)(G). See also Serrin [18] for an alternative proof. Recently, the boundedness of Riesz potentials from (G) to Orlicz-Morrey spaces was shown in [19]. This result extends [20,21]. One of the reasons why the case when p = 1 is difficult is the failure of the boundedness of the Hardy-Littlewood maximal operator M. In connection with this failure, we do not have Littlewood-Paley characterization. Due to these two difficulties, the case when p = 1 is hard to analyze. However, from the point of PDEs, we are faced with analyzing the quantity

"m M,, r^ *

in connection of the Kato condition, where V is the potential operator of the operator -A + V. See [22, Section 2], for example. Consequently, despite the difficulty arising from harmonic analysis, the case when p = 1 occurs naturally. As another evidence that the case when p = 1 is of importance, we recall that the space L appears naturally in the

following sharp maximal inequalities [23, Theorem 4.7], [24, Theorem 1.3], and [25, Theorem 1.2]: let 1 < p < œ> and X e (0, n]. Then, there exists a constant C > 0 such that

C-1 (I I Mtf\\LM+WfK )

a <~c(\\Mtf\L )

for any measurable function f, where

M"f (x) = sup j——-, yeR»>r>0\B(y, r)\

h(y,r )

\B(y,r)\

f (w) dw

dz (12)

is the sharp maximal operator due to Fefferman and Stein [26]. A disadvantage of using the Littlewood-Paley theory is that we lose the integrability of functions a little when we consider the inequality

sup|MLu <c\\f\\LhÀ

where is a Littlewood-Paley patch. By choosing a

smooth function f e Cm(R") such that Xb(o,4)\b(o,2) ^ V ^ Xb(o,8)\b(o>1) , recall that we can define the jth Littlewood-Paley patch by

Sjf(x) := F-1 [y^-]- Ff] (x)

for f e S'(R"). Note that (13) is a direct consequence of the translation invariance of the space L 1>A(R"). But this loss caused by (13) is quite big. Note that

< C sup|S,-/

jj iil,

fails. See the appendix for a proof. When p > 1, an approach using the Littlewood-Paley patch is taken effectively [27]. Indeed,

<c\\f\\^ (16)

for all f e LHowever, for the case when p = 1, due to thefactthatthe estimate (13) is essential when we consider the Littlewood-Paley patch, we prefer to avoid the Littlewood-Paley patch. See [28-43] for a huge amount of culmination of this approach.

Instead of using the Littlewood-Paley patch, we still have a good approach for the case when p = 1. Just make a closer look at the integral kernel. Our method being simple enough, there is no need to stick to the geometric structure of R". Our result relies completely only upon the positivity of the integral kernel. So, here and below, we work on a separable metric space X equipped with a nonnegative Radon measure where we do not postulate any other condition on By B(x, r), we denote the open ball centered at x of radius r > 0. While, given a point p1 and p2 in R", we write \p1 - p2| for the distance of the points p1 and p2, and we write d(x, y) for the distance of the points x and y in X. We assume that p({x}) = 0 and that 0 < p(B(x, r)) < m for x e X and r > 0 for simplicity. In the present paper, we do not postulate on ^ the "so-called" doubling condition. Recall that a Radon measure ^ is said to be doubling, if there exists a constant C>0 such that

p(B(x,2r)) < Cp(B(x,r))

for all x e supp(^)(= X) and r > 0. Otherwise ^ is said to be nondoubling. In connection with the 5r-covering lemma, the doubling condition had been a key condition in harmonic analysis.

Our aim in this paper is to show that, for the case p = 1, the operator Ia and its generalization Ip are bounded

from Morrey spaces to Orlicz-Morrey spaces, or, to generalized Holder spaces, whose definitions will be given in the next section, in the nondoubling setting. Our result extends the results in [17-21]. The definition of Ip is the following: let p be a function from (0, rn) to itself and satisfy

dt < +œ>

for all sufficiently small r > 0. We do not have to postulate the doubling condition on p. See Remark 3 for an example which fails the doubling condition. We define

,f(*) = \o

p(d (x,y))

Jg (¿(B (x, 4d (x, y))) where f e L1(G). Instead of using p(d (x, y^)

f(y)dn(y), (19)

-f(K) = \

g (x,d(x, y)))

f(y)d^(y), (20)

we discuss Ip defined above. This modification will be necessary in Lemma 9 for example. An example in [44, Section 2] shows that is less likely to be bounded in general, although there does not exist a proof. We refer to [45] for an attempt of definining fractional integral operators by using the underlying measure

Note that (18) is necessary in order that the image by Ip of Xs(x,r), the indicator functions of the balls, belongs to Lp'f(G) at least when ^ is the Lebesgue measure. Indeed, if

dt = œ>

for any sufficiently small r > 0. Then, for y e B(x, r/2) such that B(x, r) c G, we have

IpXB(x,r) (y) = f z][\\\dz

JB(x,r) \B(y,4\y-;

f p{\y-A)

iB(y,ri2) \B(y,4\y-z\

by using the spherical coordinate.

We organize the remaining part of the present paper as follows. In Section 2, we set up some notations. Section 3 is devoted to stating our main results fully based on the notations in Section 2. Some auxiliary lemmas are collected in Section 4. Finally, theorems in the present paper are proven in Section 5.

2. Notation and Terminologies

Let G be the set of all continuous functions from (0,rn) to itself with the doubling condition, that is, there exists a constant cm > 1 such that

1 < ïd c«, < <P(s)

< c« for r,s > 0 with - < - <2. (23) « 2 s

We call the smallest number cf satisfying (23) the doubling constant of (p. Note that in view of [46, page 445] and [47,

(1.2)], the doubling condition on f is a natural one. For f e G, we define the Morrey space L(1'V^(G) as follows:

(G) := [feL\oc (G) : ¡/1^ < with the norm

\\f\\L<^(G) = sup

zeG,0<r<da V (r)

fi(B(z,2r)) ¡B(z,r)

\f(x)\dl4(x).

JB(z,r)

Then, a routine argument shows that L(1'V^(G) is a Banach space. Due to the fact that R" is a geometrically doubling space, we can prove that

c-1\\f\y*\G) < suP tttt

zeG'0<r<da V(r)

■ ( J p)) f \f(x)\d^(x) (26)

< cWfWi^iG)

for all k > 1. See [48, Proposition 1.1] for a technique used to prove this inequality. Note here that if f1,f2 e G and <p1/<p2 is bounded above on (0, dG), then

L(1,<Pl) (G) c L(1'«2) (G),

in particular, if there exists a constant C > 1 such that C- (p1(r) < (p2(r) < C^1(r) for all r > 0, then

L(1'f1) (G) = L(1'f2) (G) (28)

with equivalent norms. A ball testing shows the following.

Proposition 1. The function <p1/<p2 is bounded above on (0, dG) ifL(1'fi)(G) c L(1'f2\G) when p = dx.

Here and below, we write A < B to indicate that there exists a constant C independent of Morrey functions such that A < CB. The symbol A~ B stands for A < B < A.

Proof. According to [49, Proposition A], for any ball B(x0, r) contained in G, we have

f2 (r) (29)

If L(1'fl)(G) c L(1'f2 (G), in the sense of sets, then by the closed graph theorem and the doubling condition on <p1 and <p2, we conclude

\\f\\L(i .92)(G) , 90(G)-

If we combine (29) and (30), then we obtain that <p1/<p2 is bounded above on (0, dG). □

Let us consider the family Y of all continuous, increasing, convex, and bijective functions from [0, m) to itself. For O e Y, the Orlicz space L® (G) is defined by

L® (G):=[feLic (G):|/|lO(G) < œj , (31)

IIl0(G)

:= inf -

X > 0 : I O ( ) d^ (x) < l

If O1, O2 e Y are equivalent in the sense that there exists a constant C > 1 with

O1 (C-1r) < O2 (r) < O1 (Cr) for all r > 0, then we see easily that

L®1 (G) = L®2 (G) with equivalent norms. If

O (r) = exp (rp), exp (exp (rp)), rp(log r)X or rp(log r)q(log (log r))X (r > 0) for large r > 0, then L®(G) will be denoted by exp (Lp) (G), exp exp (Lp) (G), Lp(logL)X (G) or Lp(logL)>glogL)X (G),

respectively.

For O e Y and <p e G,the Orlicz-Morrey space L(®'f) (G) is defined by

L^ (G) := [f e Lic (G) : \\f\\L^HG) < m} , (37) where

II/IIl^G)

:= sup inf {a > 0 : —-—-1-—

zeG,o<r<dG I p(B(z,2r))

XJB(zr) )d^(x)<f(r)

(see [50,51]). Then, again it is routine to prove that || • ||L(®,T)(G) is a norm and that L(®'V\G) is a Banach space. Note that the space L® is a special case of Orlicz-Morrey spaces when ^ = dx.

For f e G such that f is bounded, the generalized Holder space is defined by

A, (G) = {f :

llA„(G)

< œ} ,

\f(x)-f{y)\

'A(G) x.^SU <p{2d{x,y))

Then, ||/||A (G) is a norm modulo constants and thereby Af(G) is a Banach space. Since f is bounded, every f e Af(G) is bounded. If <p(r) ^ 0 as r I 0, then every f e Af(G) is continuous. For details, we refer to [52].

(32) 3. Main Results

In this section, we state our main theorems, whose proofs are given in Section 5.

Throughout this paper, let G be a bounded open set in X and denote by cf, the doubling constant of <p e G.

Let us begin with the following result, which is the one of Gunawantype [9].

Theorem 2. Let p : (0,m) ^ (0,m) be a measurable function such that there exist k1, k2, Cp such that

0 < 16k1 < 1 <k2 < m,

rV n(s) (41)

sup p(s)<Cp J !-^ds (r > 0).

r/2<s<r

Let f e G, and define ^ p(t)

Jfcjr s

"'(r) := (I

dt) <p

rAk2d{

>(r)+ J

G p(t)<p(t)

dt (42)

Jo t J J2k1r t

for 0 < r < dG. Then, there exists a constant C > 0 such that 1

p(B(z,4r))

\lpf M\ dp (x) < Cf (r) \\f\\L(i,f)(G) JB(z,r) \ r \ w

for z e G, 0 < r < dG and f e L(1'^) (G), where C > 0 is a constant depending only on Cp, cf, k1, and k2.

Remark 3. (1) Here it is not significant for us to choose 16; it counts that any number will do as long as it is small enough.

(2) The number 4 in the right-hand side seems to be essential. According to [44, Section 2], it can happen that the norms

Z£G,°<r<dA p(B(z,4r)) Jb(z,>-)

\f(w)\pdH(w)

JB(z,r)

zeG,°<r<dA H(B(Z,2.r)) Jb(z,j-)

\f(w)\pd^(w)

JB(z,r)

are not equivalent for 1 < p < œ>.

(3) In view of [53,Lemma 2.5],wesee that(1-A)-a/2 falls under the scope of Theorem 2. Indeed, Nagayasu and Wadade showed that the kernel p which corresponds to (1 - A)-a/2 satisfies

(0<r<l), p(r)<er (r>l). (45)

This means that we have (41) with k1 = 1/16 and k2 = 1. Note that p e G implies (41). See also [54, Remark2.2].

Remark 4. Theorem 2 is proved in [19] when G = R" and p e G. See also [21].

We now state a result for Orlicz-Morrey spaces.

Theorem 5. Let p,p : (0,x) ^ (0,x) be measurable functions such that there exist k1, k2, Cp such that 0 < 16k1 < 1 < k2 < x and that

sup p (s) < Cp

r/2<s<r

(k*r piß) Jk,r S

r/2<s<r

Let p e G. Assume

Ck2r p(s) p (s) < Cp \ tUds (r > 0)

Jk,r S

r1 p(t)p(t)

dt = x

and that p/p is continuous and decreasing. Define

fl (r) := \

k (r) :=

2"G p(t)<p(t)

2k,r t

fi (r)p(4k2r)

f(r) :=

r2k 2 i

t I •(")+ \ ' J0 t J J2kl

for 0 < r < dG. If O e Y satisfies (fi °k-1)(s)

4k2 dr

2dG p (t) f (t) 2k, r

CG = sup

O-1 (s)

: k (dg) < s < x

< x, (49)

then there exists a constant A> 0 such that

( \lpf(*)\

>B(z,r) y A\\f ||i(i,y)(G)

dp(x)<f(r) (50)

for z e G, 0 < r < dG and f e L^1,f)(G), where A > 0 is a constant depending only on Cp, Cp, cf, k1, k2, and CG.

Remark 6. Note that k is bijective from (0, dG] to [K(dG), ra) by the assumptions in the theorem. Indeed, by the definition of k above, k is a decreasing function. In addition, limr|0K(r) = ra, showing that k : (0,dG] ^ [K(dG),rn) is bijective.

Finally, we shall show a result of Gunawan type about continuity.

Theorem 7. Let p : (0, rn) ^ (0, rn) be a measurable function such that there exist k1, k2, Cp such that

r/2<s<r

0 < 16k1 < 1 <k2 < <xi, k2 p ( )

p (s) < Cp \ ^ds (r > 0).

Let f e G. Assume the following condition on p. There are 0 < 0 < 1 and C'p > 0 such that

p(d (X yy)

p(d(z, y))

^ (B (x, 4d (x, y))) ^ (B (z, 4d (z, y)))

, ( d(x,z))e p(d (x,y)) < p\d(x,y)J p(B(x,4d(x,y)))

whenever d(x,z) < d(x,y)/2. Assume in addition the Dini condition

r1 p(t)p(t)

dt < x.

r3k, f( ) = \

2rp(t)p(t), e f 2dG p (t) p (t)

-dt + t

t ¡2k,r

t1+d (54)

for 0 < r < dG,

then Ip is bounded from L^1'f^l(G) to A^(G). Moreprecisely

ML^(G) <CWfWL^(G), (55)

where C > 0 is a constant depending only on Cp, C'p, cf, k1, k2, and 0.

Note that if J^p(t)p(t)/t)dt <x and 0 < d < 1, then

e (4k2do p(t)p(t)

r e (0, dG is bounded.

-dte[0,m) (56)

4. Preliminary Lemmas

Lemma 8. Let p : (0,x) ^ (0,x) be a measurable function such that there exist k1,k2, Cp such that

r/2<s<r

Let p e G. Then

0 < 16k1 < 1 <k2 < ix,

k2 p ( )

sup p(s)<Cp \ ¡——ds (r > 0).

Jb(x„

p(d (x,y))

B(x,r) ^ (B (x, 4d (x, y))) <C, \2k2r p(t)f(t)

\f(y)\d^(y)

dt) \\f\\L(i»)(G)'

where C > 0 is a constant depending only on Cp, cf, k1, and k2.

Moreover, ifk > 0, then ( p{d{x,y))

jb(*,dg)\b(*,r) p (b (x, 4d (x, y))) d(x, y)

\f(y)\d^(y)

> " fi (,r

U 1 ti W1

r^ d')if™

where C > 0 is a constant depending only on Cp, cf, k1, k2, and k.

Proof. If y e B(x, 2'r) \ B(x, 2J-1 r) and j e Z, then a geometric observation shows

p(d (x,y))

p (B (x, 4d (x, y))) d(x, y) 1

-r sup p(s) (60)

p (B (x, 2+ r)) (2-r)k itir^T ( )

_Cf_k Pj^lds.

p(B(x,V+1 r))(2-1 r)k JHkir s

Hence,

P(d (x yy)

lB(x,2ir)\B(x>2i-1 r) p (B (x, 4d (x, y))) d(x, y)

CP f^pOO

\f(y)\dp(y)

-1 r)k hi

(2— r) JiJkjr s 1

p (B (x, 2i+1 r)) Jb(x,21 r) d

\ . \f(y)\dp(y)

Jfi(x,2i r)

_ C P(2r) j21^pO)

P (2-1 r)k jv S

\ . \f(y)W(y)

f(Vr)p(B(x,V+1 r)) Jb( p(Vr) \2'k*p(s)J

'(2-v^ J2iv~dsx|fiiM(G)'

Set d := [1 + log2(k2/k1)]. Then, by virtue of the doubling condition on f, we have

p(2r) \2-V pi)

(2J-1 r)k Jük.r s

,p(Vr) (^VpOO

Pi kjr J21k1r

(2b) -^V 5 u (2r)k J21+'-1kir 5

l. d j-2 k^ C1(2d+1k1)kl\ii

£ \2i+V p(s) ^ pOO

kir (2i+dk1 r)k 5 k^ r2i+'kir <p(s)p(s)

d+1, \k f:2i+kir <p(s)p(s)

= C1(2d+%) I P( )+;( )

V 7 J21k1r S1+k

where C1 > 0 is a constant depending only on cf, k1, and k2. Consequently, since ^({x}) = 0,

( P(d(x,y))

BMp(B(x,4d(x,y))) \f(y)dp(y) _f \ p(d(x,y))

TO /•

XCPC1 j

<CfC1 (j0 " ^f^dt) \\f\ U,(G)

r)\B(x,2-1-1 r) p (B (x, 4d (x, y)))

2"'*V p(s)p(s)

\f(y)\dp(y)

< LCPC

2k2r p(t)p(t)

L(1'^(G)

which proves (58).

We choose j0 e Z, so that dG < 2J"r < 2dG. Then, we have

p(d (x,y))

jb(*,dg)\b(*,r) p (b (x, 4d (x, y))) d(x, y)

' p(d(x,y))

\f(y)W(y)

1=1 JB(x,2

;-=1 r)\B(x,21-1 r) p (B (x, 4d (x, y))) d(x, y)k

x If (y)| dp (y)

r2i+Xr p(s)p(s)

<CpC1(2d+1k1jl\ r^dsxlfl^G

j_1 J21k1V z

<CpCl(2d+1kl)k (Q-t^dt) \\f\ U,(G).

Thus, since k,k1,d being constants, (59) follows.

Lemma 9. Let p : (0,m) ^ (0,m) be a measurable function such that there exist k1, k2, Cp such that

0 < 16k 1 < 1 <k2 < <xi,

and that

sup p(s)<Cp \ ^^ds (r > 0). (66)

r/2<s<r J^r ^

Then, for aUfeL^ (G), 1

p(B(z, 2r))

p(d (x,y))

r,r) p (B (x, 4d (x, y)))

JB(z,r) \Jb(z„

Cp(r)([k2r ^fdiWa

\f(y)\dP(y))dP(x)

1,9)(G).

Proof. By Fubini's theorem and the dyadic decomposition of the ball, we have

f (f m\fM\M>)]^)

JB(z,r)\ h(z,r) p (B (x, 4a (x, y))) J

=L|/W|

»(f ¿ftf*™*™]*™ \ JB(z,r) p (B (x, 4a (x, y))) J

< f \f(y)\

JB(z,r)

sup2- ir<s<2 i+1r

B(y,2->+1 r)\B(y,2->r) p (B (x, 2->+2r))

x dp (x)) dp (y)

\ ( Jf(y)

JB(z,r)

to f .1 \

sup2- ( ) B(y,2-'+1 r)) p(B(y,2-J+1 r))

x dp (x)) dp (y).

Since p satisfies (66), we have

p(d (x,y))

JB(z,r) \JB(z,t

B(z,r) p (B (x, 4d (x, y)))

'2-' + lk2r p(s)

\f(y)\dp(y))dp(x)

Cp \ \f(y)\(i\ " K2r ^dt)dp(y)

JB(z,r) \r^J2->+1k,r S

B(z,r) ' \'~'0)2-i+1k1r S 1

2 k2 p( )

\ * pJrdt) \ \f(y)\dp(y)

,Jo t ) JB(z,r)

< Cpf (r) p (B (z, 2r)) ( ^dt) \\f\\L(1M(G), as required.

(69) □

5. Proofs of the Theorems

We are now ready to prove our theorems.

Proof of Theorem 2. Let z e G and r e (0, dG]. By the posi-tivity of the kernel, we may assume that f > 0. We write

p(B(z,4r)) h(z,r) p 1

p (B (z, 4r))

JB(z,r) \ h

p(d (x,y))

Jb(z,^V JB(x,r) p (B (x, 4d (x, y)))

xf (y) dp (y) ) dp (x)

p(d (x,y))

p (B (z, 4r))

iß(z,r) ( iB(x,dG)\B(x,r) p (B (x, 4d (x, y))) xf (y) dp (y) ) dp (x)

p(B(z,4r))

JB(z,2r) V Jj

p(d (x yy)

JB(z,2r^ ^ )B(z,2r) p (B (x, 4d (x, y)))

xf (y) dp (y) ) dp (x)

p(d (x,yy)

p(B(z,4r))

iB(z,r) ( iB(x,da)\B(x,r) p (B (x, 4d (x, y)))

xf (y) dp (y) ) dp (x)

= h + h

for z e G and 0 < r < dG. By Lemma 9, we have h <C1f(r)(fohr ^ldt]\\fl(^(G)

< C1f(r)\\f\\L<1.9)(G). Meanwhile, by Lemma 8 we have

2 <C2 {Qdr G p^dt

IIl^KG)

< C2f(r)\\f\\L(1M(G).

Hence, it follows from (71) and (72) that 1

^(B(z,4r)) JB(z>r) p where C > 0 depends only on Cp, cf, k1, and k2. Proof of Theorem 5. By Theorem 2,wehave 1

p(B(z,4r)) JB(z>r)

for z e G and 0 < r < dG.

\ \l~p f (x)\d^(x) < C1f (r)\\f \\L(1M(G)

JB(z,r)

Let g := \/\/||/||L(m>)(g). For x e G and 0 < S < dG, since ~p/p is decreasing, we have by Lemma 8

1 p0(x)

p(d (x,yy)

Jb(x,<5) p (B (x, 4d (x, y)))

p(d (x yy)

B(x,dG)\B(x,a) p (B (x, 4d (x, y)))

g (y) dp (y)

g(y) dp (y)

p(8) \ p(d(x,y)) p(8) Ws) p(B(x,4d(x,y)))

r4k2d( J2k1S

g(y) dp (y)

G p(t)p(t)

< pS)Ippg(x) + C2f1 (8)

p(8) p(4k28)

hg(x) + C2f1 (8).

- p(4^S) "" Hence, in view of the definition of k, we have

V1 (S)

hg (x) <

■Lg(x) + C2f1 (8).

Now let 8 :_ Observe that V1 (8) _ -

k 1 (I~pg (x)) when I~pg (x) > k , dG when Ipg (x) < k (dG).

Y1 (k 1 (Ipg (x))) when 7pg (x) > k (dG), Y1 (do) when Ip,g (x) < k ,

by definition. We claim that

f1 (8) k (8)

Y1 (k 1 (ipg (x))) when ipg (x) > k (dg), Y1 (do) when Ipg (x) < k (dg).

Indeed, when Ipg(x) < K(dG), we have S = dG. Hence,

jBzr IPf(x)dp(x) < CV(r) llfllL<l*>(G), (73) (8)

K 8) Lpg (x) _ (dG) x Kd) Ipg (x) < (dG). (80)

When Ipg(x) > K(dG), we have S = k 1(Ipg(x)). Hence,

gM==*1 (K-1(^M)) ,fg{x)

k(s) ipgM (81)

= (k-1 (ipg(x))).

Consequently our claim (79) is justified. It follows from (76) and (79) that

Ipg (x) < (1 + C2)max (k-1 (Ipg (x))), f1 (dG)}.

By (49), we obtain

(f1 o K-1) (s) < CGO-1 (s) for k (dG) <s <m. (83) Hence, taking A := CG(C1 + 1)(1 + C2), we establish

\lpf(x)\

A!/!L(1^)(G)

!pg(x) < ——

max (k-1 (Ipg (x))) , Y1 (dG)}

< cG (C1 + 1)

= max (k-1 (Ipg (x))), Y1 (k-1 (k (dG)))}

= cG"(c~+7)

< max[O-1 (lpg(x)),O-1 (k^g))}

c1 + 1 .

Since ~p/p is decreasing and

(p (4^2dG) /P (4k2dG)) V1 (dG) = k (dG) , (85) we see that

T4k2dG j} (f) ^ (f)

f(r)> \

2k1dG t

> p(4k2dG) \4k2dG p(t)p(t)dt ~ p(4k2dG) J2Mg t

_ K (dG)

for all 0 < r < dG. Hence, with the aid of (74), we have 1

p (B (z,

— i o(

(z,4r)) h(z,r) \

Kf(x)\ ) d ( )

A^ )dp(x)

•<r) (G) /

C1 + 1

p(B(z,4r)) JB(z,r) 1

max {Ipg (x), k (dG)} dp (x)

JB(z,r) p

C1 + 1

( „ \\ i kg(x)dp(x)

\p(B(z,4r)) JB(z,r) H

i k (dG) dp (x) z,4r)) JB(z,r)

C1 + 1

p (B (z,

(C1f (r) + f (r)) = f (r),

which proves (50). Proof of Theorem 7. Write

Ipf(x)- Ipf(z)

i p(d(x,y))

JB(x,2d(x,z)) p (B (x, 4d (x, y)))

p(d (z,y))

JB(x,2d(x„

,z)) p(B(z,4d(z,y)))

p(d (x yy)

JG\B(x,2d(x,z)) V p (B (x, 4d (x, y)))

p(d (z,y))

p(B(z,4d(z,y)))

xf(y) dp (y).

By (58), we have i

< C{f (2d (x, z)) ||f||L(1,,)(G)

p (d (z y))

B(x,2d(x,z)) p (B (z, 4d (z, y)))

, ( p(d(z,y))

\f(y)\dp(y)

(87) □

f (y) dp (y)

f(y)dp (y)

p(d(x,y)) f( )d ( )

lB(x,2d(x,z)) p (B (x, 4d (x, y))) 11 (y)) p (y) (89)

\f(y)\dp(y)

~ JB(z,3d(x,z)) p (B (z, 4d (z, y))) < C[f(2d(x,z)) \\f\\L(1,r)(G) for x,z eG. On the other hand, we have by (52) and (59)

p(d (x yy)

p^z y))

JG\B(x,2d(x,z)) p (B (x, 4d (x, y))) p (B (z, 4d (z, y))) x\f(y)\dp(y)

e ( p(d(x,y))

< C d(x, z)e i p

G\B(x,2d(x,z)) p (B (x, 4d (x, y))) d(x, y)

x\f(y)\dp(y)

4k1d(x,z) t

< C2f(2d (x, z)) \\f\\L(1,r)(G).

dt) \\f\\L(1*)(G)

Now from (89), (90), and (91), we establish

\lpf(x)- Ipf(z)\ < Cy(2d(x,z)) \\f\\L(iM(G) (92) for x,z e G, as required. □

Appendix A. Disproof of (15)

Inequality (15) can be disproved in terms of Besov spaces and Triebel-Lizorkin spaces. Let y e Cm(R") satisfy

Xb(0,4) <f< Xb(0,8).

Define T0f := F-1 [f ■ Ff] for f e S'( R"). For parameters p e (0, x) and q e (0, x) and for f e S'(R"), the Besov

norm |

and the Triebel-Lizorkin norm IM! F are defined

:= \\T0fL + (I(2lVL)'

:= \\T0fLp +

II2VI'

respectively, and for p e (0,<m) and for f e S (R"), the Besov norm || ■ are defined by

™ and the Triebel-Lizorkin norm

:= \\T0flP + sup 2S\\Sf

JJ Wlp'

:= \\T0fW

sup \2,sSjf I

Meanwhile, by denoting P(R") the set of all polynomials, for parameters p e (1, x) and q e (1, x) and for a distribution f e S'(R"), the homogeneous Besov norm || ■ Hj* and the homogeneous Triebel-Lizorkin norm || ■ ||ps are defined

F* 1 P,<i

:= ( I (2llVL)

I \2jVi

. J=-OT

respectively. Also, for p e (1,m) and / e S'(R"), one defines

:= sup 2jS ¡VIL '

sup |2;sS:/l

respectively.

It follows from (A.7) and (A.8) that

(fe S' (R")]. (A.9)

Let 0 < p,q < œ, and s e R. The inhomogeneous Besov

space Bp ?(R") (resp. the homogeneous Besov space B

is defined to be the set of all f e S'(R") (resp. f e S'(R")/ P(R")) for which the norm ||/||B^ (resp. ||/||^ ) is finite,

when 0 < p,q < m. Likewise, for 0 < p < m, 0 < q< m and s e R, the inhomogeneous Triebel-Lizorkin space Fp^(R")

(resp. the homogeneous Triebel-Lizorkin space Fp^(R")) is defined to be the set of all f e S'(R") (resp. f e S'(R")/ P(R")) for which the norm || f|| ps (resp. ||f||y )is finite. To simplify the matters, even when we consider representatives in the function spaces £p^(R") and Fp^(R"), we forget that they are in equivalence classes, and we regard the function spaces £p„(R") and Fp„(R") assubspaces of S'(R"). Keeping this in mind, let us disprove (15). We have

sup||vlL < csup|VLm = 4

,VZ 11 "LU ,-(7 11 "L

b° <C№/A

(A.10)

from (5), (A.7), and (A.9).

However, according to [55, Theorem 11.2, (i), (11.2)], there exists f e f"/Ato(R") such that it is not represented by Ljoc(R")-functions:

f e ffj/A.œ (R")\i1oc (R")'

(A.11)

If we consider F 1 [(1 - y) • F/], where f e C™(R") is from (A.1), we can arrange that f e F°/Aœ(R") can be chosen so that supp(Ff) n B(0,4) = 0. Indeed,

F-1 [f^ Ff]eCm (Rn). (A.12)

We suppose that the Fourier support of f is away from B(0,4). Let us admit that

\\f\\P° <C\\f\\F° {feF0n/Km (R")) (A.13)

under the understanding F"/a>to(R") c S'(R"). Note also that L 1>^(R") is a subset of L^R"), hence our observation can be summarized as follows:

L 1,A (R")c4

feF "M,TO (R")\L1oc (R").

(A.14)

It then follows immediately that (15) fails since (15) implies

LU (R") 3 F°n/U

(A.15)

Inclusions (A.14) and (A.15) contradict obviously.

It remains to prove (A.13). Note that the frequency support of f does not intersect with B(0,4). Observe also that <p(2-1^) has the frequency support in £(0,4). Thus, we have

(A.8) S:f = F-1 [cp(2-J-)Ff] = 0 (j < -l), and hence

sup IM

7 L"/a

< II/IIf» + '

n//l,co

Define

Wf := F-1 [W - Ff] , W :=

sup \Sj/\

J(NU{°}

(A.16)

y + f(2-

(A.17)

In view of the size of frequency support, we conclude S0f = VT0f + VS1f. Now we invoke the following Planchrel-Polya-Nikolskii lemma.

Lemma A.1 (Planchrel-Polya-Nikols'kij [56, page 16]). Let 0 < q < 1. Assume that f e S'(R") has frequency support in Q(0, R). Then, denoting by M the Hardy-Littlewood maximal operator, we have

sup }f{X-yln <CMm(*)1h> (A.18)

yeR" (1 + R\y\)m where C is independent ofR > 0.

According to Lemma A.1 with q = 1/2 and R = 16, we conclude that

\Sof(x)\<c(J \F-1V(y)Y\Tof(x-y)\dy V Jr" 1 1

+ J |F-1 V(y)\^\SJ(x-y)\dy)

<c(M[\T0f\1/2]{x)2 + ^[|S1/|1/2](^)2)

xJ \F-1V(y)\(1 + \y\)2"dy

= C(M[\Tof\1/2](X)2 + M [|S1/|1/2] (x)2) ,

(A.19)

where for the last inequality, we invoked C

\F-1W(y)\ <

(l + \

(y e R") ' (A.20)

By the fact that \\F\\L„„ = y v|f| \\L2„,x2 and the L2n/x(Rn)-boundedness of the Hardy-Littlewood maximal operator, we conclude

\M\l^ ïc(\\M[\Tofr]\\L2nll2 + \\M[\slf\1/2]\\L2nll2)

<-c{\WT°f\U2 IL2+|isi/i1/2 IL2)

= c(\\T0fl^ + \\Sifl^ ).

(A.21)

Combining (A.16) and (A.21), we obtain the desired result.

Acknowledgment

The authors are grateful to Professor Victor Burenkov for his kind suggestion about the relations of various definitions of the Morrey norms.

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