Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 495–522

SOBOLEV EMBEDDINGS FOR VARIABLE EXPONENT RIESZ POTENTIALS ON METRIC SPACES

Toshihide Futamura, Yoshihiro Mizuta and Tetsu Shimomura

Daido Institute of Technology, Department of Mathematics Nagoya 457-8530, Japan; futamura@daido-it.ac.jp Hiroshima University, The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences, Higashi-Hiroshima 739-8521, Japan; mizuta@mis.hiroshima-u.ac.jp Hiroshima University, Graduate School of Education

Department of Mathematics, Higashi-Hiroshima 739-8524, Japan; tshimo@hiroshima-u.ac.jp

Abstract. In the metric space setting, our aim in this paper is to deal with the boundedness of Hardy–Littlewood maximal functions in generalized Lebesgue spaces L^{p( ·)} when p( ·) satisfies a log-Hölder condition. As an application of the boundedness of maximal functions, we study Sobolev’s embedding theorem for variable exponent Riesz potentials on metric space.

1. Introduction

Let X be a metric space with a metric d. Write d(x, y) = |x−y| for simplicity. We denote by B(x, r) the open ball centered at x ∈ X of radius r > 0. Let μ be a Borel measure on X. Assume that 0 < μ(B) < ∞ and there exist constants C > 0 and s ≥ 1 such that

(1.1)  \frac{μ(B′)}{μ(B)} ≥ C(\frac{r′}{r} )^{s}

for all balls B = B(x, r) and B′ = B(x′, r′) with x′ ∈ B and 0 < r′ ≤ r. Note that μ is a doubling measure on X, that is, there exists a constant C′ > 0 such that

(1.2)  μ(B(x, 2r)) ≤ C′μ(B(x, r))

for all x ∈ X and r > 0.

We define the Riesz potential of order α for a locally integrable function f on X defined by

U_{α}f(x) = ∫_{X} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y).

2000 Mathematics Subject Classification: Primary 31B15, 46E35.

Here α > 0. Following Orlicz [26] and Kováčik and Rákosník [19], we consider a positive continuous function p( ·) on X and a function f satisfying

∫_{X} |f(y)|^{p(y)} dμ(y) < ∞.

In this paper we treat p( ·) such that p > 1 on X and p satisfies a log-Hölder condition:

|p(x) − p(y)| ≤ \frac{a_{1} log(log(1/|x − y|))}{log(1/|x − y|)} + \frac{a_{2}}{log(1/|x − y|)}

whenever |x − y| < \frac{1}{4} , where a_{1} and a_{2} are nonnegative constants. If a_{1} > 0, then we can not expect the usual boundedness of maximal functions in L^{p( ·)} , according to the recent works by Diening [3], [4], Pick and Ru^{∘}žička [27] and Cruz-Uribe, Fiorenza and Neugebauer [2]. Our typical example is a variable exponent p( ·) on X such that

p(x) = p_{0} + \frac{a_{1} log(log(1/ϱ_{K}(x)))}{log(1/ϱ_{K}(x))} + \frac{a_{2}}{log(1/ϱ_{K}(x))}

when ϱ_{K}(x) is small, where p_{0} > 1, a_{1} ≥ 0, a_{2} ≥ 0 and ϱ_{K}(x) denotes the distance of x from a compact subset K of X.

Our first task is then to establish the boundedness of Hardy–Littlewood maximal functions from L^{p( ·)} to some Orlicz classes, as an extension of Harjulehto– Hästö–Pere [13] with a_{1} = 0 in metric setting and the authors’ [11, Theorem 2.4] in Euclidean setting. As an application of the boundedness of maximal functions, we establish Sobolev’s embedding theorem for variable exponent Riesz potentials on metric space; in the case a_{1} = 0, see also Diening [4] and Kokilashvili–Samko [17], [18],

In the borderline case of Sobolev’s theorem, we are concerned with exponential integrabilities of Trudinger type, which extend the results by Edmunds–Gurka– Opic [5], [6], Edmunds–Krbec [7] and the authors’ [9], [22]. We also discuss the pointwise continuity of Riesz potentials defined in the n-dimensional Euclidean space, as an extension of the authors [9], [23], [24].

For related results, see Adams–Hedberg [1], Heinonen [16], Musielak [25] and

Ru^{∘}žička [28].

2. Variable exponents

Throughout this paper, let C denote various constants independent of the variables in question.

Let G be a bounded open set in X. In this section let us assume that p( ·) is a positive continuous function on G satisfying: (p1) 1 < p_{−}(G) = inf_{G} p(x) ≤ sup_{G} p(x) = p_{+}(G) < ∞;

(p2) |p(x) − p(y)| ≤ \frac{a_{1} log(log(1/|x − y|))}{log(1/|x − y|)} + \frac{a_{2}}{log(1/|x − y|)} ,

whenever |x − y| < \frac{1}{4} , x ∈ G and y ∈ G, for some constants a_{1} ≥ 0 and a_{2} ≥ 0.

Example 2.1. Let F be a closed subset of G. For a ≥ 0 and b ≥ 0, consider

p(x) = p_{0} + ω_{a,b}(ϱ_{F} (x)),

where 1 < p_{0} < ∞, ϱ_{F} (x) denotes the distance of x from F and

ω_{a,b}(t) = \frac{a log(log(1/t))}{log(1/t)} + \frac{b}{log(1/t)}

for 0 < t ≤ r_{0} (< \frac{1}{4}); set ω_{a,b}(t) = ω_{a,b}(r_{0}) when t > r_{0} and ω_{a,b}(0) = 0. Then we can find r_{0} > 0 sufficiently small that p satisfies (p1) and (p2).

For a proof, we prepare the following result.

Lemma 2.2. Let ω be a nonnegative continuous function on the interval [0, r_{0}] such that

(i) ω(0) = 0; (ii) ω′(t) ≥ 0 for 0 < t ≤ r_{0}; (iii) ω′′(t) ≤ 0 for 0 < t ≤ r_{0} . Then

(2.1)  ω(s + t) ≤ ω(s) + ω(t) for s, t ≥ 0 and s + t ≤ r_{0} .

It is easy to find r_{0} ∈ (0, \frac{1}{4}) such that ω_{a,b} satisfies (i)–(iii) on [0, r_{0}].

Let

1/p′(x) = 1 − 1/p(x).

Then, noting that

(2.2)  \begin{aligned}p′(y) − p′(x) & = \frac{p(x) − p(y)}{(p(x) − 1)(p(y) − 1)} \\ & = \frac{p(x) − p(y)}{(p(x) − 1)^{2}} + \frac{\{p(x) − p(y)\}^{2}}{(p(x) − 1)^{2}(p(y) − 1)}, \end{aligned}

we have the following result.

Lemma 2.3. There exists a positive constant c such that

|p′(x) − p′(y)| ≤ ω_{a,c}(|x − y|) whenever x ∈ G and y ∈ G,

where a = a(x) = a_{1}(p(x) − 1)^{−2} .

3. Boundedness of maximal functions

Define the L^{p( ·)}(G) norm by

∥f∥_{p( ·)} = ∥f∥_{p( ·),G} = inf\{λ > 0 : ∫_{G}|\frac{f(y)}{λ}|^{p(y)} dμ(y) ≤ 1\}

and denote by L^{p( ·)}(G) the space of all measurable functions f on G with ∥f∥_{p( ·)} < ∞.

By the decay condition (1.1), we have

(3.1)  μ(B(x, r)) ≥ Cr^{s}

for all x ∈ G and 0 < r < d_{G}, where d_{G} denotes the diameter of G. For f ∈ L^{p( ·)}(G), define the maximal function

\begin{aligned}Mf(x) & = sup_{r>0} \frac{1}{μ(B(x, r))} ∫_{G∩B(x,r)} |f(y)| dμ(y) \\ & = sup_{0<r<d_{G}} \frac{1}{μ(B(x, r))} ∫_{G∩B(x,r)} |f(y)| dμ(y). \end{aligned}

Our first aim is to discuss the boundedness of the maximal functions.

Theorem 3.1. Let a > a_{1} when a_{1} > 0 and a = 0 when a_{1} = 0. Set A(x) = as/p(x)^{2} . If ∥f∥_{p( ·)} ≤ 1, then

∫_{G}{Mf(x)(log(e + Mf(x)))^{−A(x)}}^{p(x)} dμ(x) ≤ C.

When a_{1} = 0, Theorem 3.1 was proved by Harjulehto–Hästö–Pere [13], which is an extension of Diening [3]. For the boundedness of maximal functions in general domains, see Cruz-Uribe, Fiorenza and Neugebauer [2].

Remark 3.2. Set Φ(r, x) = {r(log(e + r))^{−A(x)}}^{p(x)} for r > 0 and x ∈ G. Then Theorem 3.1 assures the existence of C > 0 such that

∫_{G} Φ(Mf(x)/C, x) dμ(x) ≤ 1 whenever ∥f∥_{p( ·)} ≤ 1.

As in Edmunds and Rákosník [8], we define

∥f∥_{Φ} = ∥f∥_{Φ,G} = inf{λ > 0 : ∫_{G} Φ(|f(x)|/λ, x) dμ(x) ≤ 1};

then it follows that

∥Mf∥_{Φ} ≤ C∥f∥_{p( ·)} for f ∈ L^{p( ·)}(G).

Theorem 3.1 is proved along the same lines as in the authors’ [11, Theorem 2.4], but we give a proof of Theorem 3.1 for the readers’ convenience.

To complete the proof, we prepare the following lemma.

Sobolev embeddings for variable exponent Riesz potentials 499 Lemma 3.3. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)}

≤ 1. Then

Mf(x) ≤ C{Mg(x)^{1/p(x)}(log(e + Mg(x)))^{A_{1}(x)} + 1},

where g(y) = f(y)^{p(y)} and A_{1}(x) = a_{1}s/p(x)^{2} .

Proof. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)} ≤ 1. First note that

(3.2)  ∫_{G} f(y)^{p(y)} dμ(y) ≤ 1.

Then, if r ≥ r_{0} , then

(3.3) \frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ \frac{1}{μ(B(x, r))} ∫_{B(x,r)}\{1+f(y)^{p(y)}\} dμ(y) ≤ C

by our assumption.

For 0 < k ≤ 1 and r > 0, we have by Lemma 2.3

\frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y)

≤ k\{\frac{1}{μ(B(x, r))} ∫_{B(x,r)}(1/k)^{p′(y)} dμ(y) + \frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y)^{p(y)} dμ(y)\}

≤ k{(1/k)^{p′(x)+ω(r)} + F }, where F = (μ(B(x, r)))^{−1} ∫_{B(x,r)} f(y)^{p(y)} dμ(y) and ω(r) = ω_{a,c}(r) as in Example 2.1. Here, considering

k = F^{−1/{p′(x)+ω(r)}} = F^{−1/p′(x)+β(x)}

with β(x) = ω(r)/{p′(x)(p′(x) + ω(r))} when F ≥ 1, we have

\frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ 2F^{1/p(x)}F^{ω(r)/p′(x)^{2}};

if F < 1, then we can take k = 1 to obtain

\frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ 2.

500 T. Futamura, Y. Mizuta and T. Shimomura Hence it follows that

(3.4)  \frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ 2\{F^{1/p(x)}F^{ω(r)/p′(x)^{2}} + 1\}.

If r ≤ F^{−1} , then we see from (3.4) that

\frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ C\{F^{1/p(x)}(log(e + F))^{A_{1}(x)} + 1\}.

If r_{0} > r > F^{−1} , then we have by the lower bound (3.1)

\begin{aligned}F^{1/p(x)+ω(r)/p′(x)^{2}} & ≤ Cμ(B(x, r))^{−1/p(x)}r^{−sω(r)/p′(x)^{2}} \\ & × \{∫_{B(x,r)} f(y)^{p(y)} dμ(y)\}^{1/p(x)+ω(r)/p′(x)^{2}}. \end{aligned}

In view of (3.2), we find F^{1/p(x)+ω(r)/p′(x)^{2}}

≤ Cμ(B(x, r))^{−1/p(x)}(log(1/r))^{A_{1}(x)}{∫_{B(x,r)} f(y)^{p(y)} dμ(y)}^{1/p(x)+ω(r)/p′(x)^{2}} ≤ Cμ(B(x, r))^{−1/p(x)}(log(1/r))^{A_{1}(x)}{∫_{B(x,r)} f(y)^{p(y)} dμ(y)}^{1/p(x)} ≤ Cμ(B(x, r))^{−1/p(x)}(log F)^{A_{1}(x)}{∫_{B(x,r)} f(y)^{p(y)} dμ(y)}^{1/p(x)} = CF^{1/p(x)}(log F)^{A_{1}(x)}. Now we have established

(3.5)  \frac{1}{μ(B(x, r))} ∫_{B(x,r)} f(y) dμ(y) ≤ C\{F^{1/p(x)}(log(e + F))^{A_{1}(x)} + 1\}

for all r > 0 and x ∈ G, which completes the proof.

Proof of Theorem 3.1. Let p_{1}(x) = p(x)/p_{1} for 1 < p_{1} < p_{−}(G). Then Lemma 3.3 yields

{Mf(x)}^{p_{1}(x)} ≤ C{Mg(x)(log(e + Mg(x)))^{ã_{1}s/p_{1}(x)} + 1} for x ∈ G, where g(y) = f(y)^{p_{1}(y)} and ã_{1} = a_{1}/p_{1} . Letting a > a_{1} when a_{1} > 0 and a = 0 when a_{1} = 0, we set A(x) = as/p(x)^{2} . Then we can choose p_{1} so that a_{1}p_{1} ≤ a and

{Mf(x)}^{p(x)} ≤ C{Mg(x)(log(e + Mg(x)))^{A(x)p(x)/p_{1}} + 1}^{p_{1}},

which yields

{Mf(x)(log(e + Mf(x)))^{−A(x)}}^{p(x)} ≤ C{Mg(x) + 1}^{p_{1}}.

Now Theorem 3.1 follows from the boundedness of maximal functions in L^{p_{1}} (in the case of constant exponent).

Remark 3.4. Let p( ·) be a positive continuous function on G such that 1 ≤ p(x) ≤ p_{+}(G) < ∞. Then, as in Harjulehto–Hästö–Pere [13], we can prove the following weak type result for maximal functions:

|E_{f}(t)| ≤ C ∫_{G}|\frac{f(y)}{t}|^{p(y)} dμ(y)

whenever t > 0 and f ∈ L^{p( ·)}(G), where E_{f}(t) = {x ∈ G : Mf(x) ≥ t}; see also Cruz-Uribe, Fiorenza and Neugebauer [2, Theorem 1.8].

To prove this, we may assume that t = 1. We have for k > 1

\frac{1}{μ(B(x, r))} ∫_{B(x,r)} |f(y)| dμ(y)

≤ k\{\frac{1}{μ(B(x, r))} ∫_{B(x,r)}(1/k)^{p′(y)} dμ(y) + \frac{1}{μ(B(x, r))} ∫_{B(x,r)} |f(y)|^{p(y)} dμ(y)\}

≤ k{(1/k)^{(p_{+})′} + F },

where F = (μ(B(x, r)))^{−1} ∫_{B(x,r)} |f(y)|^{p(y)} dμ(y). Here, considering k = F^{−1/(p_{+})′} when F < 1, we find

1 ≤ 2F^{1/p_{+}},

so that

(\frac{1}{2})^{p_{+}} ≤ M(|f|^{p( ·)})(x) for x ∈ E_{f}(1),

which proves the required assertion.

Remark 3.5. For 0 < r < \frac{1}{2} , let
G = {x = (x_{1}, x_{2}) : 0 < x_{1} < 1, −1 < x_{2} < 1}.

Define

p(x_{1}, x_{2}) = \{ \begin{matrix}p_{0} + a_{1}(log(log(1/x_{2})))/ log(1/x_{2}) & when 0 < x_{2} ≤ r_{0}, \\ p_{0} & when x_{2} ≤ 0; \end{matrix}

and p(x_{1}, x_{2}) = p(x_{1}, r_{0}) when x_{2} > r_{0} . Setting

G(r) = {x = (x_{1}, x_{2}) : 0 < x_{1} < r, −r < x_{2} < 0},

we consider

f_{r}(y) = χ_{G(r)}(y)

and set g_{r} = f_{r}/∥f_{r}∥_{p( ·),G}, where χ_{E} denotes the characteristic function of a measurable set E. Then we claim for 0 < r < \frac{1}{2}r_{0} :

(i) ∥f_{r}∥_{p( ·),G} = r^{2/p_{0}} ; (ii) M(g_{r})(x) ≥ C_{1}r^{−2/p_{0}} for 0 < x_{1} < r and r < x_{2} < 2r;

(iii) ∫_{G}{M(g_{r})(x)(log(e + M(g_{r})(x)))^{−A(x)}}^{p(x)} dx ≥ C_{2}(log(1/r))^{2(a_{1}−a)/p_{0}} for

A(x) = 2a/p(x)^{2} , so that the conclusion of Theorem 3.1 does not hold for 0 < a < a_{1} .

4. Sobolev’s inequality

For 0 < α < s, we consider the Riesz potential U_{α}f of f ∈ L^{p( ·)}(G) defined by

U_{α}f(x) = ∫_{G} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y);

recall that s is the decay constant in (1.1). In this section, suppose p( ·) satisfies (p1), (p2) and

(p3)  p_{+}(G) < s/α.

Let

1/p^{♯}(x) = 1/p(x) − α/s.

In what follows we establish Sobolev’s inequality for α-potentials on G, as an extension of the case of constant exponent which was discussed by Hajłasz and Koskela [12] and Heinonen [16]; for the Euclidean case, see the books by Adams and Hedberg [1] and the second author [21]. In the next two sections, we are concerned with the exponential integrability, which extends the results by Edmunds, Gurka and Opic [5], [6], and the authors [22].

Now we show our result, which gives an extension of Diening [4]; for further investigations we also refer the reader to the results by Kokilashvili–Samko [17], [18], where the index α is a variable exponent.

Theorem 4.1. Letting a > a_{1} when a_{1} > 0 and a = 0 when a_{1} = 0, we set A(x) = as/p(x)^{2} . Suppose p_{+}(G) < s/α. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)} ≤ 1. Then

∫_{G}{U_{α}f(x)(log(e + U_{α}f(x)))^{−A(x)}}^{p^{♯}(x)} dμ(x) ≤ C.

In spite of the fact that the proof of Theorem 4.1 is quite similar to that of Theorem 3.4 in [11], we give a proof of Theorem 4.1 for the readers’ convenience.

For this purpose, we prepare the following two lemmas.

Lemma 4.2. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)} ≤ 1. Then

∫_{G\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ Cδ^{−s/p^{♯}(x)} log(1/δ)^{A_{1}(x)}

for x ∈ G and 0 < δ < \frac{1}{4} , where A_{1}(x) = a_{1}s/p(x)^{2} as before.

Proof. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)} ≤ 1. Then, for k > 1, we have

∫_{G\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y)

≤ k\{∫_{G\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y) + ∫_{G\B(x,δ)} f(y)^{p(y)} dμ(y)\}

≤ k\{∫_{G\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y) + 1\}.

Consider the set

E = {y ∈ G : |x − y|^{α}/(kμ(B(x, |x − y|))) > 1}.

Note here that by (1.2), (3.1) and Lemma 2.3

∫_{E\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y)

≤ ∫_{E\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(x)+ω(|x−y|)} dμ(y)

≤ C ∑_{j} ∫_{B(x,2^{j}δ)\B(x,2^{j−1}δ)}(\frac{(2^{j}δ)^{α}}{kμ(B(x, 2^{j}δ))})^{p′(x)+ω(2^{j}δ)} dμ(y)

≤ Ck^{−p′(x)−ω(δ)} ∑_{j}(2^{j}δ)^{α(p′(x)+ω(2^{j}δ))}(μ(B(x, 2^{j}δ))^{−(p′(x)+ω(2^{j}δ))+1}

≤ Ck^{−p′(x)−ω(δ)} ∑_{j}(2^{j}δ)^{(α−s)(p′(x)+ω(2^{j}δ))+s}

≤ Ck^{−p′(x)−ω(δ)} ∫_{δ}^{∞} t^{(α−s)(p′(x)+ω(t))+s}t^{−1} dt

≤ Ck^{−p′(x)−ω(δ)}δ^{(α−s)(p′(x)+ω(δ))+s}

≤ Ck^{−p′(x)−ω(δ)}δ^{p′(x)(α−s/p(x))}(log(1/δ))^{(s−α)a_{1}/(p(x)−1)^{2}}

= Ck^{−p′(x)−ω(δ)}δ^{−p′(x)s/p^{♯}(x)}(log(1/δ))^{(s−α)a_{1}/(p(x)−1)^{2}},

where ω(r) = ω_{a,c}(r). Now, letting k = δ^{−s/p^{♯}(x)}(log(1/δ))^{A_{1}(x)} , we see that

∫_{E\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y) ≤ C.

Further we find

∫_{G\E}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y) ≤ C.

Consequently it follows that

∫_{G\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ Cδ^{−s/p^{♯}(x)}(log(1/δ))^{A_{1}(x)}

for x ∈ G and 0 < δ < \frac{1}{4} , as required.

Lemma 4.3. Let f be a nonnegative measurable function on G with ∥f∥_{p( ·)} ≤ 1. Then

U_{α}f(x) ≤ C{Mf(x)^{p(x)/p^{♯}(x)}(log(e + Mf(x)))^{a_{1}α/p(x)} + 1}.

Proof. To give the required estimate, we borrow the idea of Hedberg [15]. In fact, for x ∈ G and 0 < δ < \frac{1}{4} we have by Lemma 4.1

\begin{aligned}U_{α}f(x) & = ∫_{G∩B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) + ∫_{G\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) \\ & ≤ Cδ^{α}Mf(x) + Cδ^{−s/p^{♯}(x)}(log(1/δ))^{A_{1}(x)}. \end{aligned}

Considering δ = Mf(x)^{−p(x)/s}(log(e + Mf(x)))^{a_{1}/p(x)} when Mf(x) is large enough, we obtain the required inequality.

Proof of Theorem 4.1. Let a > a_{1} > 0 or a = a_{1} = 0, and set A(x) = as/p(x)^{2} . Then Lemma 4.3 yields

{U_{α}f(x)(log(e+U_{α}f(x)))^{−A(x)}}^{p^{♯}(x)}≤ C[{Mf(x)(log(e+Mf(x)))^{−A(x)}}^{p(x)}+1],

which together with Theorem 3.1 completes the proof.

Remark 4.4. In Remark 3.5, we see that

U_{α}g_{r}(x) ≥ C_{3}r^{−2/p^{♯}(x)}(log(1/r))^{A_{1}(x)}

for 0 < x_{1} < r and r < x_{2} < 2r, where A_{1}(x) = 2a_{1}/p(x)^{2} . Hence we have

∫_{G}{U_{α}g_{r}(x)(log(e + U_{α}g_{r}(x)))^{−A(x)}}^{p^{♯}(x)} dx ≥ C_{4}(log(1/r))^{2(a_{1}−a)/p_{0}},

where A(x) = 2a/p(x)^{2} . This implies that the conclusion of Theorem 4.1 does not hold when a < a_{1} .

Remark 4.5. By Theorem 4.1 we see that U_{α}f ∈ L^{p( ·)}(G) whenever f ∈ L^{p( ·)}(G). Then, as was pointed out by Lerner [20], the inequality

∫_{G} |U_{α}f(x)|^{p(x)} dμ(x) ≤ C ∫_{G} |f(y)|^{p(y)} dμ(y)

holds whenever f ∈ L^{p( ·)}(G) if and only if p is constant, under the additional assumption that

μ(E) = sup{μ(K) | K ⊂ E, K : compact}

for every measurable set E ⊂ X. In fact, the if part is clear. We here assume that p is not constant. Then we can find numbers p_{1} and p_{2} such that 1 ≤ p_{1} < p_{2} < ∞, and both E_{1} = {x ∈ G : p(x) ≤ p_{1}} and E_{2} = {x ∈ G : p(x) ≥ p_{2}} have positive μ measure. Further by our assumption, there exist compact sets K_{i}, i = 1, 2, such that K_{i} ⊂ E_{i}. If f = kχ_{K_{1}} with k > 1, then

U_{α}f(x) ≥ Ckμ(K_{1}) for x ∈ K_{2} ,

so that

∫_{G} |U_{α}f(x)|^{p(x)} dμ(x) ≥ Ck^{p_{2}}μ(K_{2}).

On the other hand,

∫_{G} |f(x)|^{p(x)} dμ(x) ≤ k^{p_{1}}μ(K_{1}).

If the inequality holds, then we should have

k^{p_{2}} ≤ Ck^{p_{1}},

which gives a contradiction by letting k → ∞.

In the same manner, we see that the inequality

∫_{G}{Mf(x)}^{p(x)} dμ(x) ≤ C ∫_{G} |f(y)|^{p(y)} dμ(y)

holds whenever f ∈ L^{p( ·)}(G) if and only if p is constant.

Remark 4.6. Let ω(r) be a continuous function on (0, ∞) such that

ω(r) = \frac{a_{1} log(log(1/r))}{log(1/r)} + \frac{a_{2}}{log(1/r)}

for 0 < r ≤ r_{0} < \frac{1}{4} , with a_{1} > 0 and a_{2} > 0; set ω(r) = ω(r_{0}) for r > r_{0} . Consider a variable exponent p( ·) on the unit ball B in R^{n} defined by

p(x) = p_{0} + ω(ϱ(x)),

where 1 < p_{0} < n/α and ϱ(x) = 1 − |x|. Take r_{0} so small that p(x) < n/α for all x ∈ B. In view of Theorem 4.1, we see that if a > a_{1} and A(x) = an/p(x)^{2} , then

∫_{B}{U_{α}f(x)(log(e + U_{α}f(x)))^{−A(x)}}^{p^{♯}(x)} dx ≤ C

whenever f is a nonnegative measurable function on B with ∥f∥_{p( ·)} ≤ 1.

5. Exponential integrability

For fixed x_{0} ∈ G, let us assume that an exponent p(x) is a continuous function on G satisfying

(5.1)  p(x) > p_{0} when x ≠ x_{0}

and

(5.2)  |p(x) − \{p_{0} + \frac{a log(log(1/|x_{0} − x|))}{log(1/|x_{0} − x|)}\}| ≤ \frac{b}{log(1/|x_{0} − x|)}

for x ∈ B(x_{0}, r_{0}), where 0 < r_{0} < \frac{1}{4} , p_{0} = s/α, 0 < a ≤ (s − α)/α^{2} and b > 0.

Our aim in this section is to give an exponential integrability of Trudinger type. Before doing so, we prepare several lemmas. In view of (2.2) and (5.2), we have the following result.

Lemma 5.1. There exist C > 0 and 0 < r_{0} < \frac{1}{4} such that
(5.3)  p′(y) ≤ p′_{0} − ω(|x_{0} − y|)

for all y ∈ B_{0} = B(x_{0}, r_{0}), where p′_{0} = p_{0}/(p_{0} − 1) = s/(s − α) and ω is a nonnegative nondecreasing function on (0, ∞) such that

ω(r) = \frac{aα^{2}}{(s − α)^{2}} \frac{log(log(1/r))}{log(1/r)} − \frac{C}{log(1/r)}

when 0 < r ≤ r_{0} ; set ω(r) = ω(r_{0}) when r > r_{0} as before.

Lemma 5.2. If 0 < a < (s − α)/α^{2} , then

∫_{B_{0}\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y) ≤ C(log(1/δ))^{1−aα^{2}/(s−α)}

and if a = (s − α)/α^{2} , then

∫_{B_{0}\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y) ≤ C log(log(1/δ))

for x ∈ B_{0} and 0 < δ < δ_{0} , where 0 < δ_{0} < \frac{1}{4} .

Proof. First consider the case 0 < a < (s − α)/α^{2} . Let E = {y ∈ B_{0} : |x − y|^{α}/μ(B(x, |x − y|)) > 1} for fixed x ∈ B_{0} . Let j_{0} be the smallest integer such that 2^{j_{0}}δ > 2r_{0} . Since |x − y| ≤ 3|x_{0} − y| for y ∈ B_{0} \ B(x_{0}, |x_{0} − x|/2), we have by (1.2), (3.1) and (5.3)

I_{1} = ∫_{E\\{B(x_{0},|x_{0}−x|/2)∪B(x,δ)\}}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y)

≤ C ∑_{j=1}^{j_{0}} ∫_{B(x,2^{j}δ)\B(x,2^{j−1}δ)}(\frac{(2^{j}δ)^{α}}{μ(B(x, 2^{j}δ))})^{p′_{0}−ω(2^{j−1}δ/3)} dμ(y)

≤ C ∑_{j=1}^{j_{0}}(2^{j}δ)^{α(p′_{0}−ω(2^{j−1}δ/3))}(μ(B(x, 2^{j}δ)))^{−(p′_{0}−ω(2^{j−1}δ/3))+1}

≤ C ∑_{j=1}^{j_{0}}(2^{j}δ)^{(α−s)(p′_{0}−ω(2^{j−1}δ/3))+s}

≤ C ∑_{j=1}^{j_{0}}(log 1/(2^{j}δ))^{−aα^{2}/(s−α)}

≤ C ∫_{δ}^{3r_{0}}(log(1/t))^{−aα^{2}/(s−α)}t^{−1} dt ≤ C(log(1/δ))^{1−aα^{2}/(s−α)}

for 0 < δ < δ_{0} , since 1 − aα^{2}/(s − α) > 0.

Next we give an estimate for

I_{2} = ∫_{B(x_{0},|x−x_{0}|/2)\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y).

We may assume that 2|x − x_{0}| > δ. Then we see from Lemma 5.1 that if y ∈ B(x_{0}, |x − x_{0}|/2), then p′(y) ≤ p′_{0} + η, where η = C/ log(1/|x_{0} − x|). Hence we obtain by (1.1) and (3.1)

I_{2} ≤ C ∫_{B(x_{0},|x−x_{0}|/2)}(\frac{|x − x_{0}|^{α}}{μ(B(x, |x_{0} − x|/2))})^{p′(y)} dμ(y)

≤ C ∫_{B(x_{0},|x−x_{0}|/2)}\{(\frac{|x − x_{0}|^{α}}{μ(B(x, |x_{0} − x|/2))})^{p′_{0}+η} + 1\} dμ(y)

≤ Cμ(B(x_{0}, |x_{0} − x|/2))\{(\frac{|x − x_{0}|^{α}}{μ(B(x, |x_{0} − x|/2))})^{p′_{0}+η} + 1\}

≤ C{|x − x_{0}|^{α(p′_{0}+η)}μ(B(x_{0}, |x_{0} − x|/2))^{−(p′_{0}+η)+1} + 1} ≤ C.

Sobolev embeddings for variable exponent Riesz potentials 509

Thus it follows that

∫_{B_{0}\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y) ≤ C(log(1/δ))^{1−aα^{2}/(s−α)}

for 0 < δ < \frac{1}{4} , which proves the first case.

The second case a = (s − α)/α^{2} is similarly proved.

Lemma 5.3. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. If β_{1} > β = (1 − aα^{2}/(s − α))/p′_{0} = (s − α − aα^{2})/s > 0, then

(5.4)  ∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{β_{1}}

for x ∈ B_{0} and 0 < δ < δ_{0} , where 0 < δ_{0} < \frac{1}{4} .

Proof. Take p_{1} such that 1 < p_{1} < p′_{0} and β_{1} > γ = (1−aα^{2}/(s−α))/p_{1} > β. We may assume that p′(y) > p_{1} for y ∈ B_{0} .

Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. For k > 1 and 0 < δ < \frac{1}{4} , we have by Lemma 5.2

∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y)

≤ k\{∫_{B_{0}\B(x,δ)}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y) + ∫_{B_{0}\B(x,δ)} f(y)^{p(y)} dμ(y)\}

≤ k{Ck^{−p_{1}}(log(1/δ))^{1−aα^{2}/(s−α)} + 1}. Now, considering k such that k^{−p_{1}}(log(1/δ))^{1−aα^{2}/(s−α)} = 1, we have

∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{γ} ≤ C(log(1/δ))^{β_{1}},

as required.

In what follows we show that (5.4) remains true with β_{1} replaced by β = (s − α − aα^{2})/s.

Lemma 5.4. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. If β = (s − α − aα^{2})/s > 0, then

∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{β}

for x ∈ B_{0} and 0 < δ < δ_{0} , where 0 < δ_{0} < \frac{1}{4} .

510 T. Futamura, Y. Mizuta and T. Shimomura

Proof. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. Let η = (log(1/δ))^{− log log(1/δ)} for small δ, say 0 < δ < δ_{0} < \frac{1}{4} . Then note from Lemma 5.3 that

(5.5)  ∫_{B_{0}\B(x,η)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/η))^{β_{1}} ≤ C(log(1/δ))^{β}.
Letting k = (log(1/δ))^{β} and B(x) = B(x_{0}, |x_{0} − x|/2), we find
k^{ω(η/3)} ≤ C,

so that we obtain from Lemmas 5.1 and 5.2 that

∫_{B(x,η)\\{B(x,δ)∪B(x)\}}(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′(y)} dμ(y)

≤ ∫_{B(x,η)\\{B(x,δ)∪B(x)\}}\{(\frac{|x − y|^{α}}{kμ(B(x, |x − y|))})^{p′_{0}−ω(|x−y|/3)} + 1\} dμ(y)

≤ Ck^{−p′_{0}} ∫_{B_{0}\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′_{0}−ω(|x−y|/3)} dμ(y) + C

≤ Ck^{−p′_{0}}(log(1/δ))^{1−aα^{2}/(s−α)} + C ≤ C. Hence it follows from the proof of Lemma 5.3 that

(5.6)  ∫_{B(x,η)\\{B(x,δ)∪B(x)\}} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{β}.

Next we show that

(5.7)  ∫_{B(x)\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{β}.

Since a > 0, we have by the latter half of the proof of Lemma 5.2

∫_{B(x)\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y)

≤ ∫_{B(x)\B(x,δ)}(\frac{|x − y|^{α}}{μ(B(x, |x − y|))})^{p′(y)} dμ(y) + ∫_{B(x)\B(x,δ)} f(y)^{p(y)} dμ(y)

≤ C. Now we claim from (5.5), (5.6) and (5.7) that

∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(1/δ))^{β}.

Thus the proof is completed.

Lemma 5.5. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. If β = (s − α − aα^{2})/s > 0, then

U_{α}f(x) ≤ C(log(e + Mf(x)))^{β} for x ∈ B_{0} .

Proof. We see from Lemma 5.4 that

\begin{aligned}U_{α}f(x) & = ∫_{B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) + ∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) \\ & ≤ Cδ^{α}Mf(x) + C(log(1/δ))^{β}. \end{aligned}

Here, letting

δ = (Mf(x))^{−1/α}(log(e + Mf(x)))^{β/α}

when Mf(x) is large enough, we have

U_{α}f(x) ≤ C(log(e + Mf(x)))^{β},

as required.

It follows from Lemma 5.5 that

exp(C^{−1}(U_{α}f(x))^{1/β}) ≤ e + Mf(x)

whenever f is a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. By the classical fact that Mf ∈ L^{p_{−}}(B_{0}), we establish the following exponential inequality of Trudinger type.

Theorem 5.6. Let 0 < a < (s − α)/α^{2} . If β = (s − α − aα^{2})/s, then there exist positive constants c_{1} and c_{2} such that

∫_{B_{0}} exp(c_{1}(U_{α}f(x))^{1/β}) dμ(x) ≤ c_{2}

for all nonnegative measurable functions f on B_{0} with ∥f∥_{p( ·)} ≤ 1.

Remark 5.7. Let B_{0} be a ball in the n-dimensional space R^{n}. If f is a nonnegative measurable function on B_{0} such that

∫_{B_{0}} f(y)^{p(y)} dy < ∞,

then we claim by applying an idea by Hästö [14] that

(5.8)  ∫_{B_{0}} f(y)^{n/α}(log(e + f(y)))^{aα} dy < ∞.

In fact, if y ∈ E = {x ∈ B_{0} : f(x) ≥ |x_{0} − x|^{−α}(log(e + |x_{0} − x|^{−1}))^{−1}}, then

f(y)^{p(y)} ≥ Cf(y)^{n/α}(log(e + f(y)))^{aα}

so that

∫_{E} f(y)^{n/α}(log(e + f(y)))^{aα} dy < ∞,

which proves (5.8), since 0 < a < (n − α)/α^{2} . With the aid of Edmunds–Krbec [7] and the authors [22] we also obtain Theorem 5.6 in the Euclidean case.

Finally we are concerned with the case a = (s − α)/α^{2} .

Lemma 5.8. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. If a = (s − α)/α^{2} , then

U_{α}f(x) ≤ C(log(e + log(e + Mf(x))))^{p′_{0}} for x ∈ B_{0} .

Proof. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. For k > 1 and 0 < δ < δ_{0} < \frac{1}{4} , we have by applications of the arguments in the proof of Lemma 5.4

∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) ≤ C(log(log(1/δ)))^{1/p′_{0}}.

Consequently it follows that

\begin{aligned}U_{α}f(x) & = ∫_{B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) + ∫_{B_{0}\B(x,δ)} \frac{|x − y|^{α}f(y)}{μ(B(x, |x − y|))} dμ(y) \\ & ≤ Cδ^{α}Mf(x) + C(log(log(1/δ)))^{1/p′_{0}}. \end{aligned}

Here let

δ = Mf(x)^{−1/α}(log(e + log(e + Mf(x))))^{1/{αp′_{0}}}

when Mf(x) is large enough. Then we have

U_{α}f(x) ≤ C(log(e + log(e + Mf(x))))^{1/p′_{0}},

as required.

By Lemma 5.8 and the fact that Mf ∈ L^{p_{0}}(B_{0}), we establish the following double exponential inequality for f ∈ L^{p( ·)}(B_{0}).

Theorem 5.9. If a = (s − α)/α^{2} , then there exist positive constants c_{1} and c_{2} such that

∫_{B_{0}} exp(exp(c_{1}(U_{α}f(x))^{s/(s−α)})) dμ(x) ≤ c_{2}

for all nonnegative measurable functions f on B_{0} with ∥f∥_{p( ·)} ≤ 1.

Remark 5.10. In case f belongs to more general variable exponent Lebesgue spaces, we will be expected to discuss the corresponding exponential integrability as in Edmunds, Gurka and Opic [5], [6]. But we do not go into details any more.

6. Exponential integrability, II

In this section, let B = B(0, 1) be the unit ball in R^{n}. We consider a variable exponent p( ·) on B which is a continuous function on B satisfying

(6.1)  p(x) > p_{0} on B

and

(6.2)  |p(x) − \{p_{0} + \frac{a log(log(1/ϱ(x)))}{log(1/ϱ(x))}\}| ≤ \frac{b}{log(1/ϱ(x))}

when ϱ(x) < r_{0} , where 0 < r_{0} < \frac{1}{4} , a > 0, b > 0, p_{0} = n/α and ϱ(x) = 1 − |x| denotes the distance of x from the boundary ∂B.

For f ∈ L^{p( ·)}(B), the Riesz potential of order α, 0 < α < n, is defined by

U_{α}f(x) = ∫_{B} |x − y|^{α−n}f(y) dy.

If a > (n − α)/α^{2} , then we see from Theorem 7.7 below that

|U_{α}f(x) − U_{α}f(z)| ≤ C(log(1/|x − z|))^{(n−α−aα^{2})/n}

whenever x, z ∈ B and |x − z| < \frac{1}{2} ; for this fact, see also [10, Theorem 4.3].

In what follows, when 0 < a ≤ (n−α)/α^{2} , we discuss exponential inequalities of U_{α}f as in Section 5.

As in Lemma 5.1, we have the following result.

Lemma 6.1. There exist positive constants t_{0} < \frac{1}{4} and C such that

p′(x) ≤ p′_{0} − ω(ϱ(x))

for x ∈ B, where ω(t) = (aα^{2}/(n − α)^{2}) log(log(1/t))/ log(1/t) − C/ log(1/t) for 0 < t ≤ t_{0} and ω(t) = ω(t_{0}) for t > t_{0} .

Lemma 6.2. If 0 < a < (n − α)/α^{2} , then

I ≡ ∫_{B\B(x,r)} |x − y|^{(α−n)p′(y)} dy ≤ C(log(1/r))^{γ}

for all x ∈ B and 0 < r < \frac{1}{2} , where γ = 1 − aα^{2}/(n − α).

Proof. First consider the case \frac{1}{2}ϱ(x) ≤ r < \frac{1}{2} . Letting E_{1} = \{y ∈ B\B(x, r) : |ϱ(x) − ϱ(y)| > 2r}, we find by polar coordinates,

I_{1} ≡ ∫_{E_{1}} |x − y|^{(α−n)p′(y)} dy
≤ C ∫_{{t:|t−ϱ(x)|>2r}} |t − ϱ(x)|^{(α−n)(p′_{0}−ω(t))+n−1} dt
≤ C ∫_{{t:t>2r}} t^{(n−α)ω(t)−1} dt

≤ C ∫_{2r}^{1/2}(log(1/t))^{−aα^{2}/(n−α)}t^{−1} dt + C ≤ C(log(1/r))^{γ}.

Letting E_{2} = {y ∈ B \ B(x, r) : |ϱ(x) − ϱ(y)| ≤ 2r}, we find by polar coordinates,

\begin{aligned}I_{2} & ≡ ∫_{E_{2}} |x − y|^{(α−n)p′(y)} dy \\ & ≤ C ∫_{\{t:|t−ϱ(x)|≤2r\}} r^{(α−n)p′_{0}+n−1} dt ≤ Cr^{−1} ∫_{0}^{4r} dt ≤ C. \end{aligned}

Hence it follows that

∫_{B\B(x,r)} |x − y|^{(α−n)p′(y)} dy ≤ C(log(1/r))^{γ}

when \frac{1}{2}ϱ(x) ≤ r < \frac{1}{2} . In particular, we obtain

(6.3)  ∫_{B\B(x,ϱ(x)/2)} |x − y|^{(α−n)p′(y)} dy ≤ C(log(1/ϱ(x)))^{γ}.

Next consider the case 0 < r < \frac{1}{2}ϱ(x). Let E_{3} = B(x, \frac{1}{2}ϱ(x)) \ B(x, r). In view of Lemma 6.1, we find

p′(y) ≤ p′_{0} − ω(ϱ(x)) + C/ log(1/ϱ(x)) ≤ p′_{0} − ω_{1}(|x − y|)

for y ∈ E_{3} , where ω_{1}(t) = ω(t) − C/ log(1/t) for small t > 0. Hence, we see that

I_{3} ≡ ∫_{E_{3}} |x − y|^{(α−n)p′(y)} dy
≤ ∫_{E_{3}} |x − y|^{(α−n){p′_{0}−ω_{1}(|x−y|)}} dy

≤ C ∫_{r}^{ϱ(x)/2}(log(1/t))^{−aα^{2}/(n−α)}t^{−1} dt ≤ C(log(1/r))^{γ}.

In view of (6.3), we establish

∫_{B\B(x,r)} |x − y|^{(α−n)p′(y)} dy ≤ C(log(1/r))^{γ}

when 0 < r < \frac{1}{2}ϱ(x). Thus the required result is proved.

As in the proof of Lemma 5.4, we can prove the following result.

Lemma 6.3. Let f be a nonnegative function on B such that ∥f∥_{p( ·)} ≤ 1. If 0 < a < (n − α)/α^{2} and β = γ/p′_{0} = (n − α − aα^{2})/n, then

∫_{B\B(x,r)} |x − y|^{α−n}f(y) dy ≤ C(log(1/r))^{β}

whenever 0 < r < \frac{1}{2} .

We see from Lemma 6.3 that

\begin{aligned}U_{α}f(x) & = ∫_{B(x,δ)} |x − y|^{α−n}f(y) dy + ∫_{B\B(x,δ)} |x − y|^{α−n}f(y) dy \\ & ≤ Cδ^{α}Mf(x) + C(log(1/δ))^{β}. \end{aligned}

Here, letting

δ = (Mf(x))^{−1/α}(log(e + Mf(x)))^{β/α}

when Mf(x) is large enough, we have

U_{α}f(x) ≤ C(log(e + Mf(x)))^{β},

so that

exp(C^{−1}(U_{α}f(x))^{1/β}) ≤ e + Mf(x)

whenever f is a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. By the classical fact that Mf ∈ L^{p_{0}}(B), we establish the following exponential inequality of Trudinger type.

Theorem 6.4. Let 0 < a < (n − α)/α^{2} . If β = (n − α − aα^{2})/n, then there exist positive constants c_{1} and c_{2} such that

∫_{B} exp(c_{1}(U_{α}f(x))^{1/β}) dx ≤ c_{2}

for all nonnegative measurable functions f on B with ∥f∥_{p( ·)} ≤ 1.

Finally we are concerned with the case a = (n − α)/α^{2} . The following can be proved in the same way as Lemma 5.4.

Lemma 6.5. Let f be a nonnegative function on B such that ∥f∥_{p( ·)} ≤ 1. If a = (n − α)/α^{2} , then

∫_{B\B(x,r)} |x − y|^{(α−n)p′(y)}f(y) dy ≤ C(log(log(1/r)))^{(n−α)/n}

for small r > 0.

As in the proof of Theorem 6.4, we establish the following double exponential inequality for f ∈ L^{p( ·)}(B).

Theorem 6.6. If a = (n − α)/α^{2} , then there exist positive constants c_{1} and c_{2} such that

∫_{B} exp(exp(c_{1}(U_{α}f(x))^{n/(n−α)})) dx ≤ c_{2}

for all nonnegative measurable functions f on B with ∥f∥_{p( ·)} ≤ 1.

7. Continuity

Let G be a bounded open set in the n-dimensional space R^{n}, and fix x_{0} ∈ G. In this section, we deduce the continuity at x_{0} of Riesz potentials U_{α}f when f ∈ L^{p( ·)}(G) with p( ·) satisfying

|p(x) − \{\frac{n}{α} + \frac{a log(log(1/|x_{0} − x|))}{log(1/|x_{0} − x|)}\}| ≤ \frac{b}{log(1/|x_{0} − x|)},

where a > (n − α)/α^{2} , b > 0 and x runs over the small ball B_{0} = B(x_{0}, r_{0}).

Consider a positive continuous nonincreasing function φ on the interval (0, ∞) such that (φ) (log(1/t))^{−ε_{0}}φ(t) is nondecreasing on (0, r_{0}] for some ε_{0} > 0 and r_{0} > 0; set φ(r) = φ(r_{0}) for r > r_{0} . We see from condition (φ) that φ satisfies the doubling condition.

Set

Φ(r) = (∫_{0}^{r} φ(t)^{−α^{2}/(n−α)}t^{−1} dt)^{(n−α)/n}.

Our final goal is to establish the following result, which deals with the continuity of α-potentials in R^{n}.

Theorem 7.1. Let p( ·) satisfy

p(x) = \frac{n}{α} + \frac{log φ(|x_{0} − x|)}{log(1/|x_{0} − x|)} for x ∈ B_{0} = B(x_{0}, r_{0})

and f ∈ L^{p( ·)}(B_{0}). If Φ(1) < ∞, then U_{α}f is continuous at x_{0} ; in this case,

|U_{α}f(x) − U_{α}f(z)| ≤ CΦ(|x − z|)

whenever x, z ∈ B(x_{0}, \frac{1}{2}r_{0}).

Remark 7.2. Let φ(r) = (log(e + 1/r))^{a}. Then Φ(1) < ∞ if and only if a > (n − α)/α^{2} , so that Theorem 7.1 gives an extension of the authors’ [9].

For a proof of Theorem 7.1, we may assume that x_{0} = 0 without loss of generality. Before the proof we prepare the following two results.

Lemma 7.3. For x ∈ B(0, \frac{1}{2}r_{0}) and small δ > 0,

∫_{B(x,δ)} |x − y|^{p′(y)(α−n)} dy ≤ C ∫_{0}^{δ} φ(r)^{−α^{2}/(n−α)}r^{−1} dr.

Proof. First note from (2.2) and (φ) that

p′(y) ≤ p′_{0} − ω(|y|) for y ∈ B_{0} ,

where p′_{0} = n/(n − α) and ω(r) = (α^{2}/(n − α)^{2})(log φ(r))/ log(1/r) − C/ log(1/r) for 0 < r ≤ r_{0} ; set ω(r) = ω(r_{0}) for r > r_{0} . If 0 < δ ≤ \frac{1}{2}|x|, then we have

∫_{B(x,δ)} |x − y|^{p′(y)(α−n)} dy ≤ ∑_{j} ∫_{B(x,2^{−j+1}δ)\B(x,2^{−j}δ)} |x − y|^{p′(y)(α−n)} dy

≤ ∑_{j}(2^{−j}δ)^{(α−n)(p′_{0}−ω(2^{−j}δ))}σ_{n}(2^{−j+1}δ)^{n}

≤ C ∑_{j} φ(2^{−j}δ)^{−α^{2}/(n−α)}

≤ C ∫_{0}^{δ} φ(r)^{−α^{2}/(n−α)}r^{−1} dr,

where σ_{n} denotes the volume of the unit ball. Similarly, if \frac{1}{2}|x| < δ < \frac{1}{3}r_{0} , we have

\begin{aligned}∫_{B(x,δ)\B(x,|x|/2)} |x − y|^{p′(y)(α−n)} dy & ≤ C ∫_{B(0,3δ)} |y|^{p′(y)(α−n)} dy \\ & ≤ C ∫_{0}^{3δ} φ(r)^{−α^{2}/(n−α)}r^{−1} dr. \end{aligned}

Therefore it follows from the doubling property that

∫_{B(x,δ)} |x − y|^{p′(y)(α−n)} dy ≤ C ∫_{0}^{δ} φ(r)^{−α^{2}/(n−α)}r^{−1} dr

when 0 < δ < \frac{1}{3}r_{0} . Now the proof is completed.

Lemma 7.4. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. Then

∫_{B_{0}\{B(0,δ)∪B(x,δ)}} |x − y|^{α−n−1}f(y) dy ≤ Cδ^{−1}φ(δ)^{−α^{2}/n}

for x ∈ B(0, \frac{1}{2}r_{0}) and small δ > 0.

518 T. Futamura, Y. Mizuta and T. Shimomura

Proof. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. For k > 1 we have

∫_{B_{0}\{B(0,δ)∪B(x,δ)}} |x − y|^{α−n−1}f(y) dy

≤ k{∫_{B_{0}\{B(x,δ)∪B(0,δ)}}(|x − y|^{α−n−1}/k)^{p′(y)} dy
+ ∫_{B_{0}\{B(x,δ)∪B(0,δ)}} f(y)^{p(y)} dy}
≤ k{∫_{B_{0}\{B(x,δ)∪B(0,δ)}}(|x − y|^{α−n−1}/k)^{p′(y)} dy + 1}.

In view of the assumption of φ, we obtain

∫_{B_{0}\{B(x,δ)∪B(0,δ)}}(|x − y|^{α−n−1}/k)^{p′(y)} dy

≤ C{∫_{B_{0}\{B(x,δ)∪B(0,δ)}}(|x − y|^{α−n−1}/k)^{p′_{0}−ω(δ)} dy + 1}
≤ C{k^{−p′_{0}+ω(δ)} ∫_{δ}^{∞} t^{(α−n−1)(p′_{0}−ω(δ))+n}t^{−1} dt + 1}
≤ Ck^{−p′_{0}+ω(δ)}δ^{(α−n−1)(p′_{0}−ω(δ))+n}.

Considering k such that k^{−p′_{0}+ω(δ)}δ^{(α−n−1)(p′_{0}−ω(δ))+n} = 1, we see that

∫_{B_{0}\{B(0,δ)∪B(x,δ)}} |x − y|^{α−n−1}f(y) dy ≤ Cδ^{−1}φ(δ)^{−α^{2}/n},

as required.

Proof of Theorem 7.1. Let f be a nonnegative measurable function on B_{0} with ∥f∥_{p( ·)} ≤ 1. For 0 < k < 1, we have by Lemma 7.3

∫_{B(x,δ)} |x − y|^{α−n}f(y)dy

≤ k ∫_{B(x,δ)}{(|x − y|^{α−n}/k)^{p′(y)} + f(y)^{p(y)}} dy
≤ k{k^{−n/(n−α)} ∫_{B(x,δ)} |x − y|^{(α−n)p′(y)} dy + 1}
≤ k{Ck^{−n/(n−α)}Φ(δ)^{n/(n−α)} + 1}

whenever x ∈ B(0, \frac{1}{2}r_{0}) and 0 < δ < \frac{1}{2}r_{0} . Now, considering k = Φ(δ), we find

(7.1)  ∫_{B_{0}∩B(x,δ)} |x − y|^{α−n}f(y) dy ≤ CΦ(δ).

Hence, if x, z ∈ B(0, \frac{1}{2}r_{0}) and |x − z| < \frac{1}{4}r_{0} , then we have

(7.2)  ∫_{B(x,2|x−z|)} |x − y|^{α−n}f(y) dy ≤ CΦ(|x − z|).

On the other hand we have

∫_{B_{0}\B(x,2|x−z|)}||x − y|^{α−n} − |z − y|^{α−n}|f(y) dy
≤ C|x − z| ∫_{B_{0}\B(x,2|x−z|)} |x − y|^{α−n−1}f(y) dy
= C|x − z|{∫_{B_{0}\{B(x,2|x−z|)∪B(0,2|x−z|)}} |x − y|^{α−n−1}f(y) dy
+ ∫_{{B_{0}∩B(0,2|x−z|)}\B(x,2|x−z|)} |x − y|^{α−n−1}f(y) dy}.

It follows from Lemma 7.4 that

∫_{B_{0}\{B(x,2|x−z|)∪B(0,2|x−z|)}} |x − y|^{α−n−1}f(y) dy ≤ C|x − z|^{−1}φ(|x − z|)^{−α^{2}/n}.

Moreover we see from (7.1) that

\begin{matrix}∫_{\{B_{0}∩B(0,2|x−z|)\}\B(x,2|x−z|)} |x − y|^{α−n−1}f(y) dy \\ ≤ C|x − z|^{−1} ∫_{B_{0}∩B(0,2|x−z|)} |y|^{α−n}f(y) dy ≤ C|x − z|^{−1}Φ(|x − z|). \end{matrix}

Since φ(r)^{−α^{2}/n} ≤ CΦ(r) by the doubling property of φ, we obtain

∫_{B_{0}\B(x,2|x−z|)}||x − y|^{α−n} − |z − y|^{α−n}|f(y) dy ≤ CΦ(|x − z|).

Further we obtain by (7.2)

∫_{B(x,2|x−z|)} |z − y|^{α−n}f(y) dy ≤ CΦ(|x − z|).

Now, we establish

|U_{α}f(x) − U_{α}f(z)| ≤ CΦ(|x − z|),

as required.

Remark 7.5. If Φ(1) = ∞, then we can find f ∈ L^{p( ·)}(B_{0}) such that U_{α}f(0) = ∞, which means that U_{α}f is not continuous at 0.

For this purpose set

ψ(r) = ∫_{r}^{1} φ(t)^{−α^{2}/(n−α)}t^{−1} dt

and

f(y) = |y|^{−(n−α)/(p(y)−1)}ψ(|y|)^{−1}.

Take r_{0} so small that ψ(r) > e when 0 < r < r_{0} . Note that

r^{−(n−α)p/(p−1)+n} = r^{−(n−αp)/(p−1)} = φ(r)^{−α/(p−1)}

for r = |y| and p = p(y). By (φ) we have

φ(r)^{−α/(p−1)} ≥ Cφ(r)^{−α^{2}/(n−α)},

so that

\begin{aligned}U_{α}f(0) & = ∫_{B_{0}} |y|^{α−n−(n−α)/(p(y)−1)}ψ(|y|)^{−1} dy \\ & ≥ C ∫_{0}^{r_{0}} φ(t)^{−α^{2}/(n−α)}ψ(t)^{−1} dt/t = ∞ \end{aligned}

since ψ(0) = ∞ by our assumption.

On the other hand, taking a number δ such that 1 < δ < n/α and noting by (φ) that

φ(r)^{−α/(p−1)} ≤ Cφ(r)^{−α^{2}/(n−α)},

we have

\begin{aligned}∫_{B_{0}} f(y)^{p(y)} dy & = ∫_{B_{0}} |y|^{−(n−α)p(y)/(p(y)−1)}ψ(|y|)^{−p(y)} dy \\ & ≤ C ∫_{0}^{r_{0}} φ(t)^{−α^{2}/(n−α)}ψ(t)^{−δ} dt/t < ∞ \end{aligned}

since 1 < δ < n/α ≤ p(y) and ψ(0) = ∞, as required.

Finally, we consider a variable exponent p( ·) on the unit ball B such that

(7.3)  p(x) = p_{0} + \frac{log φ(ϱ(x))}{log(e/ϱ(x))}

for x ∈ B, where p_{0} = n/α; assume as above that

p(x) > p_{0} on B.
Theorem 7.6. If Φ(1) < ∞ and f ∈ L^{p( ·)}(B), then
|U_{α}f(x) − U_{α}f(z)| ≤ CΦ(|x − z|)

whenever x, z ∈ B.

Sobolev embeddings for variable exponent Riesz potentials 521

For a proof of Theorem 7.6, it suffices to show that

∫_{B(x,δ)} |x − y|^{p′(y)(α−n)} dy ≤ C ∫_{0}^{δ} φ(r)^{−α^{2}/(n−α)}r^{−1} dr

for x ∈ B and small δ > 0, as in Lemma 7.3. We obtain, in fact, this inequality in the same way as in Lemmas 6.2 and 7.3.

Remark 7.7. We do not know the best condition which assures the continuity of Riesz potentials in the metric space setting.

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Received 20 June 2005