Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.
It is sufficient to prove that there exists a constant C>0 such that
(18)1ω(B)κ∫B|Tλf(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p.

Fix a ball B=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have
(19)1ω(B)κ∫B|Tλf(x)|pω(x)dx≤C{1ω(B)κ∫B|Tλf1(x)|pω(x)dx +1ω(B)κ∫B|Tλf2(x)|pω(x)dx}=C{I1+I2}.

Using Lemmas 3 and 6, we get
(20)I1≤C1ω(B)κ∫2B|f(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p·ω(2B)κω(B)κ≤C∥f∥Lp,κ(ω)p.
We now estimate I2. We can write
(21)|Tλf2(x)|=|∫(2B)ceiλΦ(x,y)k(x-y)φ(x,y)f(y)dy|.

Now by an argument similar to the proof of Lemma 6 in [2], we choose ϕ1∈C0∞(ℝn) such that ϕ1(x)≡1, when |x|≤1, and ϕ1(x)≡0 when |x|>2. Let ϕ2=1-ϕ1 and N∈ℕ which is large enough and will be determined later. Write
(22)k(x)=kλ1(x)+kλ2(x),
where
(23)kλj(x)=k(x)ϕj(λ1/Nx), j=1,2.
Then
(24)Tλf2(x)=p·v·∫(2B)ceiλΦ(x,y)kλ1(x-y) ×φ(x,y)f(y)dy +p·v·∫(2B)ceiλΦ(x,y)kλ2(x-y) ×φ(x,y)f(y)dy:=Tλ1f2(x)+Tλ2f2(x).

Let us first estimate Tλ1f2(x). To do so, using Taylor’s expansion and the compactness of suppφ, we write
(25)Φ(x,y)=Φ(x,x)+P(x,y)+rN(x,y)
for(x,y)∈suppφ, where P(x,y) is a polynomial with deg P<N and |rN(x,y)|≤C|x-y|N with C independent of x and y. Define
(26)Rf(x)=p·v·∫(2B)ceiλP(x,y)kλ1(x-y)φ(x,y)f(y)dy.
Therefore
(27)e-iλΦ(x,x)Tλ1f2(x)-Rf(x) =∫|x-y|≤2λ-1/NeiλP(x,y)[eiλrN(x,y)-1] ×kλ1(x-y)φ(x,y)f(y)dy =∑j=0∞∫2-jλ-1/N<|x-y|≤2-j+1λ-1/NeiλP(x,y)[eiλrN(x,y)-1] ×kλ1(x-y)φ(x,y)f(y)dy ≡∑j=0∞Tλ,j1f2(x).
On Tλ,j1f2(x), by the properties of rN and k, we have
(28)|Tλ,j1f2(x)|≤C2-jNMf(x).

So we have
(29)|Tλ1f2(x)|≤C∑j=0∞2-jN|Mf(x)|+C|Rf(x)|.
By Theorem A and Lemma 9, we have
(30)∥Tλ1f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).

Now, let us turn to estimate Tλ2f2(x). We consider the following two cases.

Case 1 (λ≤1). Similar to that estimate of Tλ2 in Lemma 6 in [2], we have
(31)|Tλ2f2(x)|≤CM(f)(x).

By Lemma 9 we have
(32)∥Tλ2f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).

Case 2 (λ>1). We choose φ0∈C0∞(ℝn) such that
(33)suppφ0⊆{x∈ℝn:1<|x|≤2},ϕ2(x)=∑j=0∞φ0(2-jx).
Let
(34)kλ,j2(x)=k(x)φ0(2-jλ1/Nx).
Then
(35)Tλ2f2(x)=∫(2B)ceiλΦ(x,y)kλ2(x-y)φ(x,y)f(y)dy,∑j=0∞∫(2B)ceiλΦ(x,y)kλ,j2(x-y)φ(x,y)f(y)dy ≡∑j=0∞Tλ,j2f2(x).
For Tλ,j2, by its definition, we can get
(36)|Tλ,j2f2(x)|≤C∫2jλ-1/N<|x-y|≤2j+1λ-1/N ×1|x-y|n|f(y)|dy≤CM(f)(x).
The inequality (36) also can be seen in [2]; we omit the details here.

By Lemma 9, we have
(37)∥Tλ2f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).
Therefore
(38)I2≤C∥f∥Lp,κ(ω)p.
This finishes the proof of Theorem 1.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>.
It is sufficient to prove that there exists a constant C>0 such that
(39)1ω(B)κ∫B|Tλ*f(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p.
Fix a ball B=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have
(40)1ω(B)κ∫B|Tλ*f(x)|pω(x)dx ≤C{1ω(B)κ∫B|Tλ*f1(x)|pω(x)dx +1ω(B)κ∫B|Tλ*f2(x)|pω(x)dx} =C{J1+J2}.
Using Lemmas 3 and 7, we get
(41)J1≤C1ω(B)κ∫2B|f(x)|pω(x)dx ≤C∥f∥Lp,κ(ω)p·ω(2B)κω(B)κ ≤C∥f∥Lp,κ(ω)p.
We now estimate J2.

For each m∈ℕ and j=1,…,gm, we get
(42)ajm(x)=∫ΣΩ(x,z)Yjm(z)dσz,
where Ω(x,z)=|z|nk(x,z). Then for a.e. x∈ℝn,
(43)Ω(x,z)=∑m=1∞∑j=1gmajm(x)Yjm(z′),
where z′=z/|z| for any z∈ℝn∖{0}. By Lemma 10, we have that, for any x∈ℝn,
(44)|ajm(x)|=m-n(m+n-2)-n|∫ΣΩ(x,z)ΛnYjm(z)dσz|=m-n(m+n-2)-n|∫ΣΛnΩ(x,z)Yjm(z)dσz|≤C(n)Am-2n.

By Lemma 10 again, we can verify that, for any ϵ>0, N∈ℕ, and a.e. x∈ℝn, if |y-x|≥ϵ, then
(45)|∑m=1N∑j=1gmeiλΦ(x,y)ajm(x)Yjm((x-y)′)|x-y|nφ(x,y)f2(y)| ≤C(ϵ)A|f2(y)|.

Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that
(46)Tλ*f2(x) =limϵ→0∫|x-y|≥ϵeiλΦ(x,y)k(x,x-y)φ(x,y)f2(y)dy =limϵ→0∑m=1∞∑j=1gm∫|x-y|≥ϵeiλΦ(x,y)ajm(x)Yjm((x-y)′)|x-y|n ×φ(x,y)f2(y)dy =limϵ→0∑m=1∞∑j=1gmajm(x)∫|x-y|≥ϵeiλΦ(x,y)Yjm((x-y)′)|x-y|n ×φ(x,y)f2(y)dy.
We write
(47)Rjmf2(x) =∫|x-y|≥ϵeiλΦ(x,y)Yjm((x-y)′)|x-y|nφ(x,y)f2(y)dy.
It is easy to see that Rjmf2(x) is the oscillatory integral operator defined by (1). By Theorem 1 we have that Rjm is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have
(48)J2≤C∥f∥Lp,κ(ω)p.

This finishes the proof of Theorem 2.