Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

We prove Sobolev-type p( · ) → q( · ) -theorems for the Riesz potential operator I α in the weighted Lebesgue generalized spaces L p( · ) ( R n ,ρ) with the variable exponent p(x) and a two-parametrical power weight ﬁxed to an arbitrary ﬁnite point and to inﬁnity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L p( · ) ( S n ,ρ) on the unit sphere S n in R n + 1 .  2005 Elsevier Inc. All


Introduction
Recently, an obvious interest to the operator theory in the generalized Lebesgue spaces with variable exponent p(x) could be observed in a variety of papers, the main objects being the maximal operator, Hardy operators, singular operators and potential type operators, we refer, in particular to surveys [13,24].
In the case of maximal operators, we refer to L. Diening [5] for bounded domains in R n and to D. Cruz-Uribe, A. Fiorenza and C.J. Neugebauer [3] and A. Nekvinda [22,23] for unbounded domains, and to V. Kokilashvili and S. Samko [17] for weighted boundedness on bounded domains.
Sobolev p(·) → q(·)-theorem for potential operators on bounded domains was considered in S.G. Samko [25] and L. Diening [6], in [6] there being also treated the case of unbounded domains under the assumption that the maximal operator is bounded. Some version of the Sobolev-type theorem for unbounded domain was given in V. Kokilashvili and S. Samko [14]. The Sobolev theorem for unbounded domains in its natural form was proved by C. Capone, D. Cruz-Uribe and A. Fiorenza [1]. Another proof may be found in D. Cruz-Uribe, A. Fiorenza, J.M. Martell, and C. Perez [2] where there are also given new insights into the problems of boundedness of singular and maximal operators in variable exponent spaces.
In this paper we prove a weighted Sobolev-type theorem for the Riesz potential operator I α f (x) = R n f (y) |x − y| n−α dy, 0 < α < n, (1.1) over the whole space R n , in the weighted Lebesgue generalized spaces L p(·) (R n , ρ) with the variable exponent p(x) and power weight fixed to the origin and infinity. We prove also a similar theorem for the spherical analogue of the Riesz potential in the corresponding weighted spaces L p(·) (S n , ρ) on the unit sphere S n in R n+1 . The main results are formulated in Theorems 3.1 and 3.5. Theorem 3.5 for the spherical potential operators is derived from Theorem 3.1 for spatial potentials, while the proof of Theorem 3.1 is based on usage of the estimates obtained in [28].

The space L p(·) (R n , ρ)
By L p(·) (Ω, ρ) we denote the weighted space of functions f (x) on Ω such that where p(x) is a measurable function on Ω with values in [1, ∞) and 1 p − p(x) p + < ∞, x ∈ Ω and ρ is the weight function. This is a Banach function space with respect to the norm (see, e.g., [18]). We refer to [11,12,18,25] for basics of the spaces L p(·) with variable exponent.
We deal with Ω = R n and consider the weight fixed to the origin and infinity: We assume that the exponent p(x) satisfies the conditions observe that from (2.4) there follows that where Ω is any bounded domain in R n and N = 2 diam Ω. We treat p(x) as a function onṘ n whereṘ n is the compactification of R n by the unique infinite point. To manage with the weighted case under the consideration, we introduce an assumption on p(x) at infinity stronger than the usually considered assumption , x ∈ R n (2.6) (see, for instance, [3,26]); namely, we suppose that where p * (x) = p x |x| 2 . Condition (2.7) will be essentially used in the proof of Theorem 3.1, see the part "The term A −− " in Section 4. Namely, to be able to apply Theorem 2.3 given below, we will need the fact that after the inversion change of variables  (2.6), that is, there exist functions p(x) (and even radial ones) such that both the local log-condition and condition (2.6) are satisfied, but condition (2.7) does not hold. This is proved in Appendix A.

Theorem 2.3.
Let Ω be a bounded domain in R n and x 0 ∈Ω and let p(x) satisfy conditions (2.3) and (2.4) in Ω and ess sup x∈Ω p(x) < n α . Then the following estimate 14) Proof. Both the relations in (2.14) and (2.15) are verified directly: and similarly for the second relation in (2.14) and formula (2.15). The inequality in (2.16) is a consequence of (2.15). 2 Lemma 2.5. Let p satisfy condition (2.4). Then in the spherical layer 1 2 |x| 2 the inequality Proof. By (2.5), we have where we have used the second of the relations in (2.14). Hence (2.17) easily follows since 1 2 |x| 2. 2

The space L p(·) (S n , ρ)
We consider a similar weighted space with variable exponent on the unit sphere S n = {σ ∈ R n+1 : |σ | = 1}: where ρ β a ,β b (σ ) = |σ − a| β a · |σ − b| β b and a ∈ S n and b ∈ S n are arbitrary points on the unit sphere S n . For the variable exponent p(σ ) defined on S n we assume that Under assumption (2.18), this is a Banach space with respect to the norm

Stereographic projection
We use the stereographic projection (see, for instance, [19, p. 36]) of the sphere S n onto the space R n = {x ∈ R n+1 : x n+1 = 0} generated by the following change of variables in R n+1 : . We remind some useful formulas of passage from R n to S n : and inverse formulas of passage from S n to R n : where ξ = s(x) , σ = s(y), x, y ∈ R n+1 and e n+1 = (0, 0, . . . , 0, 1).

2)
and the exponents γ 0 and γ ∞ are related to each other by the equality In the case of constant p(x) ≡ p = const, the (p → q)-boundedness of the Riesz potential operator with the power weight |x| γ 0 is due to E.M. Stein and G. Weiss [29] without the additional condition (3.3). The general weighted case for constant p is due to B. Muckenhoupt and R. Wheeden [20]. The inequalities for the exponents γ 0 and γ ∞ in (3.2), as is well known, are necessary and sufficient for power weight to belong to the Muckenhoupt-Wheeden A pq -class. .7) and (2.10), and suppose that The statement of the corollary was proved in [1] and [2] without assumption (3.4) and under weaker assumption (2.6) instead of (2.7). . The spherical potential operator K α is bounded from the space L p(·) (S n , ρ β a ,β b ) with ρ β a ,β b (σ ) = |σ − a| β a · |σ − b| β b , where a ∈ S n and b ∈ S n are arbitrary points on the unit sphere S n , and the weight exponents β a and β b are related to each other by the connection Corollary 3.6. Under assumptions (2.18)-(2.20) the spherical potential operator K α is bounded from L p(·) (S n ) into L q(·) (S n ), 1 q(σ ) = 1 p(σ ) − α n .

Proof of Theorem 3.1
Proof. We denote We have to show that A In view of (2.3) it is easily seen that where dx so that we may separately estimate these terms. We note that the relation (3.3) will be used only in the estimation of the "mixed" terms A +− and A −+ .
The term A ++ . This term is covered by Theorem 2.3, the condition (2.12) of Theorem 2.3 being fulfilled by the first assumption in (3.2).
The term A −− . The estimation of A −− is reduced to that of A ++ by means of the simultaneous change of variables (inversion): As a result, we obtain By (2.14), we obtain dx.
The term A −+ . By the inversion change x → x * of the variable x, we have In contrast to the case of the terms A ++ and A −− , now the information about the integrability of ϕ(x) is known in terms of p(x), while h(x) should be integrated to the power q * (x), not q(x) (in the symmetrical term A +− , on the contrary, we will have to deal with q(x) preserved, but p(x) replaced by p * (x)). Fortunately, we may pass to q * (x) thanks to the properties of the inversion x * = x |x| 2 and the logarithmic smoothness of q(x) when x passes through the unit sphere. We proceed as follows. First we observe that |x| · |x * − y| |x − y| and |x| · |x * − y| 1 − |x| (4.6) for |x| 1 and |y| 1. The former of the inequalities in (4.6) was given in (2.16), the latter follows from the fact that |x * | 1 and |y| 1 and then |x * − y| |x * | − |y| = 1 |x| − |y| Here the term A 1 is finite since µ 1 > −n. For the term A 3 we have The term A 31 is finite since |x| · |x * − y| 1 2 for |x| 1 2 by (4.6) and then |h(x)| c ϕ L 1 c 1 ϕ L p(·) (R n ,ρ γ 0 ,γ∞ ) . For the term A 32 we have to show that To this end, we make use of the second inequality in (4.6) and obtain which is bounded for |x| 1 2 by Lemma 2.5. Gathering the estimates, we obtain Hence, by the first inequality in (4.6), and we are able now to apply Theorem 2.3. However, this requires the condition µ 1 Therefore, by (3.3) we may apply Theorem 2.3 which provides the necessary estimation A −+ c < ∞.

The term A +− . After the inversion change of variables in the inner integral in A +− we have
where ψ(y) = |y| −n−α ϕ(y * ) ∈ L p * (·) B + , |x| γ 1 is the same function as in (4.4). We distinguish the cases |y| 1 2 and |y| 1 2 . In the first case we make use the second of the inequalities in (4.6) in the form |y| · |x − y * | 1 − |y| 1 2 and then the estimation becomes trivial. In the case |y| 1 2 we make use of the first inequality in (4.6): |y| · |x − y * | |x − y| which gives a possibility to make use of Theorem 2.3, the passage to the exponent q * (x) = np * (x) n−αp * (x) in (4.8) is done in the same way as in the estimation of A −+ by distinguishing the cases where q(x) q * (x) and q(x) q * (x): we omit details of that passage to the exponent q * (x), they are symmetrical to those in the case of A −+ when we passed from q * (x) to q(x). We only mention that when proving the uniform boundedness of with q(x) q * (x), we may use the obvious inequality |x − y * | 1 − |x|.
When applying Theorem 2.3 with the exponents p * (x) and q * (x), according to condition (2.12) we have to assume that which gives the condition contrary to (4.7), which holds because of condition (3.3). Therefore the application of Theorem 2.3 ends the proof.
We make use of the Riesz-Thorin interpolation theorem, see Theorem 2.2, to show that the boundedness holds if instead of p(∞) = β we require that the value of 1 p(∞) does not differ much from 1 β , namely, . To avoid the condition p(∞) = β, we may interpolate between a constant p 0 > 1 and some r(·) for which the condition r(∞) = β holds. That is, we have to find θ ∈ (0, 1) and where r(x) satisfies the conditions inf x∈R n r(x) > 1, sup x∈R n r(x) < n α and r(∞) = β (4.10) (note that any log-condition for r(x) follows from the same log-condition of p(x)). Conditions (4.10) take the form , (4.11) respectively. By direct calculations, it can be proved that conditions (4.11) may be satisfied jointly with conditions p 0 ∈ (1, n α ) and θ ∈ (0, 1) if and only if assumption (3.4) holds. We prove this in Appendix B. 2

Proof of Theorem 3.5
The statement of Theorem 3.5 is derived from that of Theorem 3.1 by means of the stereographic projection. By Remark 3.4 and an appropriate rotation on the sphere, we reduce the proof to the case where a = e n+1 = (0, 0, . . . , 0, 1) where ξ = s(x), σ = s(y) and We have also the modular equivalence The direct verification shows that the corresponding intervals for the spherical weight exponents β a and β b coincide with the corresponding intervals for the spatial weight exponents γ 0 , γ ∞ : Similarly we have an equivalence between the relation (3.3) for spatial weight exponents γ 0 and γ ∞ and the relation (3.7) for spherical weight exponents, which in our case has the form where q(σ ) = np(σ ) n−αp(σ ) is the Sobolev limiting exponent on the sphere. In view of the relation (5.1) and equivalence (5.2) of norms, we then easily derive Theorem 3.5 from Theorem 3.1 after obvious recalculations.

Appendix A
To prove what was stated in Remark 2.1, it suffices to consider the case n = 1. The question we have to treat, is whether from the conditions there follows that where x, y ∈ R 1 + . This is equivalent to the following question. Let a continuous on [0, 1 2 ] function f (x) (= p( 1 x )), satisfy the log-condition everywhere beyond the origin: 3 ) The answer to this question is negative, because the only condition (A.2) may not prevent from the constant C δ in (A.1) to be tending to infinity when δ → 0. The corresponding counterexample is given in the lemma below. Proof. Let µ(x) ∈ C ∞ (R 1 ) be an even smooth "cap" with support in (−1, 1), 0 µ(x) 1, such that µ(0) = 1 and µ( 1 2 ) = 1 2 . Let also {b n } ∞ n=1 be a monotonically decreasing sequence of points in [0, 1 2 ] tending to 0 as n → ∞. We construct the "narrow" caps supported on (b n+1 , b n ). By the choice of µ(x), we have We denote ω(x) = 1 ln 1 x for brevity and construct the function f (x) in the form and the positive constants A k will be chosen later. Obviously, for any fixed x, the series in (A.5) contains one term only Under any choice of A n condition (A.1) is satisfied automatically, because for x ∈ [δ, 1 2 ], the series in (A.5) contains a finite number of terms and ω |x − a k+1 | − ω |y − a k+1 | ω |x − y| (where we took into account that the function ω(x) = 1 ln 1 x is the continuity modulus, that To satisfy condition (A.2) and show that (A.3) does not hold, we have to show that sup x∈[0, 1 2 ] G(x) < ∞ and sup x,y∈[0, 1 2 ] |f (x) − f (y)| ω(|x − y|) = ∞. A n ω(x)µ n (x)ω(|x − a n+1 |) ω(|x − a n+1 |) sup n A n ω(β n )µ n (β n ).