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I nequalities

& A pplications

of the Author

MIA-23-45 547–562

Zagreb, Croatia Volume 23, Number 2, April 2020

Nicky K. Tumalun, Denny I. Hakim and Hendra Gunawan

Inclusion between generalized Stummel classes and

other function spaces

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& Applications

Volume 23, Number 2 (2020), 547–562 doi:10.7153/mia-2020-23-45

INCLUSION BETWEEN GENERALIZED STUMMEL CLASSES AND OTHER FUNCTION SPACES

NICKYK. TUMALUN, DENNYI. HAKIM∗ ANDHENDRAGUNAWAN (Communicated by L. Pick)

Abstract. We refine the definition of generalized Stummel classes and study inclusion properties of these classes. We also study the inclusion relation between Stummel classes and other function spaces such as generalized Morrey spaces, weak Morrey spaces, and Lorentz spaces. In addition, we show that these inclusions are proper. Our results extend some previous results in [2,13].

1. Introduction and preliminaries

The definition of Stummel class was introduced in [4,13]. For 0<α <n, the Stummel class S_α=S_α(Rⁿ)is defined by

S_α:=

f ∈L¹_loc(Rⁿ):ηαf(r)0 for r0 , where

ηαf(r):= sup

x∈Rⁿ

|x−y|<r

|f(y)|

|x−y|^n−αdy, r>0.

Forα =2,S₂ is known as theStummel-Kato class. Knowledge of Stummel classes is important when one is studying the regularity properties of the solutions of some partial differential equations (see [1,2,3,5,10]).

In the mean time, the study of Morrey spaces, which were introduced by C. B. Mor- rey in [11], has attracted many researchers, especially in the last two decades. For 1p<∞ and 0λ n, theMorrey space L^p,λ =L^p,λ(Rⁿ) is defined to be the collection of all functions f ∈L_loc^p (Rⁿ)for which

f_L^p,λ := sup

x∈Rⁿ,r>0r⁻^λ^pfL^p(B(x,r))<∞, whereB(x,r):={y∈Rⁿ:|x−y|<r} and

fL^p(B(x,r)):=

|x−y|<r|f(y)|^pdy ¹_p

.

Mathematics subject classification(2010): 42B35, 46E30.

Keywords and phrases: Generalized Stummel classes, generalized Morrey spaces, generalized weak Morrey spaces, Lorentz spaces.

∗Corresponding author.

c , Zagreb

Paper MIA-23-45 547

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Note thatL^p,0=L^p. As shown in [13], one may observe thatL^1,λ ⊆S_α provided that n−λ<α<n. (For the caseα=2, this fact was proved in [5].) Conversely, ifV∈S_α for 0<α<n andηαf(r)∼r^σ for someσ>0, thenV∈L^1,n−α+σ.

Eridani and Gunawan [6] developed the concept of generalized Stummel classes and studied the inclusion relation between these classes and generalized Morrey spaces.

For 1p<∞and a measurable functionΨ:(0,∞)→(0,∞), thegeneralized Morrey space L^p,Ψ=L^p,Ψ(Rⁿ)is the collection of all functions f∈L_loc^p (Rⁿ)for which

f_L^p,Ψ:= sup

x∈Rⁿ,r>0

|B(x,r)|⁻¹^p

Ψ(r) fL^p(B(x,r))<∞,

where |B(x,r)| denotes the Lebesgue measure of B(x,r). Observe that, for Ψ(t):=

t^λ⁻^pⁿ (0λn), we haveL^p,Ψ=L^p,λ. Further works on the inclusion relation between generalized Stummel classes and Morrey spaces can be found in [8,14].

The purpose of this paper is to refine the definition of generalized Stummel classes and study the inclusion relation between these classes. We also study the inclusion relation between Stummel classes and Morrey spaces using assumptions that are different from the assumptions used in [6, 8, 14]. We give an example of a function which belongs to the generalized Stummel class but not to the generalized Morrey space. Fur- thermore, we prove that the Stummel class contains weak Morrey spaces under certain conditions. For 1p<∞and 0λn, theweak Morrey space wL^p,λ=wL^p,λ(Rⁿ) is the collection of all Lebesgue measurable functions f onRⁿ which satisfy

f_wLp,λ := sup

x∈Rⁿ,r>0r⁻^λ^pfwL^p(B(x,r))<∞, where

fwL^p(B(x,r)):=sup

t>0t|{y∈B(x,r):|f(y)|>t}|¹^p.

Observe that, by takingλ =0, we can recover the weak Lebesgue spacewL^p. In this paper, we also study the relation between Stummel classes and Lorentz spaces.

Throughout this paper we assume that Ψ:(0,∞)→(0,∞)is a measurable function. Whenever required, we consider the following conditions onΨ:

₁ 0

Ψ(t)

t dt<∞; (1)

1

A₁ Ψ(s)

Ψ(r)A₁ for 1s

r2; (2)

Ψ(r)

rⁿ A₂Ψ(s)

sⁿ for sr, (3)

whereAi>0, i=1,2, are independent ofr,s>0. The condition (2) is known as the doubling conditiononΨ. In some cases, we can weaken the doubling condition by the right doubling condition:

Ψ(s)

Ψ(r)A₃ for 1s

r2, (4)

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whereA₃is independent ofr,s>0.

In this paper, the constant c>0 that appears in the proof of all theorems may vary from line to line, and the notationc=c(α,β,... ,ζ)indicates thatc depends on α,β,...,ζ^.

2. The generalized Stummel classes

We begin this section by defining the generalized Stummel class.

DEFINITION1. For 1p <∞, we define the generalized Stummel p-class S_p,Ψ=S_p,Ψ(Rⁿ)by

S_p,Ψ:=

f ∈L_loc^p (Rⁿ):ηp,Ψf(r)0 for r0 , where

ηp,Ψf(r):=sup

x∈Rⁿ

|x−y|<r

|f(y)|^pΨ(|x−y|)

|x−y|ⁿ dy ¹_p

, r>0.

We call ηp,Ψf the Stummel p-modulusof f. Observe that the Stummel p- modulus is nondecreasing on (0,∞). For p=1, we have S_1,Ψ:=S_Ψ — the generalized Stummel class introduced in [6]. For Ψ(t):=t^α (0<α <n), we write S_p,α instead ofS_p,Ψ and ηp,α instead of ηp,Ψ. Observe that S_1,α :=S_α — the Stummel class introduced in [4,13].

The following two theorems confirm thatηp,Ψf is continuous (hence measurable) and satisfies the doubling condition.

THEOREM1. If f∈S_p,Ψ, thenηp,Ψf is continuous on(0,∞).

Proof. Let{rk} be a sequence in(0,∞)withr_k→r∈(0,∞)andx∈Rⁿ. Choose r_∗>0 such thatr,r_kr_∗ for everyk∈N. Next, for everyk∈N, define

g_k(y):=|f(y)|^pΨ(|y−x|)

|y−x|ⁿ χB(x,rk)(y) and g(y):=|f(y)|^pΨ(|y−x|)

|y−x|ⁿ χB(x,r)(y), fory∈B(x,r∗). We see that{gk}is a sequence of nonnegative measurable functions on B(x,r_∗), and g_k→g almost everywhere on B(x,r_∗). By the Dominated Convergence Theorem we obtain

|y−x|<r∗

g_k(y)dy→

|y−x|<r∗g(y)dy.

Therefore

|y−x|<rk

|f(y)|^pΨ(|y−x|)

|y−x|ⁿ dy ¹_p

→

|y−x|<r

|f(y)|^pΨ(|y−x|)

. (5)

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Letε be any positive real number. By (5), there exists k₀∈Nsuch that for all k∈N withkk₀ we have

|y−x|<r

|f(y)|^pΨ(|y−x|)

−ε<

|y−x|<rk

|f(y)|^pΨ(|y−x|)

<

|y−x|<r

|f(y)|^pΨ(|y−x|)

+ε.

Sincex∈Rⁿis arbitrary, we conclude that

ηp,Ψf(r)−εηp,Ψf(rk)ηp,Ψf(r) +ε.

Thus, we have proved thatηp,Ψf(r_k)→ηp,Ψf(r)for any sequence{r_k}in(0,∞)with r_k→r∈(0,∞). This means that ηp,Ψf is continuous on(0,∞).

THEOREM2. Let Ψ satisfy the condition (3). If f ∈S_p,Ψ, then ηp,Ψf satisfies the doubling condition.

Proof. Letx∈Rⁿandr>0. Choosem=m(n)∈Nandx₁,...,x_m∈B(x,r)such that

B(x,r)⊆ m i=1

B x_i,r

2 . Note that

|y−x|<r

|f(y)|^pΨ(|y−x|)

∑

^m

i=1

|y−xi|<₂^r

|f(y)|^pΨ(|y−x|)

=

∑

^m

i=1

I_i. (6)

Fori=1,...,m, we have

I_i=

|y−xi|<^r₂

|f(y)|^pΨ(|y−x|)

|y−x|>|y−xi|,|y−xi|<^r₂

|f(y)|^pΨ(|y−x|)

+

|y−x||y−xi|<₂^r

|f(y)|^pΨ(|y−x|)

=A_i+B_i. (7)

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By the condition (3) onΨ, we obtain

A_i=

|y−x|>|y−xi|,|y−xi|<₂^r

|f(y)|^pΨ(|y−x|)

c(p)

|y−x|>|y−xi|,|y−xi|<^r₂

|f(y)|^pΨ(|y−x_i|)

|y−x_i|ⁿ dy ¹_p

c(p)

|y−xi|<^r₂

|f(y)|^pΨ(|y−x_i|)

|y−x_i|ⁿ dy ¹_p

c(p)ηp,Ψfr 2 . It is clear that

Bi

|y−x|<^r₂

|f(y)|^pΨ(|y−x|)

ηp,Ψfr 2 . From (6) and (7), we get

|y−x|<r

|f(y)|^pΨ(|y−x|)

m(n)(c(p) +1)ηp,Ψfr 2

=c(n,p)ηp,Ψfr

2 . (8)

Since the inequality (8) holds for all x∈Rⁿ, we obtain ηp,Ψf(r)c(n,p)ηp,Ψfr

2 .

According to the fact that ηp,Ψf is nondecreasing, we conclude that ηp,Ψf satisfies the doubling condition.

As for the classical Stummel class [13], we also have more information about functions inS_p,ψ, as presented in the following theorem and its corollary.

THEOREM3. Let1p<∞,Ψsatisfy the condition (3), andΦ:(0,∞)→(0,∞) be continuous and nondecreasing. If f∈S_p,Ψ and

₁ 0

ηp,Ψf(t)p

[Φ(t)]⁻¹t⁻¹dt<∞,

then there exists a positive constant c(n,p)independent of f such that

|x−y|<r

|f(y)|^pΨ(|x−y|)

|x−y|ⁿΦ(|x−y|)dyc(n,p)

r2

0

ηp,Ψf(t)_p

[Φ(t)]⁻¹t⁻¹dt.

holds for all x∈Rⁿand r>0.

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Proof. Let f ∈S_p,Ψ. By the hypotheses, we observe that for each x∈Rⁿ and r>0, we have

|x−y|<r

|f(y)|^pΨ(|x−y|)

|x−y|ⁿΦ(|x−y|)dy=

∑

^∞

k=0

r

2k+1|x−y|<₂^r_k

|f(y)|^pΨ(|x−y|)

|x−y|ⁿΦ(|x−y|)dy

∑

^∞

k=0

Φ r 2^k+1

−1

|x−y|<₂^r_k

|f(y)|^pΨ(|x−y|)

|x−y|ⁿ dy

∑

^∞

k=0

Φ r 2^k+1

−1

ηp,Ψfr 2^k

p. (9)

Combining the inequality (9) and Theorem2, we get

|x−y|<r

|f(y)|^pΨ(|x−y|)

|x−y|ⁿΦ(|x−y|)dyc(n,p)

∑

^∞

k=0

Φ r 2^k+2

−1

ηp,Ψf r 2^k+2

p

c(n,p)

∑

^∞

k=0

^r

2k+1 r 2k+2

ηp,Ψf(t)_p

[Φ(t)]⁻¹t⁻¹dt

=c(n,p)

r2

0

ηp,Ψf(t)_p

[Φ(t)]⁻¹t⁻¹dt, which is the desired inequality.

REMARK1. If we put Θ(r):=

2r

0

ηp,Ψf(t)_p

[Φ(t)]⁻¹t⁻¹dt, r>0, then we see thatΘ(r)is nondecreasing and that lim

r→0⁺Θ(r) =0.

For f ∈S_p,Ψ, the function ηp,Ψf is continuous according to Theorem1. More- over, it is nondecreasing and lim

t→0⁺ηp,Ψf(t) =0. Accordingly, we have the next corollary which generalizes the result in [13, p. 58].

COROLLARY1. Let1p<∞. If f ∈S_p,Ψand

₁

0

ηp,Ψf(t)_p−ϑ

t⁻¹dt<∞,

for someϑ∈(0,1), then there exists a positive constant c(n,p)independent of f such that for each x∈Rⁿand r>0, we have

|x−y|<r

|f(y)|^pΨ(|x−y|)

|x−y|ⁿ

ηp,Ψf(|x−y|)_ϑ dyc(n,p)Θ(r), whereΘ(r):=₀^r²

ηp,Ψf(t)_p−ϑ t⁻¹dt .

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3. Inclusion between generalized Stummel classes

In this section, we are going to investigate the inclusion between two Stummel classes. Thefirst theorem discusses the relationship between Stummel classes with different parametersΨ. (Unless otherwise stated, we always assume that 1p<∞.) THEOREM4. Suppose thatΨ2satisfies the condition (3) and that there exist c>

0andδ>0 such thatΨ2(t)cΨ1(t)for every t∈(0,δ). Then Sp,Ψ1⊆S_p,Ψ₂. Proof. Let f ∈S_p,Ψ₁,x∈Rⁿ, andr>0. Forrδ^{, we have}

|y−x|<r

|f(y)|^pΨ2(|y−x|)

c¹^p

|y−x|<r

|f(y)|^pΨ1(|y−x|)

, whenceηp,Ψ2f(r)c¹^pηp,Ψ1f(r)0 forr0. Hence f ∈S_p,Ψ₂.

As an immediate consequence of Theorem4, we have the following corollary.

COROLLARY2. If0<αβ<n, then S_p,α⊆S_p,β.

REMARK2. For 0<α<β <n, the above inclusion is proper. Indeed, for 0<

β<n, define f :Rⁿ−→Rby the formula f(y):=

χB(y)

|y|^β|ln|y||² ¹_p

, y∈Rⁿ,

whereB:=B(0,e⁻^β²). Then f ∈S_p,β\S_p,α whenever 0<α<β. The fact that f ∈ S_p,β is proved in general in Example1. We will show here that f ∈/ S_p,α. Let 0<

r<e⁻^β². Using polar coordinates and the fact that 1/(t^β|ln(t)|²) is decreasing on (0,e⁻²^β), we have

η_p,β^f(r)_p

|y|<r

1

|y|^β|ln|y||²|y|^n−αdy

1

r^β|ln(r)|²

|y|<r

1

|y|^n−αdy

=c(n,α) 1 r^β−α|ln(r)|². Therefore

ηp,βf(r)c(n,α,p)

1 r^β^−α|ln(r)|²

¹_p

→∞, forr0. This verifies that f ∈/S_p,α.

The next theorem shows the relationship between two Stummel classes with different parametersp.

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THEOREM5. If1p₂p₁<∞andΨsatisfies (1), then S_p₁_,Ψ⊆S_p₂_,Ψ. Proof. Let f ∈S_p₁_,Ψ, x∈Rⁿ, and 0<r1. Then by H¨older’s inequality we have

|y−x|<r

|f(y)|^p²Ψ(|y−x|)

|y−x|ⁿ dy

|y−x|<r

|f(y)|^p¹Ψ(|y−x|)

|y−x|ⁿ dy ^p_p²

1

×

|y−x|<r

Ψ(|y−x|)

|y−x|ⁿ dy ₁₋^p_p²

1

=

|y−x|<r

|f(y)|^p¹Ψ(|y−x|)

|y−x|ⁿ dy ^p_p²

1

×

c(n) ^r

0

Ψ(t) t dt

₁₋^p_p²

1.

Therefore

ηp2,Ψf(r)c(n,p₁,p₂)ηp1,Ψf(r) _r

0

Ψ(t) t dt

_p¹

2−_p¹₁

0 for r0, which tells us that f ∈S_p₂_,Ψ. We conclude thatS_p₁_,Ψ⊆S_p₂_,Ψ.

As a consequence of Theorem5, we have the following corollary.

COROLLARY3. If1p2p1<∞, then Sp1,α⊆Sp2,α.

REMARK3. For 1p₂<p₁<∞, the above inclusion is proper. Indeed, for

pα1 <γ<min{_p^α₂,_pⁿ₁}, we have f(y):=|y|^−γ∈S_p₂_,α\Sp1,α. From Theorem4and Theorem5, we get the following corollary.

COROLLARY4. Suppose that 1p2p1<∞, Ψ2 satisfies the conditions (1) and (3), and there exist c>0 andδ>0 such thatΨ2(t)cΨ1(t)for every t∈(0,δ). Then S_p₁_,Ψ₁⊆S_p₂_,Ψ₂.

4. Inclusion between Stummel classes and Morrey spaces

Our next theorem gives an inclusion relation between generalized Morrey spaces and generalized Stummel classes. We also give an example of a function that belongs to the generalized Stummel class but not to the generalized Morrey space.

THEOREM6. Let 1p₂p₁<∞. Assume that Ψ1 satisfies(2)and thatΨ2

satisfies the right-doubling condition(4). If

₁

0

Ψ1(t)^p²Ψ2(t)

t dt<∞, (10)

then L^p¹^,Ψ¹ ⊆S_p₂_,Ψ₂.

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REMARK4. Let p₁=p₂=1,Ψ1(t):=t^λ⁻ⁿwhere 0λn, andΨ2(t):=t^α wheren−λ<α<n. Then the above theorem reduces to the result in [13, p. 56].

Proof of Theorem6. Let f ∈L^p¹^,Ψ¹,x∈Rⁿ, andr>0. Since Ψ2 satisfies (4), we have

|x−y|<r

|f(y)|^p²Ψ2(|x−y|)

|x−y|ⁿ dy=

∑

⁻¹

k=−∞

2^kr|x−y|<2^k+1r

|f(y)|^p²Ψ2(|x−y|)

|x−y|ⁿ dy c ⁻¹

∑

k=−∞

Ψ2(2^kr)

|B(x,2^k+1r)|

B(x,2^k+1r)|f(y)|^p² dy.

Combining the last inequality and H¨older’s inequality, we get

|x−y|<r

|f(y)|^p²Ψ2(|x−y|)

|x−y|ⁿ dyc ⁻¹

∑

k=−∞

Ψ2(2^kr)

|B(x,2^k+1r)|^p²^/p¹f_L^p²p1(B(x,2^k⁺¹r))

cf^p_L²p1,Ψ1

−1

∑

k=−∞Ψ1(2^k+1r)^p²Ψ2(2^kr). (11) Using (4) and the monotonicity of Ψ1, we get

∑

−1 k=−∞

Ψ1(2^k+1r)^p²Ψ2(2^kr)c ⁻¹

∑

k=−∞

₂^k_r 2^k⁻¹r

Ψ1(t)^p²Ψ2(t)

t dt

=c _r/2

0

Ψ1(t)^p²Ψ2(t)

t dt. (12)

We combine (11) and (12) to obtain ηp₂,Ψ2f(r)c

_r/2

0

Ψ1(t)^p²Ψ2(t)

t dt

_p¹

2 f_L^p1,Ψ1. (13)

Since ₁

0

Ψ1(t)^p²Ψ2(t)

t dt<∞, we see that lim

r→0⁺ _r/2

0

Ψ1(t)^p²Ψ2(t)

t dt=0. This fact and (13) imply lim

r→0⁺ηp2,Ψ2f(r) =0. Hence, f ∈S_p₂_,Ψ₂. This shows that L^p¹^,Ψ¹ ⊆ S_p₂_,Ψ₂.

The following example shows that the inclusion in Theorem6is proper.

EXAMPLE1. Let 1p₂p₁<∞, Ψ2 satisfy the condition (4), Ψ2(t)|ln(t)|² be nondecreasing on(0,δ)for someδ >0, andΨ1(r)^p²Ψ2(r)|ln(r)|²0 asr0.

Define f:Rⁿ−→Rby the formula f(y):=

χB(y) Ψ2(|y|)|ln|y||²

_p¹

2, y∈Rⁿ,

whereB:=B(0,δ). Then f ∈S_p₂_,Ψ₂\L^p¹^,Ψ¹.

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First we show that f ∈S_p₂_,Ψ₂. Let 0<r<min{δ,1}. Since the function f is radial and nonincreasing, the supremum in the Stummel modulus is attained at the origin, so that

ηp2,Ψ2f(r) =

|y|<r

|f(y)|^p²Ψ2(|y|)

|y|ⁿ dy _p¹

2 =

|y|<r

1

|ln|y||²|y|ⁿ dy _p¹

2 .

Converting to polar coordinates, we get

|y|<r

1

|ln|y||²|y|ⁿ dy=c _r

0

1

s(lns)²ds=− c lnr. Therefore,

ηp2,Ψ2f(r) =c

− 1 lnr

_p¹

2 .

Since lim

r→0⁺

lnr1 =0, we conclude that ηp2,Ψ2f(r)0 for r0. This proves that f∈S_p₂_,Ψ₂.

Now, we will show that f ∈/L^p¹^,Ψ¹. Let 0<r<δ^{. Since} Ψ2(t)|ln(t)|² is nondecreasing on(0,r)⊆(0,δ), we have

1 Ψ1(r)^p¹

1

|B(0,r)|

B(0,r)|f(y)|^p¹dy= 1 Ψ1(r)^p¹

1

|B(0,r)|

B(0,r)

1 Ψ2(|y|)|ln|y||²

^p_p¹

2 dy

1

Ψ1(r)^p¹|B(0,r)|

1 Ψ2(r)|lnr|²

^p_p¹

2

B(0,r)dy

=

1

Ψ1(r)^p²Ψ2(r)|lnr|² ^p_p¹

2 .

Note thatΨ1(r)^p²Ψ2(r)|lnr|²0 asr0. Then 1

Ψ1(r)^p²Ψ2(r)|lnr|² ^p_p¹

2 →∞ for r0.

We conclude that f ∈/L^p¹^,Ψ¹.

REMARK5. Let 1p₂p₁<∞,Ψ1(t):=t^λ^p⁻ⁿ¹ where 0λn, andΨ2(t):=

t^α where ^(n−λ_p₁^)p² <α<n. It can be shown that Ψ1 andΨ2 satisfy all conditions in Theorem6and Example1.

As a counterpart of Theorem6, we have the following result.

THEOREM7. Let1p₂p₁<∞and assume thatΨ1satisfies(3). If f∈S_p₁_,Ψ₁ and

ηp1,Ψ1f(r)cΨ1(r)^p¹¹Ψ2(r) (14) for someΨ2and for every r>0, then f ∈L^p²^,Ψ².

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Proof. Leta∈Rⁿandr>0. Then, by H¨older’s inequality, we have

B(a,r)|f(x)|^p² dxcrⁿ

1−^p_p²₁

B(a,r)|f(x)|^p¹ dx ^p_p²

1

= crⁿ Ψ1(r)^p^p²¹

B(a,r)

|f(x)|^p¹Ψ1(r)

rⁿ dx

^p_p²

1.

We combine (3), (14), and Definition1to obtain

B(a,r)|f(x)|^p²dx crⁿ Ψ1(r)^p^p²¹

B(a,r)

|f(x)|^p¹Ψ1(|x−a|)

|x−a|ⁿ dx ^p_p²

1

crⁿ

Ψ1(r)^p^p²¹ [ηp1,Ψ1f(r)]^p² crⁿΨ2(r)^p². Consequently,

1

|B(a,r)|Ψ2(r)

B(a,r)|f(x)|^p² dx _p¹

2 c.

Sincea andrare arbitrary, we conclude that f ∈L^p²^,Ψ².

Taking Ψ1(t):=t^α and Ψ2(t):=t^p^σ²⁻^p^α¹ where 0<α<n, 1p₂p₁<∞, and 0<σ<^α_p^p₁², we get the following corollary.

COROLLARY5. Let1p₂p₁<∞and0<α<n. If f∈S_p₁_,α andηp1,αf(r) cr^p^σ² for some0<σ<^α_p^p₁² and for every r>0, then f ∈L^p²^,n+σ−^α^p^p¹².

Next, we are going to investigate the relation between generalized Stummel classes and generalized weak Morrey spaces. The generalized weak Morrey spaces are defined as follows.

DEFINITION2. Let 1p<∞andΨ:(0,∞)→(0,∞). Thegeneralized weak Morrey spacewL^p,Ψ=wL^p,Ψ(Rⁿ)is defined to be the set of all measurable functions

f for which

f_wL^p,Ψ:= sup

a∈Rⁿ,r>0,t>0

t|{x∈B(a,r):|f(x)|>t}|^1/p Ψ(r)|B(a,r)|^1/p <∞

The inclusion between generalized Stummel classes and generalized weak Morrey spaces is given in the following theorems.

THEOREM8. Let 1p₂<p₁<∞. Assume that Ψ1 satisfies(2)and thatΨ2

satisfies(4). If

1 0

Ψ1(t)^p²Ψ2(t)

t dt<∞, then wL^p¹^,Ψ¹⊆S_p₂_,Ψ₂.

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Proof. Since p₂<p₁, by virtue of [9, Theorem 5.1], we havewL^p¹^,Ψ¹⊆L^p²^,Ψ¹. By Theorem6, we haveL^p²^,Ψ¹ ⊆S_p₂_,Ψ₂. It thus follows thatwL^p¹^,Ψ¹ ⊆S_p₂_,Ψ₂.

THEOREM9. Let1p₁p₂<∞and assume thatΨ1satisfies(3). If f∈S_p₁_,Ψ₁ and the inequality(14)holds for someΨ2and for every r>0, then f∈wL^p²^,Ψ².

Proof. The assertion follows from Theorem7and the inclusionL^p²^,Ψ²⊆wL^p²^,Ψ².

For the classical weak Morrey spaces and Stummel classes, we have the following result.

THEOREM10. For 1p₂<p₁<∞, if 0λ <n and ^(n−λ)p_p ²

1 <α<n, then wL^p¹^,λ⊆Sp2,α. Conversely, for1p<∞, if f∈S_p,α for0<α<n andηp,αf(r) cr^σ^p for someσ>0, then f ∈wL^p,n−α+σ.

Proof. Thefirst assertion follows from Theorem8by taking Ψ1(t):=t^λ^p⁻¹ⁿ, and Ψ2(t):=t^α where 0λ<n and ^(n−λ)p_p₁ ² <α<n. The second part is a consequence of Corollary5when p₁=p₂=p and the inclusionL^p,n−α+σ⊆wL^p,n−α+σ.

REMARK6. The second part of Theorem10generalizes the result in [13, p. 57].

For the case p=1, thefirst part of Theorem10does not generally hold. To see this, consider the function f(y):=|y|⁻ⁿ,y∈Rⁿ. Then f ∈wL^1,λ for 0λ<n, but f∈/S_α forn−λ<α<n.

5. Further results

In this section, we study the relation between bounded Stummel modulus classes S˜_p,α and Stummel classes. We also study the inclusion between ˜S_p,α and Lorentz spaces. For 0<α<n and 1p<∞, recall the definition of the Stummel modulus

ηp,αf(r):=sup

x∈Rⁿ

|x−y|<r

|f(y)|^p

|x−y|^n−αdy ¹_p

, r>0.

DEFINITION3. For 0<α<n and 1p<∞, we definethe bounded Stummel modulus classS˜_p,α=S˜_p,α(Rⁿ)by

S˜_p,α:=

f ∈L_loc^p (Rⁿ):ηp,αf(r)<∞ for all r>0 .

Note that the inclusions similar to Corollary2and Corollary3also hold for ˜S_p,α. Moreover, we haveS_p,α⊆S˜_p,α. This inclusion is proper due to the following example which we adapt from [1, p. 250–251].

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EXAMPLE2. Let 0<α<n and 1p<∞. For every k∈N with k3, let x_k:= (2^−k,0,...,0)∈Rⁿ and

V_k(y):=

8^α^k, y∈B(x_k,8^−k), 0, y∈/B(xk,8^−k).

DefineV:Rⁿ→Rby the formula V(y):=

∞ k=3

∑

V_k(y) ¹_p

. Since

B(x,r)|V(y)|^pdy=

∑

^∞

k=3

B(x,r)|V_k(y)|dyc(n)

∑

^∞

k=3

8^(α−n)k<∞ for everyx∈Rⁿandr>0 where c(n):=|B(0,1)|, we obtainV∈L^p_loc(Rⁿ).

We will show thatV∈S˜_p,α. Let ρk(x):=

Rⁿ

|Vk(y)|

|x−y|^n−αdy=8^αk

|y−xk|<8⁻^k

1

|x−y|^n−αdy, x∈Rⁿ. There are two cases: (i)|x−x_k|2^−2k+1, or, (ii) |x−x_k|<2^−2k+1.

Suppose that the case (i) holds, that is, |x−x_k|2^−2k+1. We have,

ρk(x)c(n)2^(α−n)k. (15)

For the case (ii)|x−x_k|<2^−2k+1, we have

ρk(x)c(n,α) (16)

wherec(n,α):=max{c(n),³_α^αc(n)}.

Given x∈Rⁿ, we havex∈/B(x_k,2^−2k+1) for allk3, orx∈B(x_j,2^−2j+1)for some j3. Assume thatx∈/B(x_k,2^−2k+1)for all k3. Hence, from (15), we have

Rⁿ

|V(y)|^p

|x−y|^n−αdy

∑

^∞

k=3ρk(x)c(n)

∑

^∞

k=3

2^(α−n)k<∞. (17) Now assume thatx∈B(x_j,2⁻²^j+1)for some j3. Since

B(x_k,2^−2k+1)

k3is a dis- joint collection, wefind that there is only one j∈N, j3, such thatx∈B(xj,2^−2j+1).

Using (15) and (16), we get

Rⁿ

|V(y)|^p

|x−y|^n−αdyc(n,α) +

∑

^∞

k=3k =j

ρk(x) (18)

c(n,α) +c(n)

∑

^∞

k=3k =j

2^(α−n)k<∞.