PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 127–138 DOI: 10.2298/PIM1206127G

GENERALIZED STUMMEL CLASS AND MORREY SPACES

Hendra Gunawan, Eiichi Nakai, Yoshihiro Sawano, and Hitoshi Tanaka

Communicated by Stevan Pilipović

Abstract. We revisit the Stummel class and its relation with Morrey spaces. We reformulate a result of Ragusa and Zamboni [11 ] and then discuss its generalization, as proposed by Eridani and Gunawan [4]. An improvement of the results previously obtained by Eridani and Gunawan is obtained and some extensions are presented.

1. Introduction

In 1971, Adams [1] studied traces of potential arising from translation invariant operators and proved the following inequality

(1.1)  ∥u · V^{1/p} : L^{p}∥ ⩽ C ∥V : L^{1,λ} ∥^{1/p} ∥∇u : L^{α} ∥

for u ∈ C_{0}^{∞} (ℝ^{d}), p = \frac{αλ}{d−α} , λ > d−α, 1 < α < d. Here V is a nonnegative function in the Morrey space L^{1,λ} = L^{1,λ} (ℝ^{d}), which we shall define below. Adams’ inequality looks like (but not the same as) Olsen’s inequality [8]

∥u · W : L^{p,λ} ∥ ⩽ C ∥W : L^{(d−λ)/α,λ} ∥ ∥(−Δ)^{α/2} u : L^{p,λ} ∥

for 1 < p < \frac{d}{α}, 0 ⩽ λ < d − αp. It is well known that if f := (−Δ)^{α/2} u ∈ L^{p,λ}, then u = (−Δ)^{−α/2} f ∈ L^{q,λ}, where \frac{1}{q} = \frac{1}{p} − \frac{α}{d−λ} [2]. Related works may also be found in [3, 12, 15].

2010 Mathematics Subject Classification: Primary 46E30; Secondary 43A15, 26A33, 35B45, 35B65.

Key words and phrases: Morrey spaces, Stummel class.

The research was carried out as part of the GCOE Program 2011 at the Department of Mathematics, Kyoto University. The second author was partly supported by Grant-in-Aid for Scientific Research (C) No. 20540167, Japan Society for the Promotion of Science. The third author was supported partially by Grant-in-Aid for Young Scientists (B) No. 21740104, Japan Society for the Promotion of Science. The last author is supported by the GCOE program at Graduate School of Mathematical Sciences, the University of Tokyo, and was supported by Fūjyukai foundation.

For 1 ⩽ p < ∞ and 0 ⩽ λ ⩽ d, the Morrey space L^{p,λ} consists of locally integrable functions f for which

∥f : L^{p,λ} ∥ := sup_{x∈ℝ^{d} , r>0} ( \frac{1}{r^{λ}} ∫_{|x−y|<r} |f(y)|^{p} dy) ^{1/p} < ∞.

Note that L^{p,0} = L^{p}, the usual Lebesgue space. For historical background of Morrey spaces, see [6].

As we learn from the definition of L^{p,λ}, the parameter p describes the local integrability, while λ seems to measure the global integrability. Hence L^{1,λ} may be used to describe the global integrability without taking into account the local integrability. The aspect can be seen from the fact that L^{1,λ} supplements L^{p,λ} in the sharp maximal inequality [13, Theorem 1.1]. The space L^{1,λ} is a function space which is difficult to grasp. For example, unlike L^{p,λ} with p > 1, it is not the case that we can characterize L^{1,λ} in terms of the Littlewood–Paley decomposition. This is because the singular integral operators like the Riesz transforms are not bounded on L^{1,λ} . Nevertheless, this space can be compared with other function spaces. This is what we do in the present paper.

Our entry point is the work of Ragusa and Zamboni [11], which offers a similar imbedding to (1.1) by assuming more hypotheses on the function V. For 0 < α < d, they define the Stummel modulus of f ∈ L_{loc}^{1} = L_{loc}^{1}(ℝ^{d}), denoted by η_{α} f, by

η_{α} f(r) := sup_{x∈ℝ^{d}} ∫_{|x−y|<r} \frac{|f(y)|}{|x − y|^{d−α}} dy, r > 0.

They then define the Stummel class S_{α} by

S_{α} := {f ∈ L_{loc}^{1} : lim_{r→0} η_{α} f(r) = 0}.

Note that the definitions also make sense for α = d. For α = 2, S_{α} is known as the Stummel–Kato class. Ragusa and Zamboni obtain the following relation between the Stummel class S_{α} and the Morrey space L^{1,λ} .

Theorem 1.1. If f belongs to L^{1,λ} , d − α < λ < d, then f belongs to S_{α} with
η_{α} f(r) ⩽ C r^{λ−d+α} ∥f : L^{1,λ} ∥, r > 0.

Conversely, if f belongs to S_{α} and η_{α} f(r) ∼ r^{β}, then f belongs to L^{1,d−α+β} .

The second part of the theorem tells us that if η_{α} f(r) behaves like a power of r, then the improvement of the integrability of u in Adams’ inequality can be seen either in terms of V in L^{1,λ} or its local Riesz potential given by η_{α} f(r). From another point of view, the above theorem provides a characterization of the Morrey space L^{1,λ} in terms of the Stummel modulus.

In this paper, we discuss a generalized version of the Stummel class and its relation with generalized Morrey spaces. Our aim is to reprove and improve the results of Eridani and Gunawan [4] by using weaker assumptions. (At the same time, an analogous result in the nonhomogeneous setting is obtained by Setya-Budhi et al. [14].) Moreover, we introduce two (generalized) Stummel classes with variable growth condition and with nonradially symmetric condition, and establish their relation with generalized Morrey spaces.

Throughout the paper, the letter C denotes a positive constant, which may vary from line to line.

2. Preliminaries

2.1. Definitions. For ψ : (0, ∞) → (0, ∞) with ∫_{0}^{1} t^{d−1} ψ(t) dt < ∞, we define

the Stummel modulus of f ∈ L_{loc}^{1} by

η_{ψ} f(r) := sup_{x∈ℝ^{d}} ∫_{|x−y|<r} |f(y)| ψ(|x − y|) dy, r > 0.

Accordingly, we define the Stummel class S_{ψ} by

S_{ψ} := {f ∈ L_{loc}^{1} : lim_{r→0} η_{ψ} f(r) = 0}.

Just like η_{α} f, we see that η_{ψ} f(r) is a nondecreasing function of r. Moreover, we note that if ψ(t) = t^{α−d} , 0 < α ⩽ d, then η_{ψ} f = η_{α} f and S_{ψ} = S_{α}. One may also observe that for 0 < α ⩽ d, we have S_{α} ⊆ S_{ψ} provided that ψ(t) ⩽ Ct^{α−d} for some positive constant C. In general, the condition ∫_{0}^{1} t^{d−1} ψ(t) dt < ∞ guarantees that S_{ψ} contains all locally bounded functions on ℝ^{d}.

Along with the generalized Stummel class, we also have the generalized Morrey spaces (as in [5, 7]). For 1 ⩽ p < ∞ and a suitable function ϕ : (0, ∞) → (0, ∞), the generalized Morrey space M^{p,ϕ} := M^{p,ϕ} (ℝ^{d}) is defined to be the space of all functions f ∈ L_{loc}^{p} for which

∥f : M^{p,ϕ} ∥ := sup_{B=B(x,r)} \frac{1}{ϕ(r)} ( \frac{1}{|B|} ∫_{B} |f(y)|^{p}dy) ^{1/p} < ∞.

Here |B| denotes the usual Lebesgue measure of B = B(x, r), which is a constant times r^{d}. We notice that if ϕ(t) = t^{(λ−d)/p} , 0 ⩽ λ ⩽ d, then M^{p,ϕ} = L^{p,λ}.

2.2. Assumptions. We say that a function φ : (0, ∞) → (0, ∞) satisfies the doubling condition if there exists a constant C > 0 such that

1 ⩽ \frac{r}{s} ⩽ 2 ⇒ \frac{1}{C} ⩽ \frac{φ(r)}{φ(s)} ⩽ C.

For example, for any a ∈ ℝ, the function φ(t) = t^{a} satisfies the doubling condition. It is not hard to see that if φ_{1} and φ_{2} satisfy the doubling condition, so do their product and quotient.

Observe that if φ satisfies the doubling condition, then we have

∫_{R}^{2R} \frac{φ(t)}{t} dt ∼ φ(R),

that is, there exists a constant C > 0 such that

\frac{1}{C} φ(R) ⩽ ∫_{R}^{2R} \frac{φ(t)}{t} dt ⩽ C φ(R), for every R > 0 .

Now, the doubling condition can be decomposed into two conditions, namely the left-doubling condition 1 ⩽ \frac{r}{s} ⩽ 2 ⇒ \frac{1}{C} ⩽ \frac{φ(r)}{φ(s)} and the right-doubling condition

1 ⩽ \frac{r}{s} ⩽ 2 ⇒ \frac{φ(r)}{φ(s)} ⩽ C.

If φ satisfies the right-doubling condition, then we have

\frac{1}{C} φ(2R) ⩽ ∫_{R}^{2R} \frac{φ(t)}{t} dt ⩽ C φ(R) for every R > 0 .

Meanwhile, if φ satisfies the left-doubling condition, then we have

\frac{1}{C} φ(R) ⩽ ∫_{R}^{2R} \frac{φ(t)}{t} dt ⩽ C φ(2R) for every R > 0 .

Note also that φ satisfies the right-doubling condition if and only if 1/φ satisfies the left-doubling condition.

Example 2.1. Consider the function φ_{1}(t) = e^{−t} , t ∈ (0, ∞). This function satisfies the right-doubling condition but fails to satisfy the left-doubling condition (and hence it does not satisfy the doubling condition). Meanwhile, the function φ_{2}(t) = e^{t}, t ∈ (0, ∞), satisfies the left-doubling condition but not the rightdoubling condition. Consequently, for any a ∈ ℝ, the function φ_{3}(t) = t^{a}e^{−t} satisfies the right-doubling condition only, while the function φ_{4}(t) = t^{a}e^{t} satisfies the left-doubling condition only.

Example 2.2. Let γ < β < 0. Define r_{1} := \frac{1}{2} and r_{j+1} := r_{j}^{γ/β} for j = 1, 2, . . ..

Then r_{j} → 0 as j → ∞. Now define the function ψ on (0, ∞) by

ψ(r) := \{ \begin{matrix}r_{j}^{γ}, & if r_{j+1} < r ⩽ r_{j}; \\ r^{γ}, & if r_{1} < r. \end{matrix}

Then r^{β} < ψ(r) ⩽ r^{γ} for 0 < r < 1. Since ψ is nonincreasing, ψ satisfies the right-doubling condition. But

\frac{ψ(r_{j} +)}{ψ(r_{j})} = \frac{r_{j−1}^{γ}}{r_{j}^{γ}} = \frac{r_{j}^{β}}{r_{j}^{γ}} = r_{j}^{β−γ} → 0, as j → ∞.

Hence ψ does not satisfy the left-doubling condition. If in the definition above we let 0 < β < γ, then the resulting function ψ satisfies the left-doubling condition but not the right-doubling condition. Moreover, the function r^{−d} ψ(r) has the same property as ψ.

In general, nonincreasing functions satisfy the right-doubling condition, while nondecreasing functions satisfy the left-doubling condition. The function ϕ in the definition of the Morrey space M^{p,ϕ} is assumed to be nonincreasing, and hence it satisfies the right-doubling condition. In addition, t^{d/p}ϕ(t) is assumed to be nondecreasing, so that — after division by t^{d/p} — the function ϕ must satisfy the left-doubling condition. Thus ϕ here satisfies the doubling condition.

In [4], the function ψ in the definition of the Stummel class S_{ψ} is also assumed to satisfy the doubling condition. We find, however, that this is not necessary: we can obtain the same result as in [4] by assuming that ψ satisfies the right-doubling condition only or the left-doubling condition only. This, of course, enlarges the coverage of the functions ψ in the definition of the Stummel class S_{ψ} .

3. Main Results

Our first result below shows the inclusion of the generalized Morrey space M^{1,ϕ} in the Stummel class S_{ψ} , under some conditions on ψ.

Theorem 3.1. Suppose that ∫_{0}^{1} t^{d−1} ψ(t)ϕ(t) dt < ∞. Then M^{1,ϕ} ⊆ S_{ψ} pro-

vided that ψ satisfies the right-doubling condition or the left-doubling condition.

Proof. The method used is the same as that employed in the proof of Theorem 1 in [10]. Let f ∈ M^{1,ϕ} . Suppose first that ψ satisfies the right-doubling condition. For x ∈ ℝ^{d} and r > 0, we have

∫_{|x−y|<r} |f(y)|ψ(|x − y|) dy = ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)| ψ(|x − y|) dy

⩽ C ∑_{j=−∞}^{−1} ψ(2^{j} r) ∫_{|x−y|<2^{j+1} r} |f(y)| dy

⩽ C ∑_{j=−∞}^{−1} (2^{j+1} r)^{d}ψ(2^{j} r) ϕ(2^{j+1} r)∥f : M^{1,ϕ} ∥.

Recall that ϕ is assumed to satisfy the doubling condition. Thus, we have

∫_{|x−y|<r} |f(y)| ψ(|x − y|) dy ⩽ C ∥f : M^{1,ϕ} ∥ ∑_{j=−∞}^{−1} (2^{j} r)^{d}ψ(2^{j} r) ϕ(2^{j} r)

⩽ C ∥f : M^{1,ϕ} ∥ ∑_{j=−∞}^{−1} ∫_{2^{j−1} r}^{2^{j} r} t^{d−1} ψ(t) ϕ(t) dt

= ∥f : M^{1,ϕ} ∥ ∫_{0}^{r/2} t^{d−1} ψ(t) ϕ(t) dt.

The last inequality implies that

η_{ψ} f(r) ⩽ C ∥f : M^{1,ϕ} ∥ ∫_{0}^{r/2} t^{d−1} ψ(t) ϕ(t) dt.

Since ∫_{0}^{1} t^{d−1} ψ(t) ϕ(t) dt < ∞, we have lim_{r→0} ∫_{0}^{r/2} t^{d−1} ψ(t)ϕ(t) dt = 0. Hence we find that lim_{r→0} η_{ψ} f(r) = 0, that is, f ∈ S_{ψ} .

Suppose now that ψ satisfies the left-doubling condition. Then, for x ∈ ℝ^{d} and r > 0, we have

∫_{|x−y|<r} |f(y)|ψ(|x − y|) dy = ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)|ψ(|x − y|) dy

⩽ C ∑_{j=−∞}^{−1} ψ(2^{j+1} r) ∫_{|x−y|<2^{j+1} r} |f(y)| dy

⩽ C ∑_{j=−∞}^{−1} (2^{j+1} r)^{d}ψ(2^{j+1} r)ϕ(2^{j+1} r)∥f : M^{1,ϕ} ∥

⩽ C ∥f : M^{1,ϕ} ∥ ∑_{j=−∞}^{−1} ∫_{2^{j+1} r}^{2^{j+2} r} t^{d−1} ψ(t)ϕ(t) dt

= C ∥f : M^{1,ϕ} ∥ ∫_{0}^{2r} t^{d−1} ψ(t)ϕ(t) dt.

The last inequality implies that

η_{ψ} f(r) ⩽ C ∥f : M^{1,ϕ} ∥ ∫_{0}^{2r} t^{d−1} ψ(t)ϕ(t) dt.

But lim_{r→0} ∫_{0}^{2r} t^{d−1} ψ(t)ϕ(t) dt = 0 gives lim_{r→0} η_{ψ} f(r) = 0, so that f ∈ S_{ψ} . □

Corollary 3.1. Suppose that for some ϵ > 0, the function t^{d−ϵ} ψ(t)ϕ(t) is almost increasing, that is, there exists a constant C > 0 such that

r ⩽ s ⇒ r^{d−ϵ} ψ(r)ϕ(r) ⩽ Cs^{d−ϵ} ψ(s)ϕ(s).

If f ∈ M^{1,ϕ} , then f ∈ S_{ψ} with

(i) η_{ψ} f(r) ⩽ C r^{d}ψ( \frac{r}{2})ϕ( \frac{r}{2}) for every r > 0, provided that ψ satisfies the right-doubling condition, or

(ii) η_{ψ} f(r) ⩽ C r^{d}ψ(2r)ϕ(2r) for every r > 0, provided that ψ satisfies the left-doubling condition.

Remark. The constant C above depends on d, ϵ, f, ψ and ϕ, but not on r.

Proof. If ψ satisfies the right-doubling condition, then we have already shown that there exists a constant C > 0 such that for every r > 0,

η_{ψ} f(r) ⩽ C ∫_{0}^{r/2} t^{d−1} ψ(t)ϕ(t) dt.

Since t^{d−ϵ} ψ(t)ϕ(t) is almost increasing, we have

∫_{0}^{r/2} t^{d−1} ψ(t)ϕ(t) dt = ∫_{0}^{r/2} t^{d−ϵ} ψ(t)ϕ(t)t^{ϵ−1} dt
⩽ Cr^{d−ϵ} ψ( \frac{r}{2})ϕ( \frac{r}{2}) ∫_{0}^{r/2} t^{ϵ−1} dt = C r^{d}ψ( \frac{r}{2})ϕ( \frac{r}{2}).

Hence η_{ψ} f(r) ⩽ C r^{d}ψ( \frac{r}{2})ϕ( \frac{r}{2}) for every r > 0. In particular, η_{ψ} f(r) ⩽ C r^{ϵ} for 0 < r < 1, so that f ∈ S_{ψ} . The second estimate for η_{ψ} f is obtained in a similar way when ψ satisfies the left-doubling condition. □

In [11], the converse of the above theorem for ψ(t) = t^{α−d} , 0 < α ⩽ d, is obtained by first showing that the Stummel modulus η_{α} f satisfies the rightdoubling condition. As we show in our next theorem, it is not necessary to do so, since we know that the Stummel modulus is nondecreasing.

Theorem 3.2. If f ∈ S_{ψ} , then f ∈ M^{1,ϕ} provided that

(i) ψ satisfies the right-doubling condition and ∫_{0}^{r} \frac{η_{ψ} f(t)}{tψ(t)} dt ⩽ Cr^{d}ϕ(r) for

every r > 0, or

(ii) ψ satisfies the left-doubling condition and ∫_{0}^{r} \frac{η_{ψ} f(4t)}{tψ(t)} dt ⩽ Cr^{d}ϕ(r) for

every r > 0.

Proof. Suppose that ψ satisfies the right-doubling condition. For a given ball B := B(x, r), where x ∈ ℝ^{d} and r > 0, we have

∫_{|x−y|<r} |f(y)| dy = ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)| dy

⩽ C ∑_{j=−∞}^{−1} \frac{1}{ψ(2^{j+1} r)} ∫_{|x−y|<2^{j+1} r} |f(y)| ψ(|x − y|) dy

⩽ C ∑_{j=−∞}^{−1} \frac{η_{ψ} f(2^{j+1} r)}{ψ(2^{j+1} r)} .

Now \frac{1}{ψ(2^{j+1} r)} ⩽ \frac{C}{ψ(t)} for 2^{j+1} r ⩽ t ⩽ 2^{j+2} r. Meanwhile, η_{ψ} f is nondecreasing, so that η_{ψ} f(2^{j+1} r) ⩽ η_{ψ} f(t) for 2^{j+1} r ⩽ t ⩽ 2^{j+2} r. Hence, from the previous inequality, we obtain

∫_{|x−y|<r} |f(y)| dy ⩽ C ∑_{j=−∞}^{−1} ∫_{2^{j+1} r}^{2^{j+2} r} \frac{η_{ψ} f(t)}{tψ(t)} dt

= C ∫_{0}^{2r} \frac{η_{ψ} f(t)}{tψ(t)} dt ⩽ C (2r)^{d}ϕ(2r) ⩽ C r^{d}ϕ(r).

The last inequality may now be rewritten as \frac{1}{ϕ(r) |B|} ∫_{B} |f(y)| dy ⩽ C. Taking the supremum over all balls B in ℝ^{d}, we obtain f ∈ M^{1,ϕ} .

We leave the proof of the second part — when ψ satisfies the left-doubling condition — to the readers. □

Corollary 3.2. Suppose that for some ϵ > 0, the function t^{ϵ}ψ(t) is almost decreasing, that is, there exists a constant C > 0 such that

r ⩽ s ⇒ r^{ϵ}ψ(r) ⩾ C s^{ϵ}ψ(s).

If f ∈ S_{ψ} , then f ∈ M^{1,ϕ} provided that

(i) ψ satisfies the right-doubling condition and η_{ψ} f(r) ⩽ Cr^{d}ψ(r)ϕ(r) for every r > 0, or

(ii) ψ satisfies the left-doubling condition and η_{ψ} f(4r) ⩽ Cr^{d}ψ(r)ϕ(r) for every r > 0.

Proof. If ψ satisfies the right-doubling condition η_{ψ} f(r) ⩽ Cr^{d}ψ(r)ϕ(r) for every r > 0, then we have

∫_{0}^{r} \frac{η_{ψ} f(t)}{tψ(t)} dt = ∫_{0}^{r} \frac{η_{ψ} f(t)}{t^{ϵ}ψ(t)} t^{ϵ−1} dt

⩽ C \frac{η_{ψ} f(r)}{r^{ϵ}ψ(r)} ∫_{0}^{r} t^{ϵ−1} dt = C \frac{η_{ψ} f(r)}{ψ(r)} ⩽ Cr^{d}ϕ(r),

for every r > 0. This is precisely the condition in the first part of Theorem 3.2. Similarly, if ψ satisfies the left-doubling condition and η_{ψ} f(4r) ⩽ C r^{d}ψ(r)ϕ(r) for every r > 0, then we obtain the condition in the second part of Theorem 3.2. □

Remark. For δ > 0, let ψ(t) ∼ t^{−d} (logt^{−1} )^{−1−δ} for small t > 0. Then one may observe that for some ϵ > 0, the function t^{ϵ}ψ(t) is almost decreasing.

4. Further Results

We shall now present some variants of the previous results.

4.1. First Variant. For 1 ⩽ p < ∞ and a suitable function ϕ : ℝ^{d}×(0, ∞) → (0, ∞), we define the generalized Morrey space M_{∗}^{p,ϕ} := M_{∗}^{p,ϕ} (ℝ^{d}) to be the space of all functions f ∈ L_{loc}^{p} for which

∥f : M_{∗}^{p,ϕ} ∥ := sup_{B=B(x,r)} \frac{1}{ϕ(x, r)} ( \frac{1}{|B|} ∫_{B} |f(y)|^{p}dy) ^{1/p} < ∞.

Next, for ψ : ℝ^{d} × (0, ∞) → (0, ∞), we define the generalized Stummel modulus by

η_{ψ}^{∗} f(r) := sup_{x∈ℝ^{d}} ∫_{|x−y|<r} |f(y)|ψ(x, |x − y|) dy, r > 0,

and the generalized Stummel class S_{ψ}^{∗} by S_{ψ}^{∗} := {f ∈ L_{loc}^{1} : lim_{r→0} η_{ψ}^{∗} f(r) = 0} . Then we have the following results, which can be proved in the same way as Theorems 3.1 and 3.2.

Theorem 4.1. Suppose that lim_{r→0} sup_{x∈ℝ^{d}} ∫_{0}^{r} t^{d−1} ψ(x, t)ϕ(x, t) dt < ∞. Then we have M_{∗}^{1,ϕ} ⊆ S_{ψ}^{∗} provided that ψ satisfies the right-doubling condition, that is, there exists a constant C > 0 such that

x ∈ ℝ^{d} and 1 ⩽ \frac{r}{s} ⩽ 2 ⇒ \frac{ψ(x, r)}{ψ(x, s)} ⩽ C

or the left-doubling condition, that is, there exists a constant C > 0 such that

x ∈ ℝ^{d} and 1 ⩽ \frac{r}{s} ⩽ 2 ⇒ \frac{1}{C} ⩽ \frac{ψ(x, r)}{ψ(x, s)} .

Example 4.1. Let α : ℝ^{d} → (0, ∞) with 0 < inf_{x∈ℝ^{d}} α(x) ⩽ sup_{x∈ℝ^{d}} α(x) ⩽ d,

and put ψ(x, t) = t^{α(x)−d} , x ∈ ℝ^{d}, t > 0. Then one may check that ψ satisfies the right- and left-doubling conditions.

Theorem 4.2. If f ∈ S_{ψ}^{∗}, then f ∈ M_{∗}^{1,ϕ} provided that

(i) ψ satisfies the right-doubling condition and ∫_{0}^{r} \frac{η_{ψ}^{∗} f(t)}{tψ(x,t)} dt ⩽ Cr^{d}ϕ(x, r) for

every x ∈ ℝ^{d} and r > 0, or

(ii) ψ satisfies the left-doubling condition and ∫_{0}^{r} \frac{η_{ψ}^{∗} f(4t)}{tψ(x,t)} dt ⩽ Cr^{d}ϕ(x, r) for

every x ∈ ℝ^{d} and r > 0.

Remark. We leave it to the readers to formulate the consequences of the above theorems that are analogous to Corollaries 3.1 and 3.2.

4.2. Second Variant. We go back to the generalized Morrey spaces M^{p,ϕ} we considered earlier, but for a nonnegative locally integrable function Ψ we now define the generalized Stummel modulus by

η_{Ψ} f(r) := sup_{x∈ℝ^{d}} ∫_{|x−y|<r} |f(y)|Ψ(x − y) dy, r > 0,

and the generalized Stummel class S_{Ψ} by S_{Ψ} := {f ∈ L_{loc}^{1} : lim_{r→0} η_{Ψ} f(r) = 0}. Notice that Ψ may not be a radial function. As in [9], the function Ψ is assumed to satisfy the following property: there are constants C, δ >0 and 0⩽ϵ<1 such that

(4.1)  sup_{R⩽|x|<2R} Ψ(x) ⩽ \frac{C}{R^{d}} ∫_{δ(1−ϵ)R⩽|y|<2δ(1+ϵ)R} Ψ(y) dy

for every R > 0. Then, we have the following inclusion of the Morrey space M^{1,ϕ} in the Stummel class S_{Ψ} .

Theorem 4.3. Suppose that Ψ satisfies (4.1). If lim_{r→0} ∫_{|x|<r} Ψ(x)ϕ(|x|) dx = 0, then M^{1,ϕ} ⊆ S_{Ψ} .

Proof. Let f ∈ M^{1,ϕ} , x ∈ ℝ^{d} and r > 0. For convenience, put c_{1} := δ(1 − ϵ) and c_{2} := δ(1 + ϵ). Then, we have

∫|x−y|<r |f(y)|Ψ(x − y) dy

= ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)|Ψ(x − y) dy

⩽ ∑_{j=−∞}^{−1} sup_{2^{j} r⩽|x−y|<2^{j+1} r} Ψ(x − y) ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)| dy

⩽ C ∑_{j=−∞}^{−1} ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} Ψ(z) dz · \frac{1}{(2^{j+1} r)^{d}} ∫_{|x−y|<2^{j+1} r} |f(y)| dy

⩽ C ∥f : M^{1,ϕ} ∥ ∑_{j=−∞}^{−1} ϕ(2^{j+1} r) ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} Ψ(z) dz.

Using the assumption that ϕ satisfies the doubling condition (when c_{2} > 1) and that ϕ is nonincreasing, we have ϕ(2^{j+1} r) ⩽ Cϕ(c_{2}2^{j+1} r) ⩽ Cϕ(|z|) for |z| < c_{2}2^{j+1} r.

Hence

ϕ(2^{j+1} r) ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} Ψ(z) dz ⩽ C ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} Ψ(z)ϕ(|z|) dz.

Next, we have ∑_{j=−∞}^{∞} χ_{[c_{1} ,2c_{2} )} (2^{j} r) ∼ 1+log_{2} \frac{1+ϵ}{1−ϵ} (the overlapping property), so that

∑_{j=−∞}^{−1} ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} Ψ(z)ϕ(|z|) dz ⩽ C ∫_{|z|<c_{2} r} Ψ(z)ϕ(|z|) dz.

Thus we obtain

∫_{|x−y|<r} |f(y)|Ψ(x − y) dy ⩽ C ∥f : M^{1,ϕ} ∥ ∫_{|z|<c_{2} r} Ψ(z)ϕ(|z|) dz.

Since the integral on the right hand side tends to 0 as r → 0, we conclude that f ∈ S_{Ψ} . □

To prove the converse, we assume that Ψ satisfies the following property: there are constants C, δ > 0 and 0 ⩽ ϵ < 1 such that

(4.2)  sup_{R⩽|x|<2R} \frac{1}{Ψ(x)} ⩽ \frac{C}{R^{d}} ∫_{δ(1−ϵ)R⩽|y|<2δ(1+ϵ)R}  \frac{1}{Ψ(y)} dy

for every R > 0.

Theorem 4.4. Suppose that Ψ satisfies (4.2). If f ∈ S_{Ψ} , then f ∈ M^{1,ϕ} provided that

∫_{|x|<r} \frac{η_{Ψ} f(κ|x|)}{|x|^{d}Ψ(x)} dx ⩽ C r^{d}ϕ(r)

for every r > 0, where κ := \frac{2}{δ(1−ϵ)} .

Proof. Let f ∈ S_{Ψ} , x ∈ ℝ^{d} and r > 0. The same as before, we let c_{1} := δ(1 − ϵ), c_{2} := δ(1 + ϵ), and put κ := \frac{2}{c_{1}} . It follows that

∫_{|x−y|<r} |f(y)| dy = ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)| dy

⩽ ∑_{j=−∞}^{−1} sup_{2^{j} r⩽|x−y|<2^{j+1} r} \frac{1}{Ψ(x − y)} ∫_{|x−y|<2^{j+1} r} |f(y)|Ψ(x − y) dy

⩽ C ∑_{j=−∞}^{−1} ∫_{c_{1} 2^{j} r⩽|z|<c_{2} 2^{j+1} r} \frac{1}{|z|^{d}Ψ(z)} dz · η_{Ψ} f(2^{j+1} r)

⩽ C ∫_{|z|<c_{2} r} \frac{η_{Ψ} f(κ|z|)}{|z|^{d}Ψ(z)} dz ⩽ C r^{d}ϕ(r),

where we have used the fact that η_{Ψ} f is nondecreasing, the overlapping property, and the doubling property of ϕ. This proves the theorem. □

5. Concluding Remarks

We have improved the results of Eridani and Gunawan [4] by using weaker assumptions. We have also found that the doubling property of the Stummel modulus is unnecessary, and used its increasing property instead. Moreover, we have added some variants and proved similar results.

We end the paper with the following proposition, which tells us that given a function in the Stummel class S_{ψ} , we can actually have extra information about its integrability.

Proposition 5.1. If f ∈ S_{ψ} and ∫_{0}^{1}[η_{ψ} f(t)]^{1−θ} t^{−1} dt < ∞, for 0 < θ < 1, then

for every x ∈ ℝ^{d} and r > 0

∫_{|x−y|<r} |f(y)|\frac{ψ(|x − y|)}{[η_{ψ} f(2|x − y|)]^{θ}} dy ⩽ ∫_{0}^{2r} [η_{ψ} f(t)]^{1−θ} t^{−1} dt,

and hence

lim_{r→0} ∫_{|x−y|<r} |f(y)|\frac{ψ(|x − y|)}{[η_{ψ} f(2|x − y|)]^{θ}} dy = 0.

Proof. Indeed, since η_{ψ} f is an increasing function of r, so are [η_{ψ} f]^{θ} and [η_{ψ} f]^{1−θ} for 0 < θ < 1. It thus follows that for every x ∈ ℝ^{d} and r > 0

∫_{|x−y|<r} |f(y)| \frac{ψ(|x − y|)}{[η_{ψ} f(2|x − y|)]^{θ} dy}

= ∑_{j=−∞}^{−1} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)|\frac{ψ(|x − y|)}{[η_{ψ} f(2|x − y|)]^{θ}} dy

⩽ ∑_{j=−∞}^{−1} \frac{1}{[η_{ψ} f(2^{j+1} r)]^{θ}} ∫_{2^{j} r⩽|x−y|<2^{j+1} r} |f(y)|ψ(|x − y|) dy

⩽ ∑_{j=−∞}^{−1} [η_{ψ} f(2^{j+1} r)]^{1−θ} ⩽ ∑_{j=−∞}^{−1} ∫_{2^{j+1} r}^{2^{j+2} r} [η_{ψ} f(t)]^{1−θ} t^{−1} dt

= ∫_{0}^{2r} [η_{ψ} f(t)]^{1−θ} t^{−1} dt.

But ∫_{0}^{1}[η_{ψ} f(t)]^{1−θ} t^{−1} dt < ∞, and so we have lim_{r→0} ∫_{0}^{2r} [η_{ψ} f(t)]^{1−θ} t^{−1} dt = 0. This

proves the proposition. □

Remark. Proposition 5.1 is also valid for the generalized Stummel classes S_{ψ}^{∗} and S_{Ψ} defined in Section 4.

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Department of Mathematics (Received 04 03 2011) Bandung Institute of Technology Bandung 40132 Indonesia

hgunawan@math.itb.ac.id

Department of Mathematics Osaka Kyoiku University Osaka 582-8582, Japan

Current address: Department of Mathematics Ibaraki University, Mito Ibaraki 310-8512, Japan]

enakai@cc.osaka-kyoiku.ac.jp
enakai@mx.ibaraki.ac.jp

Department of Mathematics Kyoto University Kyoto 606-8502, Japan

yoshihiro-sawano@celery.ocn.ne.jp

Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8194, Japan

htanaka@ms.u-tokyo.ac.jp