DYNAMICAL SYSTEMS SERIES B
Volume 25, Number 2, February 2020 pp. 671–690
SCHR ¨ODINGER TYPE OPERATORS ON LOCAL GENERALIZED
MORREY SPACES RELATED TO CERTAIN NONNEGATIVE POTENTIALS
Vagif S. Guliyev
S.M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia Department of Mathematics, Dumlupinar University, 43100 Kutahya, Turkey Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan
Ramin V. Guliyev
Department of Mathematics, Dumlupinar University, 43100 Kutahya, Turkey
Mehriban N. Omarova
Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan Baku State University, AZ1141 Baku, Azerbaijan
Maria Alessandra Ragusa∗
Dipartimento di Matematica e Informatica, Universita di Catania, Catania, Italy S.M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia
In honor of the 60-th birthday of Juan J. Nieto
Abstract. We establish the boundedness of some Schr¨odinger type operators on local generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H¨older class.
1. Introduction and main results. In this paper, we consider the Schr¨odinger differential operator
L = −∆ + V (x) on Rn, n ≥ 3,
where V (x) is a nonnegative potential belonging to the reverse H¨older class RHq
for q ≥ n/2.
A nonnegative locally Lq integrable function V (x) on Rn is said to belong to
RHq, 1 < q ≤ ∞ if there exists C > 0 such that the reverse H¨older inequality
1 |B(x, r)| Z B(x,r) Vq(y)dy !1/q ≤ C |B(x, r)| Z B(x,r) V (y)dy ! (1)
2010 Mathematics Subject Classification. Primary: 42B20, 42B25, 42B35; Secondary: 35J10, 47H50.
Key words and phrases. Fractional integral operator, Schr¨odinger differential operator, frac-tional calculus.
The research of V.S. Guliyev and M. Omarova was partially supported by the grant of 1st Azerbaijan Russia Joint Grant Competition (Grant No. EIF-BGM-4-RFTF-1/2017-21/01/1).
The research of V.S. Guliyev and M.A. Ragusa are partially supported by the Ministry of Education and Science of the Russian Federation (Agreement number N. 02.03.21.0008).
∗Corresponding author: maragusa@dmi.unict.it (Maria Alessandra Ragusa).
holds for every x ∈ Rn and 0 < r < ∞, where B(x, r) denotes the ball centered at x with radius r. In particular, if V is a nonnegative polynomial, then V ∈ RH∞.
For x ∈ Rn, the function ρ(x) is defined by (see [25,26])
ρ(x) := 1 mV(x) = sup r>0 ( r : 1 rn−2 Z B(x,r) V (y)dy ≤ 1 ) .
Obviously, 0 < mV(x) < ∞ if V 6= 0. In particular, mV(x) = 1 with V = 1 and
mV(x) ≈ 1 + |x| with V (x) = |x|2.
According to [4], the new BMO space BM Oθ(ρ) with θ ≥ 0 is defined as a set
of all locally integrable functions b such that 1 |B(x, r)| Z B(x,r) |b(y) − bB|dy ≤ C 1 + r ρ(x) θ
for all x ∈ Rn and r > 0, where b
B = |B|1 RBb(y)dy. A norm for b ∈ BM Oθ(ρ),
denoted by [b]θ, is given by the infimum of the constants in the inequalities above.
Clearly, BM O ⊂ BM Oθ(ρ).
For brevity, in the sequel we use the notations Aα,Vp,ϕ(f ; x, r) :=1 + r ρ(x) α r−n/pϕ(x, r)−1kf kLp(B(x,r)) and AW,α,VΦ,ϕ (f ; x, r) :=1 + r ρ(x) α r−n/pϕ(x, r)−1kf kW Lp(B(x,r)).
We now present the definition of generalized Morrey spaces (including weak ver-sion) associated with Schr¨odinger operator, which introduced by first author in [17].
Definition 1.1. Let ϕ(x, r) be a positive measurable function on Rn × (0, ∞),
1 ≤ p < ∞, α ≥ 0, and V ∈ RHq, q > 1. We denote by Mp,ϕα,V = Mp,ϕα,V(Rn)
the generalized Morrey space associated with Schr¨odinger operator, the space of all functions f ∈ Lploc(Rn) with finite quasinorm
kf kMα,V
p,ϕ = sup x∈Rn,r>0
Aα,Vp,ϕ(f ; x, r).
Also W Mα,V
p,ϕ = W Mp,ϕα,V(Rn) we denote the weak generalized Morrey space
associ-ated with Schr¨odinger operator,the space of all functions f ∈ W Lploc(Rn) with
kf kW Mα,V
p,ϕ = sup x∈Rn,r>0
AW,α,Vp,ϕ (f ; x, r) < ∞. Remark 1. (i) When α = 0, and ϕ(x, r) = r(λ−n)/p, Mα,V
p,ϕ (Rn) is the classical
Morrey space Lp,λ(Rn) introduced by Morrey in [22];
(ii) When ϕ(x, r) = r(λ−n)/p, Mα,V
p,ϕ(Rn) is the Morrey space associated with
Schr¨odinger operator Lα,Vp,λ(Rn) studied by Tang and Dong in [34];
(iii) When α = 0, Mα,V
p,ϕ(Rn) is the generalized Morrey space Mp,ϕ(Rn)
intro-duced by Guliyev, Mizuhara and Nakai in [12,21,23].
(iv) The generalized Morrey space associated with Schr¨odinger operator Mα,V p,ϕ (Rn) was introduced by first author in [17].
Morrey spaces Lp,λ(Rn) were originally introduced by Morrey in [22] to study the
systems. Many classical operators of harmonic analysis were studied, in Morrey-type spaces and similar classes of functions, during the last decades (see e.g. [3] [9], [28], [24], [29], [30], [36]).
Definition 1.2. Let ϕ(x, r) be a positive measurable function on Rn × (0, ∞),
1 ≤ p < ∞, α ≥ 0, and V ∈ RHq, q ≥ 1. For any fixed x0 ∈ Rn we denote by
LMα,V,{x0}
p,ϕ = LMp,ϕα,V,{x0}(Rn) the local generalized Morrey space associated with
Schr¨odinger operator, the space of all functions f ∈ Lloc
p (Rn) with finite norm
kf k LMp,ϕα,V,{x0} = supr>0A α,V p,ϕ(f ; x0, r). Also W LMα,V,{x0} p,ϕ = W LM α,V,{x0}
p,ϕ (Rn) we denote the weak local generalized
Mor-rey space associated with Schr¨odinger operator, the space of all functions f ∈ W Lloc p (Rn) with kf kW LMα,V,{x0} p,ϕ = supr>0A W,α,V p,ϕ (f ; x0, r) < ∞.
The local spaces LMα,V,{x0}
p,ϕ (Rn) and W LM α,V,{x0}
p,ϕ (Rn) are Banach spaces with
respect to the norm
kf kLMα,V,{x0} p,ϕ = supr>0A α,V p,ϕ(f ; x0, r), kf kW LMα,V,{x0} p,ϕ = supr>0A W,α,V p,ϕ (f ; x0, r), respectively.
Remark 2. (i) When α = 0, and ϕ(x, r) = r(λ−n)/p, LMα,V,{x0}
p,ϕ (Rn) is the local
(central) Morrey space LMp,λ{0}(Rn) studied in [2]; (ii) When α = 0, LMα,V,{x0}
p,ϕ (Rn) is the local generalized Morrey space V Mp,ϕ{x0}
(Rn) were introduced by first author in [12], see also [13,15,18] etc.
From [31,37], we know some Schr¨odinger type operators, such as ∇(−∆+V )−1∇ with V ∈ Bn, ∇(−∆ + V )−1/2 with V ∈ Bn, (−∆ + V )−1/2∇ with V ∈ Bn,
(−∆+V )iγ
with γ ∈ R and V ∈ Bn/2, and ∇2(−∆+V )−1where V is a nonnegative
polynomial, are standard Calder´on-Zygmund operators; see [33]. In particular, the kernels K of operators above all satisfy
|K(x, y)| ≤ Ck 1 +|x−y|ρ(x) N
1 |x − y|n
for any N ∈ N. Hence, in the rest of this paper, we always assume that T denotes the above operators.
It is well known that the boundedness of the standard Calder´on-Zygmund oper-ators and their commutoper-ators have been established on the class generalized Morrey spaces (see [14,15,16,21,23,32]). Hence, it will be an interesting question whether we can establish the boundedness of Schr¨odinger type operators on the Morrey spaces related to certain nonnegative potentials (see [17]). The main purpose of this paper is to answer the above question. More precisely, we obtain the following results.
Theorem 1.3. Let x0 ∈ Rn, V ∈ RHn/2, α ≥ 0, q ≥ n/2, 1 ≤ p < ∞ and
ϕ1, ϕ2∈ Ωα,Vp,loc satisfies the condition
Z ∞ r ess inf t<s<∞ϕ1(x0, s)s n p tnp dt t ≤ c0ϕ2(x0, r), (2)
where c0does not depend on x0and r. Then the operator T is bounded on LM α,V,{x0} p,ϕ1 to LMα,V,{x0}
p,ϕ2 for p > 1 and from LM α,V,{x0}
1,ϕ1 to W LM α,V,{x0} 1,ϕ2 .
Corollary 1. Let V ∈ RHn/2, α ≥ 0, 1 ≤ p < ∞ and ϕ1, ϕ2∈ Ωα,Vp satisfies the
condition Z ∞ r ess inf t<s<∞ϕ1(x, s)s n p tnp dt t ≤ c0ϕ2(x, r), (3)
where c0does not depend on x and r. Then the operator T is bounded on Mp,ϕα,V1 to Mα,V
p,ϕ2 for p > 1 and from M α,V
1,ϕ1 to W M α,V 1,ϕ2.
Theorem 1.4. Let x0 ∈ Rn, V ∈ RHn/2, α ≥ 0, 1 < p < ∞ and ϕ1, ϕ2 ∈ Ω α,V p,loc
satisfies the condition Z ∞ r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p tnp dt t ≤ c0ϕ2(x0, r), (4) where c0 does not depend on x0 and r. If b ∈ BM Oθ(ρ), then the operator [b, T ]
is bounded from LMα,V,{x0} p,ϕ1 to LM α,V,{x0} p,ϕ2 and from LM α,V,{x0} Φ,ϕ1 to W LM α,V,{x0} 1,ϕ2 , where Φ(t) = t ln(e+t), kf k
LMΦ,ϕα,V,{x0} = kΦ(|f |)kLM1,ϕα,V,{x0}, and C does not depend
on f .
Corollary 2. Let V ∈ RHn/2, α ≥ 0, 1 < p < ∞ and ϕ1, ϕ2∈ Ωα,Vp satisfies the
condition Z ∞ r 1 + lnt r ess inft<s<∞ϕ1(x, s)s n p tnp dt t ≤ c0ϕ2(x, r), (5)
where c0 does not depend on x and r. If b ∈ BM Oθ(ρ), then the operator [b, T ]
is bounded from Mp,ϕα,V1 to Mp,ϕα,V2 and from MΦ,ϕα,V
1 to W M α,V
1,ϕ2, where kf kMΦ,ϕα,V =
kΦ(|f |)kMα,V 1,ϕ
, and C does not depend on f .
Remark 3. Note that, Theorems1.3 and1.4 in the case of V ≡ 0 was proved in [15, Theorems 4 and 6].
Definition 1.5. Let L = −4 + V with V ∈ RHn/2. The fractional integral
associated with L is defined by
IβLf (x) = L−β/2f (x) = Z ∞
0
e−tL(f )(x) tβ/2−1dt for 0 < β < n. Let b ∈ BM Oθ(ρ). The commutator of IβL is defined by
[b, IβL]f (x) = b(x)IβLf (x) − IβL(bf )(x).
In this paper, we consider the boundedness of the fractional integral operator IL β
on the generalized Morrey spaces Mp,ϕα,V(Rn). When b belongs to the new BM O
space BM Oθ(ρ), we also show that [b, IβL] is bounded on M α,V
p,ϕ (Rn) to Mq,ϕα,V(Rn).
Theorem 1.6. Let x0 ∈ Rn, V ∈ RHn/2, α ≥ 0, 1 < p < n/β, 1/q = 1/p − β/n
and ϕ1∈ Ωα,Vp,loc, ϕ2∈ Ωα,Vq,loc satisfies the condition
Z ∞ r ess inf t<s<∞ϕ1(x0, s)s n p tnq dt t ≤ c0ϕ2(x0, r), (6)
where c0 does not depend on x0 and r. Then the operator IβL is bounded on
LMα,V,{x0}
p,ϕ1 to LM
α,V,{x0}
q,ϕ2 for p > 1 and from LM α,V,{x0} 1,ϕ1 to W LM α,V,{x0} n n−β,ϕ2 . Corollary 3. Let V ∈ RHn/2, α ≥ 0, 1 < p < n/β, 1/q = 1/p − β/n and
ϕ1∈ Ωα,Vp , ϕ2∈ Ωα,Vq satisfies the condition
Z ∞ r ess inf t<s<∞ϕ1(x, s)s n p tnq dt t ≤ c0ϕ2(x, r), (7)
where c0 does not depend on x and r. Then the operator IβL is bounded on M α,V p,ϕ1 to Mq,ϕα,V2 for p > 1 and from M
α,V 1,ϕ1 to W M α,V n n−β,ϕ2. Theorem 1.7. Let x0 ∈ Rn, V ∈ RHn/2, α ≥ 0, 1 < p < n/β, 1/q = 1/p − β/n and ϕ1∈ Ω α,V p,loc, ϕ2∈ Ω α,V
q,loc satisfies the condition
Z ∞ r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p tnq dt t ≤ c0ϕ2(x0, r), (8) where c0 does not depend on x0 and r. If b ∈ BM Oθ(ρ), then the operator [b, IβL]
is bounded from LMα,V,{x0} p,ϕ1 to LM α,V,{x0} q,ϕ2 and from LM α,V,{x0} Ψ,ϕ1 to W LM α,V,{x0} n n−β,ϕ2 , where Ψ(t) = [t ln(e + tβ/n)]n/(n−β), and C does not depend on f .
Remark 4. Note that, Theorems1.6 and1.7 in the case of V ≡ 0 was proved in [16, Theorems 3.6 and 4.5].
Corollary 4. Let V ∈ RHn/2, α ≥ 0, 1 < p < n/β, 1/q = 1/p − β/n and
ϕ1∈ Ωα,Vp , ϕ2∈ Ωα,Vq satisfies the condition
Z ∞ r 1 + lnt r ess inft<s<∞ϕ1(x, s)s n p tnq dt t ≤ c0ϕ2(x, r), (9)
where c0 does not depend on x and r. If b ∈ BM Oθ(ρ), then the operator [b, IβL] is
bounded from Mp,ϕα,V1 to M α,V q,ϕ2 and from M α,V Ψ,ϕ1 to W M α,V n n−β,ϕ2.
Remark 5. Note that, Corollaries 1, 2, 3 and 4 in the case of ϕ(x, r) = r(λ−n)/p
was proved in [34, Theorems 1.1, 1.2, 1.3 and 1.4].
In this paper, we shall use the symbol A . B to indicate that there exists a universal positive constant C, independent of all important parameters, such that A ≤ CB. Moreover, A ≈ B means that A . B and B . A.
2. Some preliminaries. We would like to recall the important properties con-cerning the critical function.
Lemma 2.1. [31] Let V ∈ RHn/2. For the associated function ρ there exist C and k0≥ 1 such that C−1ρ(x)1 +|x − y| ρ(x) −k0 ≤ ρ(y) ≤ Cρ(x)1 +|x − y| ρ(x) 1+k0k0 (10) for all x, y ∈ Rn.
Lemma 2.2. [1] Suppose x ∈ B(x0, r). Then for k ∈ N we have
1 1 + ρ(x)2kr N . 1 1 +ρ(x2kr 0) N/(k0+1).
We give some inequalities about the new BMO space BM Oθ(ρ).
Lemma 2.3. [4] Let 1 ≤ s < ∞. If b ∈ BM Oθ(ρ), then
1 |B| Z B |b(y) − bB|sdy 1/s ≤ [b]θ 1 + r ρ(x) θ0
for all B = B(x, r), with x ∈ Rn and r > 0, where θ0 = (k0+ 1)θ and k0 is the
constant appearing in (10).
Lemma 2.4. [4] Let 1 ≤ s < ∞, b ∈ BM Oθ(ρ), and B = B(x, r). Then
1 |2kB| Z 2kB |b(y) − bB|sdy 1/s ≤ [b]θk 1 + 2 kr ρ(x) θ0 for all k ∈ N, with θ0 as in Lemma2.3.
Let Kβ be the kernel of IβL. The following result give the estimate on the kernel
Kβ(x, y).
Lemma 2.5. [5] If V ∈ RHn/2, then for every N , there exists a constant C such
that |Kβ(x, y)| ≤ C 1 +|x−y|ρ(x) N 1 |x − y|n−β. (11)
Finally, we recall a relationship between essential supremum and essential infi-mum.
Lemma 2.6. [35] Let f be a real-valued nonnegative function and measurable on E. Then ess inf x∈E f (x) −1 = ess sup x∈E 1 f (x).
It is natural, first of all, to find conditions ensuring that the spaces LMα,V,{x0} p,ϕ
and Mα,V
p,ϕ are nontrivial, that is consist not only of functions equivalent to 0 on Rn.
Lemma 2.7. Let x0∈ Rn, ϕ(x, r) be a positive measurable function on Rn×(0, ∞),
1 ≤ p < ∞, α ≥ 0, and V ∈ RHq, q ≥ 1. If sup t<r<∞ 1 + r ρ(x0) α r− n p ϕ(x0, r) = ∞ for some t > 0, (12) then LMα,V,{x0}
p,ϕ (Rn) = Θ, where Θ is the set of all functions equivalent to 0 on
Proof. Let (13) be satisfied and f be not equivalent to zero. Then kf kL p(B(x0,t)) > 0, hence kf kLMα,V,{x0} p,ϕ ≥ supt<r<∞ 1 + r ρ(x0) α ϕ(x0, r)−1r− n pkf kL p(B(x0,r)) ≥ kf kLp(B(x0,t)) sup t<r<∞ 1 + r ρ(x0) α ϕ(x0, r)−1r− n p. Therefore kf kLMα,V,{x0} p,ϕ = ∞.
Lemma 2.8. [1] Let ϕ(x, r) be a positive measurable function on Rn × (0, ∞),
1 ≤ p < ∞, α ≥ 0, and V ∈ RHq, q ≥ 1. (i) If sup t<r<∞ 1 + r ρ(x) α r− n p
ϕ(x, r) = ∞ for some t > 0 and for all x ∈ R
n, (13) then Mα,V p,ϕ(Rn) = Θ. (ii) If sup 0<r<τ 1 + r ρ(x) α
ϕ(x, r)−1 = ∞ for some τ > 0 and for all x ∈ Rn, (14) then Mα,V
p,ϕ(Rn) = Θ.
Remark 6. We denote by Ωα,Vp,loc the sets of all positive measurable functions ϕ on Rn× (0, ∞) such that for all t > 0,
sup x∈Rn 1 + r ρ(x) α r− n p ϕ(x, r) L ∞(t,∞) < ∞. Moreover, we denote by Ωα,V
p the sets of all positive measurable functions ϕ on
Rn× (0, ∞) such that for all t > 0,
sup x∈Rn 1 + r ρ(x) α r−np ϕ(x, r) L∞(t,∞) < ∞, and sup x∈Rn 1 + r ρ(x) α ϕ(x, r)−1 L∞(0,t) < ∞.
For the non-triviality of the spaces LMα,V,{x0}
p,ϕ (Rn) and Mp,ϕα,V(R
n) we always
assume that ϕ ∈ Ωα,Vp,locand ϕ ∈ Ωα,V
p , respectively.
3. Proof of Theorem1.3. To prove Theorem1.3, we first investigate the follow-ing local estimate.
Theorem 3.1. Let 1 < p < ∞, then the inequality kT f kLp(B(x0,r)). r n p Z ∞ 2r kf kLp(B(x0,t)) tnp dt t (15)
holds for any ball B(x0, r) and for all f ∈ L p loc(R
n).
Moreover, for p = 1 the inequality kT f kW L1(B(x0,r)). r nZ ∞ 2r kf kL1(B(x0,t)) tn dt t (16)
holds for any ball B(x0, r) and for all f ∈ L1loc(R n).
Proof. For arbitrary x0∈ Rn, set B = B(x0, r) and λB = B(x0, λr) for any λ > 0.
We write f as f = f1+ f2, where f1(y) = f (y)χB(x0,2r)(y), and χB(x0,2r) denotes the
characteristic function of B(x0, 2r). Then
kT f kLp(B(x0,r))≤ kT f1kLp(B(x0,r))+ kT f2kLp(B(x0,r)).
Since f1∈ Lp(Rn) and from the boundedness of T on Lp(Rn), 1 < p < ∞ (see [31])
it follows that kT f1kLp(B(x0,r)). kf kLp(B(x0,2r)). r n pkf k Lp(B(x0,2r)) Z ∞ 2r dt tnp+1 . rnp Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . (17)
To estimate kT f2kLp(B(x0,2r)) obverse that x ∈ B, y ∈ (2B)
c implies 1
2|x0− y| ≤
|x − y| ≤ 3
2|x0− y|. Then by Lemma2.6for all x ∈ B(x0, r) we have
|T f2(x)| ≤ Z (2B)c |K(x, y)f (y)|dy . Z (2B)c 1 1 + |x−y|ρ(x)N |f (y)| |x − y|ndy . Z (2B)c 1 1 + |x0−y| ρ(x) N |f (y)| |x0− y|n dy.
By H¨older’s inequality and Lemma2.3we get |T f2(x)| . 1 1 +ρ(x)2r N Z (2B)c |f (y)| |x0− y|n dy . 1 1 +ρ(x)2r N ∞ X k=1 (2k+1r)−n Z 2k+1B |f (y)|dy . 1 1 +ρ(x)2r N ∞ X k=1 kf kLp(2k+1B)(2 k+1r)−1−n p Z 2k+1r 2kr dt . 1 1 +ρ(x2r 0) N/(k0+1) ∞ X k=1 Z 2k+1r 2kr kf kLp(B(x0,t)) tnp dt t . 1 1 +ρ(x2r 0) N/(k0+1) Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . Then sup x∈B(x0,r) |T f2(x)| . 1 1 +ρ(x2r 0) N/(k0+1) Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . (18) Taking N ≥ k0+ 1, then kT f2kLp(B(x0,r)). r n p Z ∞ 2r kf kLp(B(x0,t)) tnp dt t (19) holds for 1 ≤ p < ∞.
Let p = 1. From the weak (1, 1) boundedness of T and (17) it follows that: kT f1kW L1(B(x0,r)). kT f1kW L1(Rn). kf1kL1(Rn) . kf kL1(B(x0,2r)). r nkf k L1(B(x0,2r)) Z ∞ 2r dt tn+1 . rn Z ∞ 2r kf kL1(B(x0,t)) tn dt t . (20)
Then by (20) and (19) we get the inequality (16). This completes the proof of Theorem3.1. Proof of Theorem 1.3. From Lemma2.6, we have
1 ess inf t<s<∞ϕ1(x0, s)s n p = ess sup t<s<∞ 1 ϕ1(x0, s)s n p .
Note the fact that kf kLp(B(x0,t)) is a nondecresing function of t, and LM α,V,{x0} p,ϕ1 , then 1 + t ρ(x0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p . ess supt<s<∞ 1 + t ρ(x0) α kf kLp(B(x0,t)) ϕ1(x0, s)s n p . sup 0<s<∞ 1 + s ρ(x0) α kf kLp(B(x0,s)) ϕ1(x0, s)s n p . kf kLMp,ϕ1α,V,{x0}. (21)
Since α ≥ 0, and (ϕ1, ϕ2) satisfies the condition (7), then
Z ∞ 2r kf kLp(B(x0,t)) tnp dt t = Z ∞ 2r 1 +ρ(xt 0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p ess inf t<s<∞ϕ1(x0, s)s n p 1 + ρ(xt 0) α tnp dt t . kf kLMp,ϕ1α,V,{x0} Z ∞ r ess inf t<s<∞ϕ1(x0, s)s n p 1 +ρ(xt 0) α tnp dt t . kf kLMp,ϕ1α,V,{x0} 1 + r ρ(x0) −αZ ∞ r ess inf t<s<∞ϕ1(x0, s)s n p tnp dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −α ϕ2(x0, r). (22)
Then by Theorem5.1we get kT f kLMα,V,{x0} p,ϕ2 . sup r>0 1 + r ρ(x0) α ϕ2(x0, r)−1r−n/pkT f kLp(B(x0,r)) . sup r>0 1 + r ρ(x0) α ϕ2(x0, r)−1 Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . kf kLMp,ϕ1α,V,{x0}.
4. Proof of Theorem1.4. Similar to the proof of Theorem3, it suffices to prove the following result.
Theorem 4.1. Let V ∈ RHn/2, b ∈ BM Oθ(ρ). If 1 < p < ∞, then the inequality
k[b, T ]f kLp(B(x0,r)). [b]θr n p Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnp dt t (23)
holds for any ball B(x0, r) and for all f ∈ L p loc(R
n).
Moreover, the inequality
k[b, T ]f kW L1(B(x0,r)). [b]θr n Z ∞ 2r 1 + lnt r kf kL Φ(B(x0,t)) tn dt t (24)
holds for any ball B(x0, r) and for all f ∈ L1loc(Rn), where Φ(t) = t ln(e + t).
Proof. We write f as f = f1+ f2, where f1(y) = f (y)χB(x0,2r)(y). Then
k[b, T ]f kLp(B(x0,r))≤ k[b, T ]f1kLp(B(x0,r))+ k[b, T ]f2kLp(B(x0,r)). By the boundedness of [b, T ] on Lp(Rn) (see [31]) and (17) we get
k[b, T ]f1kLp(B(x0,r)). [b]θkf kLp(B(x0,2r)) . [b]θr n p Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . [b]θr n p Z ∞ 2r 1 + ln t r kf kL p(B(x0,t)) tnp dt t . (25)
We now turn to deal with the term k[b, T ]f2kLp(B(x0,r)). For any given x ∈ B(x0, 2r) we have [b, T ]f2(x) ≤ Z (2B)c
|K(x, y)| |b(x) − b(y)| |f (y)|dy ≤ |b(x) − b2B| T f2(x) + T ((b − b2B)f2)(x) . Then by (18), Lemma2.3, and taking N ≥ (k0+ 1)θ we get
k(b(x) − b2B)T f2kLp(B(x0,r)). [b]θr n p 1 + 2r ρ(x0) θ−N/(k0+1)Z ∞ 2r kf kLp(B(x0,t)) tnp dt t . [b]θr n p Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnp dt t . (26)
Finally, let us estimate kT ((b − b2B)f2)kLp(B(x0,r)). By (11), Lemma2.2 and (18) we have T ((b − b2B)f2)(x) . Z (2B)c 1 1 + |x−y|ρ(x) N |b(y) − b2B||f (y)| |x0− y|n dy . ∞ X k=1 1 (2kr)n1 + 2kr ρ(x) N Z 2k+1B |b(y) − b2B||f (y)|dy . ∞ X k=1 1 (2kr)n1 + 2kr ρ(x0) N/(k0+1) Z 2k+1B |b(y) − b2B||f (y)|dy.
Note that Z 2k+1B |b(y) − b2B||f (y)|dy . Z 2k+1B |b(y) − b2B|p 0 dy 1/p0 kf kLp(B(x0,2k+1r)) . [b]θk 1 + 2 kr ρ(x0) θ0 (2kr)p0nkf k Lp(B(x0,2k+1r)). Then sup x∈B(x0,r) T ((b − bB)f2)(x) . [b]θ ∞ X k=1 k 1 + ρ(x2kr 0) N/(k0+1)−θ0(2 kr)−n pkf k Lp(B(x0,2k+1r)) . [b]θ ∞ X k=1 k(2kr)−npkf k Lp(B(x0,2k+1r)). [b]θ ∞ X k=1 k Z 2k+1r 2kr kf kLp(B(x0,t)) tnp dt t . Since 2kr ≤ t ≤ 2k+1r, then k ≈ lnrt. Thus
sup x∈B(x0,r) T ((b − bB)f2)(x) . [b]θ ∞ X k=1 k Z 2k+1r 2kr kf kLp(B(x0,t)) tnp dt t . [b]θ ∞ X k=1 Z 2k+1r 2kr lnt r kf kLp(B(x0,t)) tnp dt t . [b]θ Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnp dt t . Then kT ((b − b2B)f2)kLp(B(x0,r)). [b]θr n p Z ∞ 2r 1 + lnt r kf kLp(B(x0,t)) tnp dt t . (27)
Combining (25), (26) and (27), the proof of Theorem4.1in the case of p > 1 is completed.
To deal with this result in the case p = 1, we split f as above by f = f1+ f2,
which yields
k[b, T ]f kW L1(B)≤ k[b, T ]f1kW L1(B)+ k[b, T ]f2kW L1(B).
From the boundedness of [b, T ] from LΦ(Rn)to W L1(Rn) ( see [27]) it follows that: k[b, T ]f1kW L1(B)≤ k[b, T ]f1kW L1
. kbk∗kf1kLΦ = kbk∗kf kLΦ(2B).
For the last term k[b, T ]f2kW L1(B), by homogeneity it is enough to assume λ = 1 and hence, we only need to prove that
x ∈ B : [b, T ]f2(x) > 1 . |B| Z ∞ 2r kf kLΦ(B(x0,t))|B(x0, t)| −1dt t for all B = B(x0, r). In fact, by [27, Lemma 1.5], we get
{x ∈ B : [b, T ]f2(x) > 1} ≤ sup t>0 1 Φ(1 t) {x ∈ R n: [b, T ]f2(x) > t} = sup t>0 1 Φ(1 t) {x ∈ B : M (Φf2)(x) > t}
where Φ(t) = t ln(e + t). We use the Fefferman-Stein maximal inequality Z {x∈Rn:M f (x)>t} φ(t)dx ≤ C t Z Rn |f (x)|M φ(x)dx, for any functions f and φ ≥ 0. This yields
x ∈ B : M (Φf2)(x) > t . 1 t Z x∈Rn:Φf2(x)>t χB(x)dx . 1t Z Rn Φf2(x)M χB(x)dx. Then x ∈ B : [b, T ]f2(x) > 1 . sup t>0 1 Φ(1 t) x ∈ B : M (Φf2)(x) > t . sup t>0 1 tΦ(1t) Z B Φf (x)M χB(x)dx. Theorem4.1is completed.
Proof of Theorem 1.4. Since f ∈ LMα,V,{x0}
p,ϕ1 and (ϕ1, ϕ2) satisfies the condition (4), by (33) we have Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnp dt t = Z ∞ 2r 1 + ρ(xt 0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p 1 +ρ(xt 0) α tnp dt t . kf kLMα,V,{x0} p,ϕ1 Z ∞ 2r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p 1 + ρ(xt 0) α tnp dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −αZ ∞ r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p tnp dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −α ϕ2(x0, r). (28)
Then from Theorem4.1 we get k[b, T ]f kLMα,V,{x0} p,ϕ2 . [b]θsup r>0 1 + r ρ(x0) α ϕ2(x0, r)−1 Z ∞ 2r 1 + lnt r kf kLp(B(x0,t)) tnp dt t . [b]θkf kLMα,V,{x0} p,ϕ1 .
5. Proof of Theorem1.6. We first prove the following conclusions
Theorem 5.1. Let V ∈ RHn/2. If 1 < p < n/β, 1/q = 1/p − β/n then the
inequality kIL β(f )kLq(B(x0,r)). r n q Z ∞ 2r kf kLp(B(x0,t)) tnq dt t
holds for any ball B(x0, r) and for all f ∈ Lploc(R n).
Moreover, for p = 1 the inequality kIL β(f )kW L n n−β(B(x0,r)). r n−β Z ∞ 2r kf kL1(B(x0,t)) tn−β dt t holds for any ball B(x0, r) and for all f ∈ L1loc(Rn).
Proof. For arbitrary x0∈ Rn, set B = B(x0, r) and λB = B(x0, λr) for any λ > 0.
We write f as f = f1+ f2, where f1(y) = f (y)χB(x0,2r)(y), and χB(x0,2r) denotes the
characteristic function of B(x0, 2r). Then
kIβL(f )kLq(B(x0,r))≤ kI L
βf1kLq(B(x0,r))+ kI L
βf2kLq(B(x0,r)).
Since f1 ∈ Lp(Rn) and from the boundedness of IβL from Lp(Rn) to Lq(Rn) (see
[33]) it follows that kIL βf1kLq(B(x0,r)). kf kLp(B(x0,2r)). r n qkf k Lp(B(x0,2r)) Z ∞ 2r dt tnq+1 . rnq Z ∞ 2r kf kLp(B(x0,t)) tnq dt t . (29) To estimate kIL βf2kLp(B(x0,r)), obverse that x ∈ B, y ∈ (2B) c implies |x − y| ≈
|x0− y|. Then by (11) we have
sup x∈B |IL βf2(x)| ≤ Z (2B)c |Kβ(x, y)f (y)|dy . Z (2B)c |f (y)| |x0− y|n−βdy . ∞ X k=1 (2k+1r)−n+β Z 2k+1B |f (y)|dy. By H¨older’s inequality we get
sup x∈B |IβLf2(x)| . ∞ X k=1 kf kLp(2k+1B)(2 k+1 r)−1−np+β Z 2k+1r 2kr dt . ∞ X k=1 Z 2k+1r 2kr kf kLp(B(x0,t)) tnq dt t . Z ∞ 2r kf kLp(B(x0,t)) tnq dt t . (30) Then kIL βf2kLq(B(x0,r)). r n q Z ∞ 2r kf kLp(B(x0,t)) tnq dt t (31)
holds for 1 ≤ p < n/β. Therefore, by (29) and (31) we get kIL β(f )kLq(B(x0,r)). r n q Z ∞ 2r kf kLp(B(x0,t)) tnq dt t (32) holds for 1 ≤ p < n/β.
When p = 1, by the boundedness of IL
β from L1(Rn) to W L n n−β(R n), we get kIL βf1kW L n n−β(B(x0,r)). kf kL1(B(x0,2r)). r n−β Z ∞ 2r kf kL1(B(x0,t)) tn−β dt t . By (31) we have kIL βf2kW L n n−β(B(x0,r)) ≤ kIL βf2kL n n−β(B(x0,2r)). r n−β Z ∞ 2r kf kL1(B(x0,t)) tn−β dt t .
Then kIL β(f )kW L n n−β(B(x0,r)). r n−βZ ∞ 2r kf kL1(B(x0,t)) tn−β dt t .
Proof of Theorem 3. Note the fact that kf kLp(B(x0,t)) is a nondecresing function of t, and f ∈ LMα,V,{x0} p,ϕ1 , then 1 +ρ(xt 0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p . ess supt<s<∞ 1 + ρ(xt 0) α kf kLp(B(x0,t)) ϕ1(x0, s)s n p . sup 0<s<∞ 1 + ρ(xs 0) α kf kLp(B(x0,s)) ϕ1(x0, s)s n p . kf kLMp,ϕ1α,V,{x0} . Since α ≥ 0, and (ϕ1, ϕ2) satisfies the condition (6), then
Z ∞ 2r kf kLp(B(x0,t)) tnq dt t = Z ∞ 2r 1 +ρ(xt 0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p ess inf t<s<∞ϕ1(x0, s)s n p 1 + ρ(xt 0) α tnq dt t . kf kLMα,V,{x0} p,ϕ1 Z ∞ 2r ess inf t<s<∞ϕ1(x0, s)s n p 1 +ρ(xt 0) α tnq dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −αZ ∞ r ess inf t<s<∞ϕ1(x0, s)s n p tnq dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −α ϕ2(x0, r). (33)
Then by Theorem5.1we get kIL β(f )kLMα,V,{x0} q,ϕ2 . supr>0 1 + r ρ(x0) α ϕ2(x0, r)−1 Z ∞ 2r kf kLp(B(x0,t)) tnq dt t . kf kLMα,V,{x0} p,ϕ1 .
Let q = n−βn , similar to the estimates of (33) we have Z ∞ 2r kf kL1(B(x0,t)) tn−β dt t . kf kLM1,ϕ1α,V,{x0} 1 + r ρ(x0) −α ϕ2(x0, r).
Thus by Theorem5.1we get kIL β(f )kW LMα,V,{x0} n n−β,ϕ2 . sup r>0 1 + r ρ(x0) α ϕ2(x0, r)−1 Z ∞ 2r kf kL1(B(x0,t)) tn−β dt t . kf kLMα,V,{x0} 1,ϕ1 .
6. Proof of Theorem 4. As the proof of Theorem 3, it suffices to prove the following result.
Theorem 6.1. Let V ∈ RHn/2, b ∈ BM Oθ(ρ). If 1 < p < n/β, 1/q = 1/p − β/n
then the inequality k[b, IL β(f )]kLq(B(x0,r)). [b]θr n q Z ∞ 2r 1 + ln t r kf kL p(B(x0,t)) tnq dt t (34)
holds for any f ∈ Lploc(Rn).
Moreover, the inequality k[b, IL β(f )kW L n n−β(B(x0,r)) . [b]θr n−β Z ∞ 2r 1 + ln t r kf kL Ψ(B(x0,t)) tn−β dt t (35)
holds for any ball B(x0, r) and for all f ∈ L1loc(R
n), where
Ψ(t) = [t ln(e+tβ/n)]n/n−β. Proof. We write f as f = f1+ f2, where f1(y) = f (y)χB(x0,2r)(y). Then
k[b, IL β](f )kLq(B(x0,r))≤ k[b, I L β]f1kLq(B(x0,r))+ k[b, I L β]f2kLq(B(x0,r)). By the boundedness of [b, IL
β] on Lp(Rn) to Lq(Rn) (see [34]) and (29) we get
k[b, IL β]f1kLq(B(x0,r)). [b]θkf kLp(B(x0,2r)) . [b]θr n q Z ∞ 2r kf kLp(B(x0,t)) tnq dt t . [b]θr n q Z ∞ 2r 1 + ln t r kf kL p(B(x0,t)) tnq dt t . (36)
We now turn to deal with the term k[b, IL
β]f2kLq(B(x0,r)). For any given x ∈ B(x0, r) we have
|[b, IL
β]f2(x)| ≤ |b(x) − b2B| |IβLf2(x)| + |IβL((b − b2B)f2)(x)|.
Then by (30), Lemma2.3, and taking N ≥ (k0+ 1)θ we get
k(b(x) − b2B)IβLf2kLq(B(x0,r)) . [b]θr n q 1 + 2r ρ(x0) θ−N/(k0+1)Z ∞ 2r kf kLp(B(x0,t)) tnq dt t . [b]θr n q Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnq dt t . (37)
Finally, let us estimate kIL
β((b − b2B)f2)kLq(B(x0,r)). By (11), Lemma 2.2 and (30) we have sup x∈β |IL β((b − b2B)f2)(x)| . Z (2B)c 1 1 + |x−y|ρ(x) N |b(y) − b2B||f (y)| |x0− y|n−β dy
. ∞ X k=1 1 (2kr)n−β1 + 2kr ρ(x) N Z 2k+1B |b(y) − b2B||f (y)|dy . ∞ X k=1 1 (2kr)n−β1 + 2kr ρ(x0) N/(k0+1) Z 2k+1B |b(y) − b2B||f (y)|dy. Note that Z 2k+1B |b(y) − b2B||f (y)|dy . Z 2k+1B |b(y) − b2B|p 01/p0 kf kLp(B(x0,2k+1r)) . [b]θk 1 + 2 kr ρ(x0) θ0 (2kr)p0nkf k Lp(B(x0,2k+1r)). Then sup x∈B |IL β((b − b2B)f2)(x)| . [b]θ ∞ X k=1 k(2kr)−n p+β 1 + ρ(x2kr 0) N/(k0+1)−θ0kf kLp(B(x0,2k+1r)) . [b]θ ∞ X k=1 k(2kr)−nqkf k Lp(B(x0,2k+1r)) . [b]θ ∞ X k=1 k Z 2k+1r 2kr kf kLp(B(x0,t)) tnq dt t . Since 2kr ≤ t ≤ 2k+1r, then k ≈ lnt r. Thus sup x∈B |IβL((b − b2B)f2)(x)| . [b]θ ∞ X k=1 k Z 2k+1r 2kr kf kLp(B(x0,t)) tnq dt t . [b]θ ∞ X k=1 Z 2k+1r 2kr lnt r kf kLp(B(x0,t)) tnq dt t . [b]θ Z ∞ 2r 1 + ln t r kf kL p(B(x0,t)) tnq dt t . Then kIβL((b − b2B)f2)kLq(B(x0,r)). [b]θr n q Z ∞ 2r 1 + ln t r kf kLp(B(x0,t)) tnq dt t . (38) Combining (36), (37) and (38), in the case of 1 < p < n/β, 1/q = 1/p − β/n the proof of Theorem6.1is completed.
In the case p = 1 and q = n/(n − β) the proof of the inequlity (35) goes along the came line as that of the case 1 < p < n/β, 1/q = 1/p − β/n once we use the lemmatas 2.7 and 2.9 in [10].
Thus the proof of Theorem6.1is completed. Proof of Theorem4. Since f ∈ LMα,V,{x0}
p,ϕ1 and (ϕ1, ϕ2) satisfies the condition (8), by (33) we have Z ∞ 2r 1 + ln t r kf kL p(B(x0,t)) tnq dt t
= Z ∞ 2r 1 + ρ(xt 0) α kf kLp(B(x0,t)) ess inf t<s<∞ϕ1(x0, s)s n p 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p 1 +ρ(xt 0) α tnq dt t . kf kLMα,V,{x0} p,ϕ1 Z ∞ 2r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p 1 + ρ(xt 0) α tnq dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −αZ ∞ r 1 + lnt r ess inft<s<∞ϕ1(x0, s)s n p tnq dt t . kf kLMα,V,{x0} p,ϕ1 1 + r ρ(x0) −α ϕ2(x0, r). (39)
Then from Theorem6.1 we get k[b, IL β]f kLMα,V,{x0} q,ϕ2 . [b]θsup r>0 1 + r ρ(x0) α ϕ2(x0, r)−1 Z ∞ 2r 1 + lnt r kf kL p(B(x0,t)) tnq dt t . [b]θkf kLMα,V,{x0} p,ϕ1 .
7. The Calder´on-Zygmund inequality. For the open set Ω ⊂ Rn, x0 ∈ Ω,
1 ≤ p < ∞ and V ∈ RHn/2, we say f ∈ LM α,V,{x0} p,ϕ (Ω) and f ∈ Mp,ϕα,V(Ω), if kf kp LMp,ϕα,V,{x0}(Ω) = sup r>0 (1 + r/ρ(x0))αϕ(x0, r)−1 Z B(x0,r)∩Ω |f (x)|pdx < ∞ and kf kp Mp,ϕα,V(Ω) = sup x0∈Ω, r>0 (1 + r/ρ(x0))αϕ(x0, r)−1 Z B(x0,r)∩Ω |f (x)|pdx < ∞, respectively.
In this section, we consider the behavior of the solution of the following Sch¨ondinger equation (−∆ + V )u = f (x), a.e. x ∈ Ω, where f ∈ LMα,V,{x0} p,ϕ (Ω), ϕ ∈ Ωα,Vp,loc or f ∈ M α,V p,ϕ(Ω), ϕ ∈ Ω α,V p .
Theorem 7.1. Let Ω be an open set in Rn, x
0 ∈ Ω, V ∈ RHn/2, α ∈ (−∞, ∞),
1 < p < n/2, 1/p − 1/q = 2/n, and ϕ1∈ Ωα,Vp,loc, ϕ2∈ Ωα,Vq,loc satisfies the condition
(6). If f ∈ LMα,V,{x0}
p,ϕ1 (Ω), then there exists a function u ∈ LM α,V,{x0} q,ϕ2 (Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore, kukLMα,V,{x0} q,ϕ2 (Ω). kf kLMp,ϕ1α,V,{x0}(Ω). (40) Proof. From the proof of Theorem3, we have
kukLMα,V,{x0} q,ϕ2 (Ω). kI L 2f kLMα,V,{x0} q,ϕ2 (Ω). kf kLMp,ϕ1α,V,{x0}(Ω). Thus, (40) hold.
Corollary 5. Let Ω be an open set in Rn, V ∈ RHn/2, α ∈ (−∞, ∞), 1 < p <
n/2, 1/p − 1/q = 2/n, and ϕ1 ∈ Ωα,Vp , ϕ2 ∈ Ωα,Vq satisfies the condition (7). If
f ∈ Mα,V
p,ϕ1(Ω), then there exists a function u ∈ M α,V q,ϕ2(Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore, kukMα,V q,ϕ2(Ω). kf kMp,ϕ1α,V(Ω).
Theorem 7.2. Let Ω be an open set in Rn, x0 ∈ Ω, V ∈ RHn/2, α ∈ (−∞, ∞),
1 < p < n, 1/p − 1/q = 1/n, and ϕ1∈ Ωα,Vp,loc, ϕ2∈ Ωα,Vq,loc satisfies the condition (6).
If f ∈ LMα,V,{x0}
p,ϕ1 (Ω), then there exists a function Du ∈ LM α,V,{x0} q,ϕ2 (Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore, kDukLMα,V,{x0} q,ϕ2 (Ω). kf kLMp,ϕ1α,V,{x0}(Ω). (41) Proof. From page 543 in [31], we know
|DxΓ(x, y)| ≤ CN 1 +|x−y|ρ(x) N 1 |x − y|n−1, (42)
where Γ(x, y) is the fundamental solution for L = −∆ + V .
Using (42) and adapting the argument for Theorem3, we then have kDukLMα,V,{x0}
q,ϕ2 (Ω). kDL −1f k
LMq,ϕ2α,V,{x0}(Ω). kf kLMp,ϕ1α,V,{x0}(Ω).
Thus, (41) hold.
Corollary 6. Let Ω be an open set in Rn, V ∈ RH
n/2, α ∈ (−∞, ∞), 1 < p < n,
1/p − 1/q = 1/n, and ϕ1 ∈ Ωα,Vp , ϕ2 ∈ Ωα,Vq satisfies the condition (7). If f ∈
Mp,ϕα,V1(Ω), then there exists a function Du ∈ M α,V q,ϕ2(Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore, kDukMα,V q,ϕ2(Ω). kf kMp,ϕ1α,V (Ω). Theorem 7.3. Let Ω be an open set in Rn, x
0 ∈ Ω, V ∈ RHn/2, α ∈ (−∞, ∞),
1 < p < ∞, and ϕ1, ϕ2 ∈ Ωα,Vp,loc satisfies the condition (2). If f ∈ LM α,V,{x0} p,ϕ1 (Ω), then there exists a function D2u ∈ LMα,V,{x0}
p,ϕ2 (Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore,
kD2uk
LMp,ϕ2α,V,{x0}(Ω). kf kLMp,ϕ1α,V,{x0}(Ω). (43)
Proof. From the proof of Theorem1, we have kD2uk
LMq,ϕ2α,V,{x0}(Ω). kD 2L−1f k
LMp,ϕ2α,V,{x0}(Ω). kf kLMp,ϕ1α,V,{x0}(Ω).
Corollary 7. Let Ω be an open set in Rn, V ∈ RHn/2, α ∈ (−∞, ∞), 1 < p < ∞,
and ϕ1, ϕ2∈ Ωα,Vp satisfies the condition (3). If f ∈ Mp,ϕα,V1(Ω), then there exists a function D2u ∈ Mα,V p,ϕ2(Ω), such that (−∆ + V )u = f (x), a.e. x ∈ Ω. Furthermore, kD2uk Mp,ϕ2α,V(Ω). kf kMp,ϕ1α,V(Ω).
Remark 7. Note that, Corollaries5,6and7in the case of ϕ(x, r) = r(λ−n)/p was
proved in [34, Theorem 4.1].
Acknowledgments. We would like to express our gratitude to professor Juan J. Nieto to give us the opportunity to dedicate him these results.
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Received February 2018; revised October 2018.
E-mail address: vagif@guliyev.com
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