Variation inequalities related to Schrödinger operators on weighted Morrey spaces

Definition 1.1

A locally integrable function b in ℝⁿ is BMO_θ(γ) provided that

for all x ∈ ℝⁿ and r > 0, where b_B = ∣B∣⁻¹ ∫_B∣b(x) − b_B∣dx. We denote for b ∈ BMO_θ(γ)

It is easy to check that BMO(ℝⁿ) = BMO₀(γ) ⊂ BMO_θ(γ) ⊂ BMO_θ′(γ) for 0 ≤ θ ≤ θ′. Set BMO_∞(γ) = ∪_θ>0BMO_θ(γ). Then BMO_∞(γ) is larger than BMO(ℝⁿ) in general (see [12]).

For b ∈ BMO_∞(γ) and ℓ = 1, ⋯, n, the commutators Rb,ℓL and Rb,ℓL,∗ are defined by

It was shown in [12] that, for every b ∈ BMO_∞(γ) and ℓ = 1, ⋯, n, the operators Rb,ℓL (resp., Rb,ℓL,∗ ) are bounded on L^p(ℝⁿ), provided that 1 < p < p₀ (resp., p0′ < p < ∞) with 1/p₀ = (1/s−1/n)₊ for V ∈ RH_s, s ≥ n/2. In [7], Betancor et al obtained the following point-wise representations of the commutator operators by a principal value integral:

and

Moreover, the authors in [7] proved that for every b ∈ BMO_∞(γ) and ℓ = 1, ⋯, n, the variation operators Vq(Rb,ℓ,εL)(resp.,Vq(Rb,ℓ,εL,∗)) associated with the family of truncations {Rb,ℓ,εL}ε>0(resp.,{Rb,ℓ,εL,∗}ε>0) are bounded from L^p(ℝⁿ) into itself, provided that 1 < p < p₀ (resp., p0′ < p < ∞) with 1/p₀ = (1/s−1/n)₊ for V ∈ RH_s, s ≥ n/2. In [9], the above results have been extend to the weighted Lebesgue space.

On the other hand, in order to extend the boundedness of Schrödinger type operators in Lebesgue spaces, Pan and Tang [13] introduced the following weighted Morrey spaces related to the non-negative potential V, denoted by Lα,V,ωp,λ (ℝⁿ).

Definition 1.2

Let k > 1, p ∈ [1, ∞), α ∈ (−∞, +∞) and λ ∈ [0, 1). For f ∈ Llocp (ℝⁿ) and V ∈ RH_q (q > 1), we say f ∈ Lα,V,ωp,λ (ℝⁿ) provided that

where B = B(x₀, r) denote a ball with centered at x₀ and radius r, γ(x₀) is the critical radius at x₀ defined as in (1.5) and the weight function ω ∈ Apγ,∞ .

Clearly, when α = 0 or V = 0, ω = 1 and 0 < λ < 1, the spaces Lα,V,ωp,λ (ℝⁿ) are the classical Morrey spaces L^p,λ(ℝⁿ), which were introduced by Morrey [14] in 1938 and were subsequently found to have many important applications to the elliptic equations (see [15, 16, 17, 18, 19, 20]), the Navier-Stokes equations (see [21, 22, 23]) and the Schrödinger equations (see [24, 25, 26, 27]) etc. When α = 0 or V = 0 and 0 < λ < 1, the spaces Lωp,λ (ℝⁿ) is first introduced in [28], where ω ∈ A_p(ℝⁿ)(Muckenhoupt weights class). It is easy to see that Lα,V,ωp,λ (ℝⁿ) ⊂ Lωp,λ (ℝⁿ) for α > 0 and Lωp,λ (ℝⁿ) ⊂ Lα,V,ωp,λ (ℝⁿ) for α < 0. In [29], the authors established the Lα,Vp,λ (ℝⁿ)-boundedness of the Riesz transforms Vq(RℓL),Vq(RℓL,∗), and the corresponding commutators with V ∈ B_n/2.

Based on the above, it is a natural and interesting question whether we can establish the Lα,V,ωp,λ (ℝⁿ)-boundedness of the variation operators aforementioned in Schrödinger setting. The main purpose of this paper is to answer this question. Our results can be formulated as follows:

Theorem 1.3

Let ℓ = 1, ⋯ n, q > 2 and V ∈ RH_s. Assume α ∈ (−∞, +∞) and λ ∈ (0, 1). Then

1. If n/2 ≤ s < n and p₀ is such that 1/p₀ = 1/s−1/n, then the variation operator Vq(Rl,εL,∗) is bounded on Lα,V,ωp,λ (ℝⁿ) for p0′ ≤ p < ∞ and ω ∈ Ap/p0′γ,∞.

2. If n/2 ≤ s < n and p₀ is such that 1/p₀ = 1/s−1/n, then the variation operator Vq(Rl,εL) is bounded on Lα,V,ωp,λ (ℝⁿ) for 1 < p ≤ p₀ and ω−1p−1∈Ap′/p0′γ,∞.

3. If s ≥ n, then the variation operators Vq(Rl,εL,∗) and Vq(Rl,εL) are bounded on Lα,V,ωp,λ (ℝⁿ) for 1 < p < ∞ and ω ∈ Apγ,∞

Theorem 1.4

Let ℓ = 1, ⋯, n. Assume that q > 2 and V ∈ RH_s. (i) If n/2 ≤ s < n then for p = 1, η > 0 and ωp0′∈A1γ,∞,

(ii) If s ≥ n, then for p = 1, η > 0 and ω ∈ A1γ,∞,

and

holds for all balls B, where C is independent of x, r, η and f.

Theorem 1.5

Let ℓ = 1, ⋯, n, q > 2, b ∈ BMOθγ with θ > 0 and V ∈ RH_s, Assume that α ∈ (−∞, +∞) and λ ∈ (0, 1).

1. If n/2 ≤ s < n and p₀ is such that 1/p₀ = 1/s−1/n, then the variation operator Vq(Rb,l,εL,∗) is bounded on Lα,V,ωp,λ (ℝⁿ) for p0′ ≤ p < ∞ and ω ∈ Ap/p0′γ,∞.

2. If n/2 ≤ s < n and p₀ is such that 1/p₀ = 1/s−1/n, then the variation operator Vq(Rb,l,εL) is bounded on Lα,V,ωp,λ (ℝⁿ) for 1 < p ≤ p₀, with ω−1p−1∈Ap′/p0′γ,∞.

3. If s ≥ n, then the variation operator Vq(Rb,l,εL,∗) and Vq(Rb,l,εL) are bounded on Lα,V,ωp,λ (ℝⁿ) for 1 < p < ∞ and ω ∈ Apγ,∞.

Remark 1.6

In [8], it was proved that if V is a nonnegative polynomial, then V ∈ RH_s for any 1 < s < ∞. Therefore, as special cases of our results, the corresponding ones to the Hermite operator: H = −Δ+∣x∣² hold. This can be regarded as the generalization of the corresponding results in [2, 30].

The rest of this paper is organized as follows. In Section 2, we will recall some properties of the function γ and some basic facts concerning weights Apγ , which will play a crucial roles in our arguments. In Section 3, we will prove Theorem 1.3 and 1.4, the proof of Theorem 1.5 will be given in Section 4. Throughout this paper, the letter C always denotes a positive constant that is independent of main parameters involved but whose value may differ from line to line. We use f ≲ g to denote f ≤ Cg. If f ≲ g ≲ f, we write f ∼ g. For any t ∈ (0, ∞), we denote the ball B(x, tr) by tB. Given any p ∈ [1, ∞], p′ = pp−1 denote its conjugate index. In particular, it should be pointed out that these weighted Morrey spaces in Definition 1.2 are equivalent for different k > 1.

Lemma 2.1

(cf. [8]) If V ∈ RH_n/2, then there exist c₀ and l₀ ≥ 1 such that for all x, y ∈ ℝⁿ,

In particular, γ(x) ∼ γ(y) if ∣x − y∣ < Cγ(x), and the ball B(x, γ(x)) is called critical.

According to [31], we recall a new class of weights Apγ,∞=⋃θ≥0Apγ,θ for p ≥ 1, where Apγ,θ (p > 1) is the set of those weights satisfying

and A1γ,θ is the set of those weights ω such that

for every ball B = B(x, r).

Clearly, the classes Apγ,θ are increasing with θ and for θ = 0, they are just the Munkenhoupt classes A_p. From [32], we know that the following properties for Apγ,θ hold.

Lemma 2.2

([32]) Let 1 < p < ∞, 0 < θ < ∞. Then

1. Ap1γ,θ⊂Ap2γ,θ for 1 ≤ p₁ < p₂ < ∞;

2. ω ∈ Apγ,θ if and only if ω−1p−1∈Ap′γ,θ, where 1/p+1/p′ = 1;

3. if ω ∈ Apγ,θ for 1 ≤ p < ∞, then there exists a constant such that for any κ > 1,

   ω(κB(x0,r))≤C(1+κrγ(x0))(l0+1)θω(B(x0,r)). (2.4)

Lemma 2.3

([32]) Let 0 < θ < ∞, 1 ≤ p < ∞. If ω ∈ Apγ,θ , then there exist positive constant δ > 1, σ and C such that

for any measurable subset E of a ball B(x₀, r).

Lemma 2.4

([32]) If ω ∈ Apγ,∞ , 1 ≤ p < ∞, then there exist positive constant β, τ and C such that

for every ball B = B(x, r).

Lemma 2.5

([32]) If ω ∈ Apγ,∞ , 1 ≤ p < ∞, then there exist positive constant ϵ > 0 such that ω ∈ Ap−ϵγ,∞ for every ball B = B(x, r).

Lemma 3.1

[8] Let l = 1, ⋯, n, V ∈ RH_s with s > n/2. Then:

1. For every N ∈ ℕ, there exist C > 0 such that

   |Rℓ∗(x,z)|≤C(1+|x−z|/γ(x))−N|x−z|n−1(∫B(z,|x−z|4)V(u)|u−z|n−1du+1|x−z|). (3.1)

   Moreover, the last inequality also holds with γ(x) replaced by γ(z).

2. When s > n, the term involving V can be dropped from inequalities (3.1).

Proof of Theorem 1.3

To prove (i), without loss of generality, we may assume α < 0 and ω ∈ Ap/p0′γ,∞. Pick any ball B := B(x₀, r) and write f(x) = f₀(x) + ∑i=1∞ f_i(x), where f₀ = fχ_2B, f_i = fχ_{2ⁱ⁺¹B∖2ⁱB} for i ≥ 1. By the weighted L^p-boundedness of Vq(Rℓ,εL) (f) (see [9]). Hence, we have

From (1.3), we have

In the term last but one, we used that ∥χ_{εi+1<∣x−y∣<εi}∥_Fq ≤ 1.

Now it follows from Lemma 3.1 that

For term A₁, using Lemma 2.1, we have

Now, we will estimate the term A₂. Using (2.1) and Hölder’s inequality, we can write

If we choose ϱ such that 1ϱ+1p0+1p=1. Then by Hölder’s inequality, we have

Using the boundedness of the 1-th Euclidean fractional integral 𝓙: L^s → L^p₀ with 1/p₀ = 1/s−1/n, we obtain that

Recall that V ∈ B_s for some s > 1 implies that V satisfies the doubling condition, i.e., there exist constants μ ≥ 1 and C such that,

holds for every ball B and t > 1. Therefore

Since ω ∈ Ap/p0′γ,θ and p/ϱ = p/ p0′ − 1, we get

Go back to (3.10), we have

This together with (3.4) by choosing N which is big enough, we get

For part (ii), we note that the adjoint Rℓ,εL of Rℓ,εL,∗, when V ∈ B_s, with n/2 ≤ s < n. We write

By proceeding as the previous proofs (i), we can prove that Vq(Rℓ,εL) is bounded on Lα,V,ωp,λ (ℝⁿ) for 1 < p ≤ p₀, with ω−1p−1∈Ap′/p0′γ. We omit the details.

The proof of part (iii) can be given analogously as in (i) and (ii), we leave the part to the part to the interested readers and complete the proof of therorem 1.3.□

Proof of Theorem 1.4

As for the case p = 1, by replacing (3.2) with the corresponding weak estimate. By the Vq(Rℓ,εL) (f) is bounded of weighted weak type (1,1) (see [10]), we have