Regularity Estimates for the Oblique Derivative Problem on Non-smooth Domains II

Pengfei Guan^{∗} Department of Mathematics McMaster University Hamilton, Ontario L8S 4K1 Canada

Eric Sawyer^{†} Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 CANADA

1 Some special operators

In the previous paper [GSI] in this journal, we considered the oblique derivative problem on nonsmooth domains, and reduced the question of regularity of the solutions to the boundedness properties of certain pseudodifferential operators with limited smoothness. In this paper we will apply the technique of symbol splitting to these operators, and ultimately reduce their boundedness on Sobolev spaces to weighted norm inequalities for the Hardy operator. For example, one of the basic operators we consider in the third subsection below is of the form

Ku(x, t) = (2π)^{− \frac{n}{2}} ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′.

One obvious feature of this operator is that it is a pseudodifferential operator in the xvariables, parameterized by t and t′. So we will also consider extending the rough ψdo calculus to pseudodifferential operators with parameters in the second subsection below. But first, we collect the necessary material on rough symbols.

∗Supported in part by NSERC grant OGP0046732 and a Sloan Fellowship.

†Research supported in part by NSERC grant OGP0005149

1.1 Rough pseudodifferential operators

In this subsection we will define the symbol classes C^{d}S_{γ,δ}^{m}, recall the mapping properties of their associated operators in the Hölder, L^{p} Sobolev and Besov scale of spaces, and describe the symbol splitting which permits a calculus for the composition of two such operators. In particular, this calculus is crucial for the estimates in the remainder of this section and in section 2.

Definition 1 A symbol σ : R^{n} × R^{n} → R belongs to the symbol class C^{M}S_{ρ,δ}^{m} (where m is real, 0 ≤ ρ, δ ≤ 1 , and M is a nonnegative integer) if for all multi-indices α and β with |α| ≤ M, there are constants C_{α,β} such that

|∂_{x}^{α}∂_{ξ}^{β}σ(x, ξ)| ≤ C_{α,β}(1 + |ξ|^{2})^{^{\frac{1}{2}} (m+δ|α|−ρ|β|)}.  (1)

If 0 < μ < 1, then σ belongs to C^{M+μ}S_{ρ,δ}^{m} if in addition to (1), we have

|∂_{ξ}^{β}σ(x + h, ξ) − ∑_{ℓ=0}^{M} \frac{(h · ∇_{x})^{ℓ}}{ℓ!} ∂_{ξ}^{β}σ(x, ξ)| ≤ C_{M,β} h^{M+μ} (1 + |ξ|^{2})^{^{\frac{1}{2}} (m+δ(M+μ)−ρ|β|)}. (2)

These symbol classes are treated for example in M. Taylor’s book [Tay]. Here C^{d} denotes the usual Lipschitz space of continuous functions whose derivatives of order ⌊d⌋ are bounded (when d is an integer), or satisfy a Hölder condition of order d−[d] (when d is not an integer). Let Λ^{s} be the usual Hölder space, denoted by C_{∗}^{s} in [Tay]. Of course Λ^{s} = C^{s} for s not an integer, and consists of those f with D^{α}f in the Zygmund class for |α| = s − 1, when s is a positive integer.

Definition 2 A symbol σ : R^{n} × R^{n} → R belongs to O_{I}^{m} if its associated operator

σ(f)(x) = ∫_{R^{n}} e^{ix·ξ}σ(x, ξ)f̂(ξ)dξ

admits a bounded extension from H_{p,comp}^{s+m} to H_{p,loc}^{s} (respectively Λ_{comp}^{s+m} to Λ_{loc}^{s}) for all s in the interval I (respectively I ∩ (0, ∞)) and all 1 < p < ∞. The symbol σ belongs to \overline{O}_{I}^{m} if in addition, its associated operator is bounded from Λ_{comp}^{t+m} to Λ_{loc}^{t} where t is the right endpoint of the interval I.

The symbol classes arising naturally in the oblique derivative problem are C^{d}S_{1,δ}^{m} (see subsection 1.2 ) and so we restrict attention to the classes C^{d}S_{1,δ}^{m} for the remainder of this section. In [CM], R. Coifman and Y. Meyer show that C^{d}S_{1,0}^{m} ⊂ O_{{0}}^{m} for d > 0 (a special case of Proposition 9, p. 38), and G. Bourdaud [Bou, Bou] then extended this to the following result (see e.g. section 2.1 of [Tay]).

Theorem 3 C^{d}S_{1,δ}^{m} ⊂ \overline{O}_{(−(1−δ)d,d)}^{m} for all m real, all d > 0 and 0 ≤ δ < 1.

We now recall the technique of symbol splitting (see (1.3.21) and (1.3.15) of [Tay]).

Proposition 4 Given τ ∈ C^{d}S_{1,δ}^{m}, we can write τ = τ^{♯} + τ^{♭} with τ^{♯} ∈ S_{1,γ}^{m} and τ^{♭} ∈ C^{d}S_{1,γ}^{m+d(δ−γ)}. Moreover, there are the following improved estimates:

∇_{x}^{ℓ}τ^{♯} ∈ \{ \begin{matrix}S_{1,γ}^{m+|ℓ|δ} & for 0 ≤ ℓ ≤ d \\ S_{1,γ}^{m+γ(|ℓ|−d)+δd} & for ℓ > d \end{matrix} .  (3)

This symbol splitting permits the use of τ^{♯} in the classical asymptotic formula for the composition of two smooth pseudodifferential operators, namely

σ ∘ τ − ∑_{ℓ=0}^{M} \frac{1}{i^{ℓ}ℓ!}∇_{ξ}^{ℓ}σ · ∇_{x}^{ℓ}τ ∈ C^{d}S_{1,δ}^{m_{1}+m_{2}+(M+1)(δ−1)} for all M ≥ 0,  (4)

where σ ∈ C^{d}S_{1,δ}^{m_{1}} and τ ∈ S_{1,δ}^{m_{2}} with 0 ≤ δ < 1 and d > 0. Note that the standard proofs in the smooth case apply even when σ is rough and τ is smooth (see e.g. section 3 of Chapter VI of [St2]). As a consequence of (4) and Proposition 4 we obtain

Proposition 5 Let σ ∈ C^{d}S_{1,δ}^{m_{1}} and τ ∈ C^{M+d}S_{1,δ}^{m_{2}} where 0 ≤ δ < 1, d > 0 and m_{1} < M +d. Suppose d > 1. Then with γ = 1 − \frac{(d−1)(1−δ)}{M+d}, we have

σ ∘ τ − ∑_{ℓ=0}^{M} \frac{1}{i^{ℓ}ℓ!}∇_{ξ}^{ℓ}σ · ∇_{x}^{ℓ}τ ∈ \overline{O}_{(−d(1−γ),min\{d,M+d−m_{1}\})}^{m_{1}+m_{2}+(M+1)(δ−1)}.  (5)

Suppose 0 < d ≤ 1. Then with γ = 1 − \frac{ϵ}{M+1} for any ϵ > 0, we have

σ ∘ τ − ∑_{ℓ=0}^{M} \frac{1}{i^{ℓ}ℓ!}∇_{ξ}^{ℓ}σ · ∇_{x}^{ℓ}τ ∈ O_{(−d(1−γ),min\{d,M+d−m_{1}\})}^{m_{1}+m_{2}+(M+1)(δ−1)+(1−d)(γ−δ)+ϵ}.  (6)

1.2 Pseudodifferential operators with parameters

Define t_{0} = −1 if ℓ^{→} satisfies case (I), and t_{0} = 0 if ℓ^{→} satisfies case (II). We now define our symbol classes for operators with parameters. We will be working exclusively with the cases ρ = 1 and 0 ≤ δ < 1.

Definition 6 Let g be a nonnegative function on (0, 1). We say that the symbol σ(x, t, t′, ξ) ∈ C^{M}S_{1,δ,g}^{m} (where M ≥ 0 is an integer) if for all multi-indices α, β and nonnegative integers s, s′, with|α| + s + s′ ≤ M, there are constants C_{α,β,γ} such that

|∂_{t}^{s}∂_{t′}^{s′}∂_{x}^{α}∂_{ξ}^{β}σ(x, t, t′, ξ)| ≤ C_{α,β,s,s′} g (|t − t′|) (1 + |ξ|)^{m+δ|α|−|β|+s+s′}  (7)

for |ξ| ≥ 1, x ∈ R^{n} and t, t′ ∈ (−1, 1). If 0 < μ < 1, then σ belongs to C^{M+μ}S_{1,δ,g}^{m} if in addition to (7), we have for 0 ≤ s + s′ ≤ M,

\begin{matrix}|∂_{t}^{s}∂_{t′}^{s′}∂_{ξ}^{β}σ(x + h, t, t′, ξ) − ∑_{ℓ=0}^{M−s−s′} \frac{(h·∇_{x})^{ℓ}}{ℓ!} ∂_{t}^{s}∂_{t′}^{s′}∂_{ξ}^{β}σ(x, t, t′, ξ)| \\ ≤ C_{M,β,s,s′} g (|t − t′|) h^{M−s−s′+μ} (1 + |ξ|^{2})^{^{\frac{1}{2}} (m+δ(M+μ)−|β|)}. \end{matrix}  (8)

Definition 7 We say that the symbol σ(x, t, t′, ξ) ∈ O_{I,g}^{m} if its associated operator,

σf(x, t, t′) = ∫_{R^{n}} e^{ix·ξ}σ(x, t, t′, ξ)f̃(ξ, t′)dξ,

satisfies ∥σf(·, t, t′)∥_{H_{p}^{s}(I^{n})} ≤ C g (|t − t′|) ∥f(·, t, t′)∥_{H_{p}^{s+m}(I^{n})} for all f (·, t, t′) ∈ H_{p}^{s+m}(I^{n}), s ∈ I and t, t′ ∈ (−1, 1).

Lemma 8 If σ ∈ C^{M+μ}S_{1,δ,g}^{m}, then σ ∈ O_{I,g}^{m} where I = (−(1 − δ)(M + μ), M + μ).

Proof. Apply Theorem 3.

Although we assume σ is compactly supported in the x variables, g is not necessarily continuous. Various choices of g will be used later.

Proposition 9 Let σ ∈ O_{I,g}^{m} , m ≥ 0, 0 ∈ I. If g is integrable on (0, 1), then the operator

σf(x, t) = ∫_{t_{0}}^{t} ∫_{R^{n}} e^{ix·ξ}σ (x, t, t′, ξ) f̃(ξ, t′)dξdt′,

is bounded from H_{p}^{m}(R^{n} × (−1, 1)) to L_{p}(R^{n} × (−1, 1)). In particular, this holds for σ ∈ C^{μ}S_{1,δ,g}^{m}, μ > 0.

Proof. We have

∫_{−1}^{1} ∫_{R^{n}} |σf(x, t)|^{p} dxdt = ∫_{−1}^{1} ∫_{R^{n}} |∫_{t_{0}}^{t} σ (x, t, t′, D_{x}) f (·, t′) (x)dt′|^{p} dxdt
≤ ∫_{−1}^{1} ∫_{R^{n}} \{∫_{t_{0}}^{t} |g (|t − t′|)|^{− \frac{p}{p′}} |σ (x, t, t′, D_{x}) f (·, t′) (x)|^{p} dt′\} dx
× (∫_{−1}^{1} |g (|t − t′|)| dt′)^{\frac{p}{p′}} dt

by Hölder’s inequality. Since the operators σ (x, t, t′, ξ) are in O_{I,g}^{m} by Lemma 8, the above is bounded by

≤ C ∫_{−1}^{1} ∫_{R^{n}} ∫_{t_{0}}^{t} |g (|t − t′|)|^{− \frac{p}{p′} +p} ∥f (x, t′) ∥_{H_{p}^{m}(R_{x}^{n})}^{p}dt′dt
≤ C ∫_{−1}^{1} ∫_{−1}^{1} |g (|t − t′|)| ∥f (x, t′) ∥_{H_{p}^{m}(R_{x}^{n})}^{p}dt′dt
≤ C ∥g∥_{1} ∫_{−1}^{1} ∥f (x, t′) ∥_{H_{p}^{m}(R_{x}^{n})}^{p}dt′
≤ C∥u∥_{H_{p}^{m}(R_{x}^{n}×(−1,1))}^{p},

where \frac{1}{p′} + \frac{1}{p} = 1 and ∥ · ∥_{H_{p}^{m}(R_{x}^{n})} denotes the norm in H_{p}^{m}(R^{n}).

Corollary 10 If φ is a zero order ψdo in R^{n+ℓ} with support in {ξ_{n+1} < λ ∑_{i=1}^{n} |ξ_{i}|^{2}} for some λ > 0, and if σ is a symbol in C^{M+μ}S_{1,δ,g}^{m} with g integrable on (0, 1), then σ ∘ φ is bounded from H_{p}^{s+m} (R^{n} × (−1, 1)) to H_{p}^{s} (R^{n} × (−1, 1)) for all s ∈ (− (1 − δ) M, M].

Proof. This follows from the identity

σ ∘ φ = ∑_{j} ∂_{x_{j}} ∘ σ ∘ ∂_{x_{j}}△^{−1} ∘ φ + ∑_{j} σ ∘ ∂_{x_{j}}△^{−1} ∘ φ

where σ_{j} ∈ C^{M+μ−1}S_{1,δ,g}^{m+δ}. By Proposition 9, we now see that σ ∘ φ is bounded from H_{p}^{m−1+δ} (R^{n} × (−1, 1)) to H_{p}^{−1+δ} (R^{n} × (−1, 1)). A simple induction procedure and an interpolation argument imply that σ∘φ is bounded from H_{p}^{s+m} (R^{n} × (−1, 1)) to H_{p}^{s} (R^{n} × (−1, 1)) for all s ∈ (− (1 − δ) M, M].

For all of the special operators we consider, we will obtain the above conclusion for the extended range s < M + μ. See Proposition 18 and Lemma 19 below.

1.3 Symbol estimates for K, T and R

Now we turn to estimating some of the special operators appearing in the solution to the oblique derivative problem. We consider the following operators that arise in the section 2 of [GSI]:

Ku(x, t) = (2π)^{− \frac{n}{2}} ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}K (x, t, t′, ξ) ũ^{∗} (ξ, t′) dξdt′,

T u(x, t) = (2π)^{− \frac{n}{2}} ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}K (x, t, t′, ξ) a (x, t′) Q (x, t′, ξ, a (x, t′)) ũ^{∗} (ξ, t′) dξdt′, Ku(x, t) = −(2π)^{− \frac{n}{2}} △_{x}^{−1} ∫_{|ξ|≥1} e^{ix·ξ}ρ̃(x, t_{0})e^{−Ã(x,t,t_{0},ξ)} |ξ|^{2} ũ^{∗} (ξ, t′) dξ,

(9)

along with the commutator

R = K ∘ ãQ − KãQ,  (10)

where Q (x, t, ξ, s) is smooth and satisfies

\begin{matrix}C^{−1} |ξ| ≤ ReQ ≤ C |ξ| , \\ |\frac{∂^{β}}{∂x^{β}} \frac{∂^{α}}{∂ξ^{α}} \frac{∂^{ℓ}}{∂t^{ℓ}} \frac{∂^{m}}{∂s^{m}} Q| ≤ C_{β,α,ℓ,m} (1 + |ξ|)^{1+(|β|+ℓ)δ−|α|} , \end{matrix}  (11)

(a_{i,j} (t, x))_{i,j} > 0, and

\begin{matrix}Ã(x, t, t′, ξ) = ∫_{t′}^{t} a(x, θ)Q (x, θ, ξ, a (x, θ)) dθ, \\ A(x, t, t′) = ∫_{t′}^{t} a(x, θ)dθ, \\ K(x, t, t′, ξ) = ρ̃(x, t)e^{−Ã(x,t,t′,ξ)}. \end{matrix}  (12)

We use here ã as an abbreviation for a (x, t′) and a for a (x, t). We have

\begin{matrix}a ∈ C^{λ+2}, \\ a (x, ±1) ≠ 0, \\ and either a ≥ 0 or ta ≥ 0. \end{matrix}  (13)

Also, u^{∗} = ρ^{∗}u , ũ^{∗}(ξ, t′) = ∫_{R^{n}} e^{−ix·ξ}u^{∗}(x, t′)dx, u_{0}^{∗} (x) = ρ^{∗} (x, 0) u_{0} (x) , ũ_{0}^{∗}(ξ) = ∫_{R^{n}} e^{−ix·ξ}u_{0}^{∗}(x)dx, and t_{0} = −1 if a satisfies case (I), while t_{0} = 0 if a satisfies case (II). The function ρ^{∗} ∈ C_{c}^{∞} (R^{n} × R) is identically 1 in a neighbourhood of the set where a vanishes. The choice of t_{0} is crucial. With the above choice, a(x, θ) keeps the same sign in (t′, t) for t′ ∈ (t_{0}, t), and Re (Ã(x, t, t′, ξ)) ≥ 0 and A(x, t, t′) ≥ 0 for t′ ∈ (t_{0}, t).

One of the features in our estimates is evident when we consider the operator T in (9)

- namely that the presence of a with e^{−A} in the symbol results in greater gain than is first apparent. A simple but effective means for realizing this is the following lemma.

Lemma 11 For a satisfying (13), we have

\begin{matrix}|∇_{x}a (x, t)|^{2} ≤ C |a (x, t)| , \\ |∇_{x}A (x, t, t′)|^{2} ≤ C |t − t′| A (x, t, t′) , t_{0} < t′ < t, \\ |a (x, t)|^{2} ≤ CA (x, t, t_{0}) \end{matrix}  (14)

Proof. The first two inequalities are proved in ([GS]; Lemmas 4.3 and 4.4). The last inequality follows from the facts that A (x, t, t_{0}) ≥ 0 for all (x, t) and a (x, t) = \frac{∂}{∂t}A (x, t, t_{0}).

The following lemma is crucial for obtaining the gain of 2 − \frac{1}{p(λ+3)} derivatives from f in Theorem 4 of §1 (see Lemma 5.13 of [GS] when k is an integer and see also [Koo], p. 203 for related results).

Lemma 12 For each real number k ≥ 1, there is a constant C_{k} > 0, such that

max_{t∈I} |f′(t)|

≤ C_{k} \{\frac{1}{|I|} max_{t_{1},t_{2}∈I} |f(t_{1}) − f(t_{2})| + max_{t_{1},t_{2}∈I} |f(t_{1}) − f(t_{2})|^{1− \frac{1}{k}} ∥f∥C^{k}(I)^{\frac{1}{k}}\} (15)

for all intervals I and functions f ∈ C^{k}(I).

Proof. Suppose first that k is a positive integer. Fix z ∈ I, let P (t) = f′(z)(t − z) + ... +\frac{f^{(k−1)}(t)}{(k−1)!}(t − z)^{k−1}, and let r(t) = f(t) − f(z) − P (t). Let J be an interval such that z ∈ J ⊂ I, and |J| = min{|I|, δ}, where

δ = (\frac{max_{t_{1},t_{2}∈I} |f(t_{1}) − f(t_{2})|}{∥f∥_{C^{k}(I)}})^{\frac{1}{k}} .  (16)

Since P is a polynomial of degree k − 1, there is C_{k} > 0, independent of J and P such that max_{t∈J} |P′(t)| ≤ C_{k} \frac{1}{|J|} max_{t∈J} |P (t)|. Then using Taylor’s formula to estimate r(t), we have

|f′(z)| = |P′(z)| ≤ max_{t∈J} |P′(t)| ≤ C_{k} \frac{1}{|J|} max_{t∈J} |P (t)|
≤ C_{k} \frac{1}{|J|} [max_{t∈J} |f(t) − f(z)| + max_{t∈J} |r(t)|]
≤ C_{k} \frac{1}{|J|} [max_{t_{1},t_{2}∈J} |f(t_{1}) − f(t_{2})| + (∥f∥_{C^{k}(I)}) |J|^{k}]
≤ C_{k} \frac{1}{|J|} max_{t_{1},t_{2}∈I} |f(t_{1}) − f(t_{2})| + (∥f∥_{C^{k}(I)}) δ^{k−1},

which yields (15) using |J| = min {|I| , δ}.

If k is not an integer, then the above argument goes through using P (t) = f′(z)(t−z)+... +\frac{f^{([k])}(t)}{[k]!}(t − z)^{[k]} and r(t) = f(t) − f(z) − P (t), together with the inequality max_{t∈J} |r(t)| ≤

C (∥f∥_{C^{k}(I)}) |J|^{k}.

Remark The inequality max_{t∈J} |r(t)| ≤ C (∥f∥_{C^{k}(I)}) |J|^{k} used above fails with Λ^{k} in place

of C^{k} when k is an integer. For example, f (t) = ∑_{k=o}^{∞} \frac{e^{i2^{k}t}}{2^{2k}} ∈ Λ^{2}, but

|Re (f(t) − f(0) − f′(0)t)| ≥ c (log_{2} \frac{1}{t}) t^{2}.

Lemma 13 Suppose a ∈ C^{k−1} and t′ ∈ (t_{0}, t). Then

max_{θ∈(t′,t)} |a(x, θ)| ≤ C_{k} \{\frac{A(x, t, t′)}{|t − t′|} + A(x, t, t′)^{1− \frac{1}{k}} ∥a(x, ·)∥C^{k−1}((t′,t))^{\frac{1}{k}}\} .

In particular, there is C′_{k} such that

|a(x, θ)| ≤ C′_{k} (\frac{A(x, t, t′)}{|t − t′|})^{1− \frac{1}{k}} for all θ ∈ (t′, t).  (17)

Proof. Since a(x, θ) keeps the same sign in (t′, t) if t′ ∈ (t_{0}, t), the lemma follows from Lemma 12 directly.

The next lemma is the basis for Proposition 15 which in turn yields boundedness properties for K and T and their associated operators. We emphasize that in this section, we obtain boundedness results valid for H_{p}^{s} for all 1 < p < ∞. Sharp boundedness results for K and T are obtained in section 2.

Lemma 14 Let Q and Ã be as in (11) and (12). For t′ ∈ (t_{0}, t), |ξ| ≥ 1, s_{1}+s_{2}+|α| ≤ λ+2, and μ ∈ Z_{+}, we have with δ = max \{\frac{1}{2}, \frac{1}{λ+1}\},

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β}Ã(x, t, t′, ξ)| \\ ≤ C_{s_{1},s_{2},α,β}(1 + |ξ|)^{δ|α|−|β|+s_{1}+s_{2}} (Ã^{δ|α|+1}(x, t, t′, ξ) + 1) , \end{matrix}  (18)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β}(a(x, t′)Q^{μ}(x, t′, ξ, a (x, t′)))| \\ ≤ C_{s_{1},s_{2},α,β,k} |t − t′|^{−1+ \frac{1}{λ+3}} (1 + |ξ|)^{μ−1+δ|α|−|β|+s_{1}+s_{2}+ \frac{1}{λ+3}} (Ã^{1− \frac{1}{λ+3}} (x, t, t′, ξ) + 1) , \end{matrix} (19)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β}Ã_{x_{j}}(x, t, t′, ξ))| \\ ≤ C_{s_{1},s_{2},α,β,q}(1 + |ξ|)^{δ(|α|+1)+s_{1}+s_{2}−|β|} (1 + Ã(x, t, t′, ξ)^{δ}) \\ for s_{1} + s_{2} + |α| ≤ λ + 1, \end{matrix}  (20)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (a(x, t′)Q(x, t′, ξ)Ã_{x_{j}}(x, t, t′, ξ))| \\ ≤ C_{s_{1},s_{2},α,β} |t − t′|^{− \frac{1}{2} + \frac{1}{λ+3}} (1 + |ξ|)^{δ(|α|+1)+s_{1}+s_{2}−|β|+ \frac{1}{λ+3}} (1 + Ã(x, t, t′, ξ)^{1− \frac{1}{λ+3} + \frac{1}{2}} ) \\ for s_{1} + s_{2} + |α| ≤ λ + 1, \end{matrix} (21)

Proof. If s_{1} + s_{2} > 0 or |α| ≥ 2, (18) is trivial since Q satisfies (11). Thus we need only check the case s_{1} = s_{2} = 0, |α| = 1 of (18) and this follows from Lemma (11) if we note that Ã^{1/2} ≤ Ã^{3/2} + 1. The proofs for (19), (20) and (21) are similar and follow from Lemma 11 and Lemma 13.

Proposition 15 Let b(x, t, t′, ξ) be C^{k} in the variables x, t, t′ for some k > 0, be C^{∞} in the variable ξ and satisfy for some N > 0,

\begin{matrix}|\frac{∂^{α}}{∂x^{α}} \frac{∂^{β}}{∂ξ^{β}} \frac{∂^{s_{1}}}{∂t^{s_{1}}} \frac{∂^{s_{2}}}{∂t^{′s_{2}}} b| ≤ C_{α,β,s_{1},s_{2}} \{(1 + |ξ|)^{δ(|α|+s_{1}+s_{2})} + Ã(x, t, t′, ξ)^{|α|+s_{1}+s_{2}+N} \} (1 + |ξ|)^{−|β|} , \\ |α| + s_{1} + s_{2} ≤ k. \end{matrix}

(22)

For t′ ∈ (t_{0}, t), |ξ| ≥ 1, s_{1} + s_{2} + |α| ≤ min \{λ + 2, k\}, and μ ∈ Z_{+}, with δ = max \{\frac{1}{2}, \frac{1}{λ+1}\}, we have

|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (b(x, t, t′, ξ)K(x, t, t′, ξ)) | ≤ C_{s_{1},s_{2}α,β}(1 + |ξ|)^{s_{1}+s_{2}+δ|α|−|β|},  (23)
\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (a(x, t′)b(x, t, t′, ξ)Q^{μ}(x, t′, ξ)K(x, t, t′, ξ)) | \\ ≤ C_{s_{1},s_{2},α,β,k} |t − t′|^{−1+ \frac{1}{λ+3}} (1 + |ξ|)^{μ−1+s_{1}+s_{2}+δ|α|−|β|+ \frac{1}{λ+3}} , \end{matrix}  (24)

and for |α| + s_{1} + s_{2} ≤ min {k, (λ + 1)}, we have

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (b(x, t, t′, ξ)\frac{∂}{∂x_{j}} a (x, t′) K(x, t, t′, ξ)) | \\ ≤ C_{s_{1},s_{2}α,β} |t − t′|^{− \frac{1}{2} \frac{λ+2}{λ+3}} (1 + |ξ|)^{s_{1}+s_{2}+δ|α|−|β|− \frac{1}{2} \frac{λ+2}{λ+3}} , \end{matrix}  (25)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (b(x, t, t′, ξ)\frac{∂}{∂t}a (x, t′) K(x, t, t′, ξ)) | \\ ≤ C_{s_{1},s_{2}α,β} |t − t′|^{−1+ \frac{1}{2(λ+3)}} (1 + |ξ|)^{s_{1}+s_{2}+δ|α|−|β|− \frac{1}{2} \frac{λ+2}{λ+3}} \end{matrix}  (26)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (b(x, t, t′, ξ)Ã_{x_{j}}(x, t, t′, ξ)K(x, t, t′, ξ)) | \\ ≤ C_{s_{1},s_{2},α,β}(1 + |ξ|)^{s_{1}+s_{2}+δ(|α|+1)−|β|}, \end{matrix}  (27)

\begin{matrix}|∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β} (a(x, t′)Q(x, t′, ξ)b(x, t, t′, ξ)Ã_{x_{j}}(x, t, t′, ξ)K(x, t, t′, ξ)) | \\ ≤ C_{s_{1},s_{2},α,β} |t − t′|^{− \frac{1}{2} + \frac{1}{λ+3}} (1 + |ξ|)^{s_{1}+s_{2}+δ(|α|+1)−|β|+ \frac{1}{λ+3}} . \end{matrix}  (28)

Proof. Inequalities (23), (24), (27) and (28) are consequences of Lemma 14. To see this for (23), note that the Liebniz formula gives

\begin{matrix}∂_{t}^{s_{1}}∂_{t′}^{s_{2}}∂_{x}^{α}∂_{ξ}^{β}e^{−Ã(x,t,t′,ξ)} \\ = ∑ C_{α′,β′,s′_{1},s′_{2}} (∏_{j=1}^{n} ∂_{t}^{s_{1,j}}∂_{t′}^{s_{2,j}}∂_{x}^{α′_{j}}∂_{ξ}^{β′_{j}} Ã(x, t, t′, ξ)) e^{−Ã(x,t,t′,ξ)} \end{matrix}

where the summation is taken over all α′ = (α′_{1}, ..., α′_{n}), β′ = (β′_{1}, ..., β′_{n}), s′_{1} = (s_{1,1}, ..., s_{1,n}), s′_{2} = (s_{2,1}, ..., s_{2,n}) with |α′| = |α|, |β′| = |β|, |s′_{1}| = s_{1}, |s′_{2}| = s_{2}. By (18)

| ∏_{j=1}^{n} ∂_{t}^{s_{1,j}}∂_{t′}^{s_{2,j}}∂_{x}^{α_{j}}∂_{ξ}^{β_{j}} Ã|e^{−Ã}
≤ C ∏_{j=1}^{n} (Ã^{δ|α_{j}|+1} + 1) (1 + |ξ|)^{δ|α_{j}|−|β_{j}|+s_{1,j}+s_{2,j}}e^{−Ã}
≤ C(1 + |ξ|)^{δ|α|−|β|+s_{1}+s_{2}} (A^{δ|α|+|β|+s_{1}+s_{2}} + 1) e^{−Ã} ≤ C(1 + |ξ|)^{δ|α|−|β|+s_{1}+s_{2}}.

as required. The proofs for (24), (27) and (28) are similar.

As for (25), we use (14) and (17) for k ≤ λ + 3 to obtain

|\frac{∂}{∂x_{j}} a (x, t′)| ≤ C |a (x, t′)|^{\frac{1}{2}} ≤ C (\frac{A (x, t, t′)}{|t − t′|})^{\frac{1}{2} − \frac{1}{2(λ+3)}} = C (\frac{A (x, t, t′)}{|t − t′|})^{\frac{1}{2} \frac{λ+2}{λ+3}} ,

while for (26), we use (15) with f = a to obtain

\begin{aligned}|\frac{∂}{∂t}a (x, t′)| & ≤ C (\frac{max_{t′≤θ′≤θ≤t}|a(x,θ)−a(x,θ′)|}{|t−t′|})^{1− \frac{1}{λ+2}} \\ & ≤ C (\frac{max_{t′≤θ≤t}|a(x,θ)|}{|t−t′|})^{\frac{1}{2}} \\ & ≤ C (\frac{A(x,t,t′)}{|t−t′|})^{^{\frac{1}{2}}(1− \frac{1}{λ+3})} |t − t′|^{− \frac{1}{2}} by (17), \\ & ≤ C |t − t′|^{−1+ \frac{1}{2(λ+3)}} A (x, t, t′)^{\frac{1}{2} \frac{λ+2}{λ+3}} \end{aligned}  (29)

and the result now follows easily.

We must also deal with commutators of the above operators with D_{x} and this introduces factors of Ã_{x} into the symbol. Thus we set for m ≥ 0 an integer and δ = max \{\frac{1}{2}, \frac{1}{λ+1}\},

G_{m}u (x, t) = ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{−mδ} b(x, t, t′, ξ) (Ã_{x}(x, t, t′, ξ))^{m} K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′,
H_{m}u (x, t) = ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}a(x, t′)Q(x, t′, ξ) |ξ|^{−mδ} b(x, t, t′, ξ) (Ã_{x}(x, t, t′, ξ))^{m} ×
K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′,
J_{m}u (x, t) = ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{−mδ} b(x, t, t′, ξ) (Ã_{x}(x, t, t′, ξ))^{m} Da (x, t′)
×K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′,

(30)

and in Case II, we define corresponding “Poisson” operators for G_{m} and H_{m},

\begin{aligned}G_{m}u (x, t) & = & ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{−mδ} b(x, t, ξ) (Ã_{x}(x, t, 0, ξ))^{m} K(x, t, 0, ξ)ũ^{∗}(ξ, t)dξ, \\ H_{m}u (x, t) & = & ∫_{|ξ|≥1} e^{ix·ξ}a(x, t)Q(x, t, ξ) |ξ|^{−mδ} b(x, t, ξ) (Ã_{x}(x, t, 0, ξ))^{m} × \\ & & K(x, t, 0, ξ)ũ^{∗}(ξ, t)dξ. \end{aligned} (31)

Here we assume that b(x, t, t′, ξ) is C^{k} in the variables x, t, t′ for some k > 0, and is C^{∞} and homogeneous of degree zero in the variable ξ. Moreover, in order to obtain boundedness up to the cap λ + 2, we assume that b(x, t, t′, ξ) satisfies

\begin{matrix}F or any first order constant coefficient differential \\ operator D, there is an integer M such that \end{matrix}

Db(x, t, t′, ξ) = ∑_{j=1}^{M} b′_{j} (x, t, t′) b_{j}(x, t, t′, ξ),  (32)

where b′_{j} ∈ C^{k−1} and where b_{j} satisfies the same conditions as b.

Remark If σ ∈ C^{k}S_{1,δ,g}^{0} for k > 0, Corollary 10 only yields σ ∈ O_{(−(1−δ)[k],[k])}. As we will see below, property (32) permits us to easily obtain the improved range O_{(−(1−δ)[k],k)}.

Lemma 16 Let b(x, t, t′, ξ) be C^{k} in the variables x, t, t′ for some k ≥ 1, be C^{∞} in the variable ξ and satisfy both (22) and (32). Let Z_{m} denote either of the operators G_{m}, H_{m} or J_{m} in (30). Then there is an integer M such that

D ∘ Z_{m} = Z̃ ∘ |D_{x}| + ∫_{t_{0}}^{t} ∑_{j=1}^{M} ∫_{t_{0}}^{θ_{1}} b_{j} (x, t, θ) Z_{j} (x, t, t′, D_{x}) |D_{x}|^{1−δ} dt′dθ + Z_{0} (x, t, D_{x}) (33)

where Z̃, Z_{j}^{i} have the same form as Z_{m} and Z_{m+1} respectively, with the same smoothness in x, t, t′, where b_{j} = 0 if m = 0 or Z_{m} = G_{m} or H_{m}, and b_{j}^{i} ∈ C^{min{λ,k−1}} otherwise, and where

Z_{0}u = ∑_{i=1}^{k′} ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{k′−i} b_{0}^{i}(x, t, ξ)ũ^{∗}(ξ, t)dξ,  (34)

where b_{0}^{i}(x, t, ξ) is C^{min{λ+2,k}} in the variables x, t, and is C^{∞} in the variable ξ and satisfies (22). The corresponding conclusions hold if Z is one of the operators G_{m} or H_{m} with the corresponding formula as follows:

D ∘ Z = Z̃ ∘ |D_{x}| + ∑_{j=1}^{M} ∫_{0}^{t} b_{j}^{i} (x, t, θ) Z_{j} (x, t, D_{x}) |D_{x}|^{1−δ} dθ.  (35)

Proof. We need only note that Q |ξ|^{−1} satisfies property (32) and then compute. The operator Z_{0} in (33) arises from the action of ∂_{t} on Z in the case when at least one of the D_{i} involves ∂_{t}, while b_{j}^{i} arises from the action of ∂_{x_{i}} on b \binom{}{Ã_{x}}^{m}.

The next definition extends the mapping properties of Definition 7 to the parameter variable.

Definition 17 An operator W belongs to O_{I}^{m}, where I is an interval, if W admits a bounded extension from H_{p,comp}^{s+m} to H_{p,loc}^{s} and from Λ_{comp}^{s} to Λ_{loc}^{s} for all s in the interval I and all 1 < p < ∞. The operator W belongs to \overline{O}_{I}^{m} if in addition, it is bounded from Λ_{comp}^{t} to Λ_{loc}^{t} where t is the right endpoint of the interval I.

Proposition 18 Let φ be a zero order ψdo with constant coefficients and support in the cone {|τ| < C |ξ|}. Let b(x, t, t′, ξ) be C^{k} in the variables x, t, t′ for some k ≥ 1, and be C^{∞} in the variable ξ and satisfy both (22) and (32). Then

G_{0} ∘ φ, G_{0} ∘ φ ∈ O_{I}^{0}, H_{0} ∘ φ ∈ OI^{\frac{1}{λ+3}} , H_{0} ∈ OI^{\frac{1}{2}} , J_{0} ∘ φ ∈ O_{I}^{− \frac{1}{4}}

with I = (−1, min {λ + 2, k}) and

G_{m} ∘ φ, G_{m} ∈ O_{I′}^{0}, H_{m} ∘ φ ∈ OI′^{\frac{1}{λ+3}} , H_{m} ∈ OI′^{\frac{1}{2}}, J_{0} ∘ φ ∈ O_{I′}^{− \frac{1}{2} \frac{λ+2}{λ+3}}

with I′ = (−1, min {λ + 1, k}) for m ≥ 1. In particular,

\begin{matrix}K ∘ φ, K ∈ O_{(−1,λ+2)}^{0}, T ∘ φ ∈ O(−1,λ+2)^{\frac{1}{λ+3}}, ∂_{t} ∘ K, D_{x} ∘ K − K ∘ D_{x} ∈ O(−1,λ+1)^{\frac{1}{2}}, \\ K ∘ a ∘ φ ∈ O_{(−1,λ+2)}^{\frac{1}{λ+3} −1}, K ∘ Da ∘ φ ∈ O_{(−1,λ+1)}^{− \frac{1}{2} \frac{λ+2}{λ+3}}. \end{matrix}

In fact, we have much stronger statements for T and K ∘ a, namely

T ∘ φ ∈ O(−1,λ+2)^{\frac{1}{p(λ+3)}}, K ∘ a ∘ φ ∈ O_{(−1,λ+2)}^{\frac{1}{p(λ+3)} −1}.

Proof. Since min {λ + 2, k} ≥ 1, we have by Lemma 16 formulas (33) and (35). All of the assertions, apart from the last three concerning K ∘ Da, T and K ∘ a, follow from Proposition 9, Corollary 10, Proposition 15 (with μ = 1 for H_{m}), and the fact that C^{α} ⊂ H_{p}^{β} for 0 ≤ β < α. To handle K ∘ Da ∘ φ we note that

K ∘ Da ∘ φ = K (Da) ∘ φ + (K ∘ Da − K (Da)) ∘ φ
= K (Da) ∘ φ + (K ∘ Da^{♯} − K (Da^{♯})) ∘ φ + O_{(0,λ+1)}^{−1}
= K (Da) ∘ φ + O_{(0,λ+1)}^{−1},

by applying (3) to Da^{♯}. Since K (Da) ∘ φ ∈ O_{(−1,λ+1)}^{− \frac{1}{2} \frac{λ+2}{λ+3}}, so is K ∘ Da ∘ φ.

Finally, we use Theorem 31 from the next section to obtain the improvement T ∘ φ ∈

O(−1,λ+2)^{\frac{1}{p(λ+3)}}. Alternatively, we could note that T : Λ^{s} → Λ^{s} by Corollary 24 below, and that

T : B_{1+ϵ}^{s+ \frac{1}{λ+3} ,1+ϵ} → B_{1+ϵ}^{s,1+ϵ} for all ϵ > 0 by what we just proved. Interpolation then yields

T ∘ φ ∈ O_{(−1,λ+2)}^{\frac{1}{p(λ+3)} +ϵ′}. Similar arguments apply to show that K ∘ a ∘ φ ∈ O_{(−1,λ+2)}^{\frac{1}{p(λ+3)} −1}.

We now dispose of the operator R in (10).

Lemma 19 Let μ_{p} = min \{1 − δ − \frac{1}{p(λ+3)}, \frac{λ+2}{2(λ+3)}\} where δ = max \{\frac{1}{2}, \frac{1}{λ+1}\}. Let φ be

supported in the nonelliptic cone as in Proposition 18. Then R ∘ φ ∈ O_{[0,λ+2)}^{−μ_{p}}.

Proof. We begin by decomposing R ∘ φ as

R ∘ φ = (K ∘ ãQ̃ − KãQ̃) ∘ φ
= (K ∘ ã − Kã) ∘ Q̃ ∘ φ + (Kã ∘ Q̃ − KãQ̃) ∘ φ
= I + II.

Write ã = ã^{♯} + ã^{♭} where ã^{♭} ∈ C^{λ+2}S_{1,δ}^{−(λ+2)δ}. Now let φ_{1} be homogeneous of degree zero and satisfy φ_{1} = 1 on the support of φ. Then there exists φ̃ of the same type such that

\begin{aligned}Q̃ ∘ φ & = Q̃ ∘ φ_{1} ∘ φ = φ_{1} ∘ Q̃ ∘ φ − \{∑_{1≤|ℓ|≤N} (∇_{ξ}^{ℓ}φ_{1}) · (∇_{x}^{ℓ}Q̃)\} ∘ φ + O_{[0,λ+2)}^{−(1−δ)N} \\ & = φ̃ ∘ O_{[0,λ+2)}^{1} + O_{[0,λ+2)}^{−(1−δ)N}. \end{aligned}

Then we have

I = (K ∘ ã^{♯} − Kã^{♯}) ∘ φ̃ ∘ O_{[0,λ+2)}^{1} + O_{[0,λ+2)}^{1−(λ+2)δ}.

Now we have

(K ∘ ã^{♯} − Kã^{♯}) ∘ φ̃ = ∇_{ξ}K · ∇_{x}ã^{♯} ∘ φ̃ + O_{[0,λ+2)}^{−2}

by (3). Also,

\begin{matrix}∇_{ξ}K · ∇_{x}ã^{♯} ∘ φ̃ = ∇_{ξ}K · ∇_{x}ã ∘ φ̃ − ∇_{ξ}K · ∇_{x}ã^{♭} ∘ φ̃ = ∇_{ξ}K · ∇_{x}ã ∘ φ̃ + O_{[0,λ+1)}^{−2− \frac{1}{λ+1}} , \\ ∇_{ξ}K · ∇_{x}ã ∈ C^{λ+1}S_{1,δ}^{− \frac{1}{2} \frac{λ+2}{λ+3} −1}, \end{matrix}

by (25). Thus ∇_{ξ}K · ∇_{x}ã^{♯} ∘ φ̃ ∈ O_{[0,λ+1)}^{− \frac{1}{2} \frac{λ+2}{λ+3} −1}, and to obtain the improvement in the cap up to λ + 2, i.e. ∇_{ξ}K · ∇_{x}ã^{♯} ∘ φ̃ ∈ O_{[0,λ+2)}^{− \frac{1}{2} \frac{λ+2}{λ+3} −1}, we compose with D:

\begin{matrix}D_{x} ∘ ∇_{ξ}K · ∇_{x}ã^{♯} = ∇_{ξ}K · ∇_{x}ã^{♯} ∘ D_{x} + (∇_{x}∇_{ξ}K) ã^{♯} + ∇_{ξ}K · ∇_{x}^{2}ã^{♯}, \\ D_{t} ∘ ∇_{ξ}K · ∇_{x}ã^{♯} = KaQ∇_{ξ}Ãã^{♯} − Ka∇_{ξ}Qã^{♯}. \end{matrix}

Thus ∇_{ξ}K · ∇_{x}ã^{♯} ∘ φ̃ ∈ O_{[0,λ+2)}^{− \frac{1}{2} \frac{λ+2}{λ+3} −1} and we get

I ∈ O_{[0,λ+2)}^{− \frac{1}{2} \frac{λ+2}{λ+3}} .

For term II, we write Q̃ = Q̃^{♯} + Q̃^{♭} where Q̃^{♯} = Q (x, t′, ξ, ã^{♯}) and Q̃^{♭} = Q̃ − Q̃^{♯} = ã^{♭}Q̃′ ∈ C^{λ+2}S_{1,δ}^{1−(λ+2)δ} upon applying Taylor’s formula to the final variable in Q̃. Then

II = (Kã ∘ Q̃^{♯} − KãQ̃^{♯}) ∘ φ + (Kã ∘ Q̃^{♭} − KãQ̃^{♭}) ∘ φ
= \{∇_{ξ} (Kã) · ∇_{x}Q̃^{♯} + O_{[0,λ+2)}^{−1+ \frac{1}{p(λ+3)} +1+2(δ−1)}\} ∘ φ + O_{[0,λ+2)}^{\frac{1}{λ+3} −(λ+2)δ}
= ∇_{ξ} (Kã) · ∇_{x}Q̃^{♯} ∘ φ + O_{[0,λ+2)}^{\frac{1}{p(λ+3)} +2(δ−1)}

Finally, ∇_{ξ} (Kã) · ∇_{x}Q̃^{♯} ∘ φ ∈ O_{[0,λ+2)}^{−1+ \frac{1}{p(λ+3)} −1+(1+δ)} = O_{[0,λ+2)}^{\frac{1}{p(λ+3)} +δ−1} and so R ∘ φ ∈ O_{[0,λ+2)}^{−μ_{p}} as required.

Remark If \overrightarrow{ℓ} satisfies A_{p,α}^{∓} on Γ, etc. as in part (A) of Theorem 8 of [GSI], then R ∘ φ ∈ O_{[0,λ+2)}^{−(1−δ−α)} since Theorem 31 in section 2 below (or more precisely its proof) applies to show that ∇_{ξ} (Kã) · ∇_{x}Q̃^{♯} ∘ φ ∈ O_{[0,λ+2)}^{−(1−δ−α)}.

1.4 Boundedness of operators on Λ^{s}

Here we discuss the behaviour of the operators K, K and T on the Hölder spaces Λ^{s}, s > 0. We begin by considering the the following types of operator with δ = max \{\frac{1}{2}, \frac{1}{λ+1}\}:

G_{m,σ}u (x, t) = ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{−σ−mδ} b(x, t, t′, ξ) (Ã_{x}(x, t, t′, ξ))^{m} K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′,
H_{m,σ}u (x, t) = ∫_{t_{0}}^{t} ∫_{|ξ|≥1} e^{ix·ξ}a(x, t′)Q(x, t′, ξ) |ξ|^{−σ−mδ} b(x, t, t′, ξ) (Ã_{x}(x, t, t′, ξ))^{m} ×
K(x, t, t′, ξ)ũ^{∗}(ξ, t′)dξdt′,
G_{m,σ}u (x, t) = ∫_{|ξ|≥1} e^{ix·ξ} |ξ|^{−σ−mδ} b(x, t, ξ) (Ã_{x}(x, t, t_{0}, ξ))^{m} K(x, t, t_{0}, ξ)ũ^{∗}(ξ)dξ,

where σ ∈ R, m ∈ Z^{+} and b ∈ C^{ℓ}S_{1,δ}^{0}. Let G_{m,σ}(x, x′, t, t′), H_{m,σ}(x, x′, t, t′) and G_{m,σ}u (x, x′, t) be the distribution kernels of G_{m,σ}, H_{m,σ} and G_{m,σ} respectively. It is then easy to show (see also [Gu]) the following.

Lemma 20 If c (x, t, t′, ξ) satisfies |∂_{ξ}^{α}c (x, t, t′, ξ)| ≤ C_{α} (1 + Ã (x, t, t′, ξ))^{N} (1 + |ξ|)^{−|α|+σ} for some N ≥ 0 and 0 ≤ |α| ≤ [σ] + n + 1, if Z is an operator of the form

Zu (x, t, t′) = ∫_{|ξ|≥1} e^{ix·ξ}c(x, t, t′ξ)K(x, t, t′, ξ)ũ^{∗}(ξ)dξ,

and if Z (x, x′, t, t′) is the distribution kernel of Z, then for |x − x′| + A (x, t, t′) > 0,

|Z (x, x′, t, t′)| ≤ C ( max_{\begin{matrix}|α|≤[γ]+n+1 \\ |ξ|=1 \end{matrix}} |∂_{ξ}^{α}c(x, t, t′, ξ)|) \frac{1}{(|x − x′| + A(x, t, t′))^{n+σ}}  (36)

Proof. We consider two cases: (i) A (x, t, t′) ≥ |x − x′| and (ii) A (x, t, t′) ≤ |x − x′|. In case (i), we use K = e^{−_{A}} where \frac{1}{c}A |ξ| ≤ Re \binom{}{A} ≤ cA |ξ| to obtain

|∫_{|ξ|≥1} e^{i(x−x′)·ξ}c (x, t, t′ξ) e^{−_{A}(x,t,t′ξ)}dξ| ≤ ∫_{|ξ|≥1} |ξ|^{γ} e^{−A|ξ|}dξ ≤ CA^{−n−σ}.

In case (ii), choose ρ ∈ C_{c}^{∞} (R_{+}) so that ρ = 0 on (0, \frac{1}{2}) and ρ = 1 on [1, ∞). Now

Z (x, x′, t, t′) = ∫_{|ξ|≥1} e^{i(x−x′)·ξ}c (x, t, t′ξ) ρ (|x − x′| |ξ|) e^{−_{A}(x,t,t′ξ)}dξ

+ ∫_{|ξ|≥1} e^{i(x−x′)·ξ}c (x, t, t′ξ) {1 − ρ (|x − x′| |ξ|)} e^{−_{A}(x,t,t′ξ)}dξ

= I + II,

and we have

\begin{aligned}|II| & ≤ ∫_{|ξ|≥1} |ξ|^{σ} \{1 − ρ (|x − x′| |ξ|)\} dξ \\ & ≤ ∫_{|x−x′|^{−1}≥|ξ|≥1} |ξ|^{σ} dξ ≤ |x − x′|^{−σ−n} . \end{aligned}

As for I, we may assume (x − x′) = (|x − x′| , 0, ..., 0), and we then have

I = ∫_{|ξ|≥1} (\frac{1}{i |x − x′|})^{k} \{∂_{ξ_{1}}^{k}e^{i(x−x′)·ξ}\} c (x, t, t′ξ) ρ (|x − x′| |ξ|) e^{−_{A}(x,t,t′ξ)}dξ.

Now perform integration by parts to get

\begin{matrix}I = ∫_{|ξ|≥1} e^{i(x−x′)·ξ} ∑_{j=1}^{k} \frac{c_{j,k}}{|x−x′|^{j}} ∂_{ξ_{1}}^{j}ρ (|x − x′| |ξ|) \frac{1}{|x−x′|^{k−j}} ∂_{ξ_{1}}^{k−j} \{c (x, t, t′ξ) e^{−_{A}(x,t,t′ξ)}\} dξ \\ + ∫_{|ξ|≥1} e^{i(x−x′)·ξ} \frac{c_{k}}{|x−x′|^{k}} ρ (|x − x′| |ξ|) ∂_{ξ_{1}}^{k} \{c (x, t, t′ξ) e^{−_{A}(x,t,t′ξ)}\} dξ. \end{matrix}

Since |∂_{ξ_{1}}^{j}ρ (|x − x′| |ξ|)| ≤ C_{j} |x − x′|^{j} for \frac{1}{2|x−x′|} ≤ |ξ| ≤ \frac{1}{|x−x′|}, and 0 otherwise, we conclude that

\begin{matrix}|I| ≤ C ∫_{\frac{1}{2|x−x′|} ≤|ξ|≤ \frac{1}{|x−x′|}} ∑_{j=1}^{k} \frac{1}{|x−x′|^{k−j}} (max\begin{matrix}1≤|α|≤k \\ |ξ|=1 \end{matrix} |∂_{ξ}^{α}c(x, t, t′, ξ)|) |ξ|^{σ−k+j} dξ \\ +C ∫_{\frac{1}{2|x−x′|} ≤|ξ|} ∑_{j=1}^{k} \frac{1}{|x−x′|^{k}} (max\begin{matrix}1≤|α|≤k \\ |ξ|=1 \end{matrix} |∂_{ξ}^{α}c(x, t, t′, ξ)|) |ξ|^{σ−k} dξ. \end{matrix}

If we choose k = [σ] + n + 1, then the integral in the second term on the right is convergent, and we obtain the desired estimate for term I.

Lemma 21 There is a constant C independent of σ, x, x′, t and t′ such that for x ≠ x′,

|G_{m,σ}(x, x′, t, t′)| ≤ C (max\begin{matrix}|α|≤[σ]+n+1 \\ |ξ|=1 \end{matrix} |∂_{ξ}^{α}b(x, t, t′, ξ)|) \begin{matrix}|A_{x}(x,t,t′)| & ^{m} \\ (|x−x′|+A(x,t,t′)) & ^{n+ \frac{m}{2} −σ} \end{matrix} ,

|H_{m,σ}(x, x′, t, t′)| ≤ C (max\begin{matrix}|α|≤[σ]+n+1 \\ |ξ|=1 \end{matrix} |∂_{ξ}^{α}b(x, t, t′, ξ)|) \begin{matrix}|A_{x}(x,t,t′)| & ^{m} & |a(x,t′)| \\ (|x−x′|+A(x,t,t′)) & ^{n+1+ \frac{m}{2} −σ} \end{matrix} .

|G_{m,σ}(x, x′, t)| ≤ C (max\begin{matrix}|α|≤[σ]+n+1 \\ |ξ|=1 \end{matrix} |∂_{ξ}^{α}b(x, t, ξ)|) \begin{matrix}|A_{x}(x,t,t_{0})| & ^{m} \\ (|x−x′|+A(x,t,t_{0})) & ^{n+ \frac{m}{2} −σ} \end{matrix} ,

Proof. This is a direct consequence of Lemma 20 and the inequality

max_{|ξ|=1} |∂_{ξ}^{α}Ã_{x}(x, t, t′, ξ)| ≤ C_{α} |A_{x} (x, t, t′)| .

We now have

Lemma 22 If b satisfies (22) with k = 1 and 0 < σ < 1, then there is a positive constant C_{σ} such that for every ϵ ∈ R, ϵ ≠ 0 and |t| < 1, the following estimates hold where Z_{σ} is either G_{m,σ} or H_{m,σ}:

∫_{0}^{t} ∫_{|x−x′|<|ϵ|} |Z_{σ}(x, x′, t, t′)| dx′dt′ ≤ C_{σ} |ϵ|^{σ} ,  (37)

∫_{0}^{t} ∫_{B} |Z_{σ}(x + h, x′, t, t′) − Z_{σ}(x, x′, t, t′)| dx′dt′ ≤ C_{σ} |h|^{σ} ,  (38)

and if in addition a(x, t)a(x, t + ϵ) ≥ 0, then

∫_{t}^{t+ϵ} ∫_{B} |Z_{σ}(x, x′, t, t′)| dx′dt′ ≤ C_{σ} |ϵ|^{σ} ,  (39)

∫_{0}^{t} ∫_{B} |Z_{σ}(x, x′, t + ϵ, t′) − Z_{σ}(x, x′, t, t′)| dx′dt′ ≤ C_{σ} |ϵ|^{σ} .  (40)

If Z_{σ} is G_{m,σ}, we have

∫_{|x−x′|<|ξ|} |Z_{σ}(x, x′, t)| dx′ ≤ C_{σ} |ϵ|^{σ} ,
∫_{B} |Z_{σ}(x + h, x′, t) − Z_{σ}(x, x′, t)| dx′ ≤ C_{σ} |h|^{σ} ,

and

∫_{B} |Z_{σ}(x, x′, t + ϵ) − Z_{σ}(x, x′, t)| dx′ ≤ C_{σ} |ϵ|^{σ} .

Proof. We prove only the estimates for H_{0,σ}, the cases for G_{0,σ} and G_{0,σ} being easier, and the cases for G_{m,σ}, H_{m,σ} and G_{m,σ} being similar, but using |A_{x}| ≤ C\sqrt{A}. We have

∫_{0}^{t} ∫_{|x−x′|<|ϵ|} |H_{0,σ}(x, x′, t, t′)| dx′dt′ ≤ C ∫_{0}^{t} ∫_{|x−x′|<|ϵ|} \frac{|a(x,t′)|}{(|x−x′|+A(x,t,t′))^{n+1−σ}} dx′dt′
≤ C ∫_{0}^{t} ∫_{0}^{|ϵ|} \frac{|a(x,t′)|}{(r+A(t,t′,x))^{2−σ}} drdt′
≤ C ∫_{0}^{|ϵ|} |∫_{0}^{t} \frac{\frac{d}{dt′} A(x,t,t′)}{(r+A(x,t,t′))^{2−σ}} dt′| dr

since a(x, t′) keeps the same sign in (0, t), and thus we have

\begin{aligned}∫_{0}^{t} ∫_{|x−x′|<|ϵ|} |H_{0,σ}(x, x′, t, t′)| dx′dt′ & ≤ C ∫_{0}^{|ϵ|} \{(r + A(x, t, t))^{−1+σ} + (r + A(x, t, 0))^{−1+σ}\} dr \\ & ≤ C_{σ} |ϵ|^{σ} , \end{aligned}

which proves (37).

To obtain (38) we write

∫_{0}^{t} ∫_{B} |H_{0,σ}(x + h, x′, t, t′) − H_{0,σ}(x, x′, t, t′)| dx′dt′

= ∫_{0}^{t} {∫_{|x−x′|≤4|h|} + ∫_{{x′∈B:|x−x′|≥4|h|}}} |H_{0,σ}(x + h, x′, t, t′) − H_{0,σ}(x, x′, t, t′)| dx′dt′ = I + II.

¿From (37) we have I ≤ C_{σ} |h|^{σ}. To estimate II, fix h and set

F (x, x′, t, t′, s) = H_{0,σ}(x + sh, x′, t, t′).

Then by Lemma 21 we have

II = ∫_{0}^{t} ∫_{\{x′∈B:|x−x′|≥4|h|\}} |∫_{0}^{1} \frac{d}{ds}F (x, x′, t, t′, s)ds| dx′dt′

≤ ∫_{0}^{t} ∫_{\{x′∈B:|x−x′|≥4|h|\}} ∫_{0}^{1} Ch \{\frac{|a(x+sh,t′)|}{(|x−x′|+A(x+sh,t,t′))^{n+2−σ}} + \frac{1}{(|x−x′|+A(x+sh,t,t′))^{n+1−σ}} \} dsdx′dt′

= III + IV.

Now we have

\begin{aligned}III & = Ch ∫_{|x−x′|≥4|h|} ∫_{0}^{1} |∫_{0}^{t} \frac{\frac{d}{dt′} A(x+sh,t,t′)}{(|x−x′|+A(x+sh,t,t′))^{n+2−σ}} dt′| dsdx′ \\ & ≤ Ch ∫_{\{x′∈B:|x−x′|≥4|h|\}} ∫_{0}^{1} \frac{1}{|x−x′|^{n+1−σ}} dsdx′ ≤ C_{σ}hh^{−1+σ} = C_{σ}h^{σ}, \end{aligned}

and similarly,

IV ≤ Ch ∫_{0}^{t} ∫_{\{x′∈B:|x−x′|≥4|h|\}} ∫_{0}^{1} \frac{1}{|x − x′|^{n+1−σ}} dsdx′dt′ ≤ C_{σ}h^{σ}.

The proofs for (39) and (40) are similar. We need only notice that a(x, t) keeps the same sign in the interval [t, t + ϵ] (or [t + ϵ, t]) if a(x, t)a(x, t + ϵ) ≥ 0.

Combining the estimates in Lemma 22 with a standard argument as in [Gu], we obtain the following proposition.

Proposition 23 Let 0 < σ ≤ λ. If b ∈ C^{2+λ}, then G_{m,σ} ∘φ, H_{m,σ} ∘φ and G_{m,σ} are bounded from Λ_{comp}^{s} to Λ_{loc}^{s+σ} provided 0 < s + σ ≤ λ + 1.

Corollary 24 If λ > 0, then K, K ∘ φ, T ∘ φ, Kã |D_{x}| ∘ φ and KaQ ∘ φ are bounded on Λ^{s} for 0 ≤ s ≤ λ + 2.

Proof. By Proposition 23 K, K ∘ φ, T ∘ φ, Kã |D_{x}| ∘ φ and KaQ ∘ φ are bounded on Λ^{s} for 0 ≤ s ≤ λ + 1. Then if Z is one of K, K, or T, we have ∂_{t} ∘ Z = Z′ ∘ |D_{x}| + Z̃ where Z̃ = b (x, t, D_{x}) Q (x, t, D_{x}) with b ∈ C^{λ+2}S_{1,0}^{0} and ∂_{x} ∘ Z = Z ∘ ∂_{x} + Z̃′, where Z′ has the form G_{0,0}, H_{0,0} or G_{0,0}, and where Z̃′ has the form G_{0,1}, H_{0,1} or G_{0,1}. The Corollary now follows from Proposition 23.

We can also now improve the conclusions of Proposition 18 to include boundedness on Λ^{s} up to the cap.

Proposition 25 For 1 < p ≤ ∞, we have

K,K ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{0},
T ∘ φ : Λ^{s} → Λ^{s}, 0 < s ≤ λ + 2,
T ∘ φ : B_{p}^{s+ \frac{1}{p(λ+3)}} → B_{p}^{s}, −1 < s ≤ λ + 2,
K ∘ a ∘ φ : Λ^{s} → Λ^{s+1}, 0 < s ≤ λ + 1,
K ∘ a ∘ φ : B_{p}^{s+ \frac{1}{p(λ+3)}} → B_{p}^{s+1}, −1 < s ≤ λ + 1,
K ∘ Da ∘ φ ∈ \overline{O}_{(−1,λ+1)}^{− \frac{1}{4}},
R ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{−μ_{p}},

where μ_{p} = min \{1 − δ + \frac{1}{p(λ+3)}, \frac{λ+2}{2(λ+3)}\} for 1 < p < ∞ and 0 < μ_{∞} < min \{1 − δ, \frac{λ+2}{2(λ+3)}\},

δ = max \{\frac{1}{2}, \frac{1}{λ+1}\}.

Proof. For K, K and T, the assertions follow from Proposition 18 and Corollary 24. As

for K ∘ a, we write a = a^{♯} + a^{♭} with a^{♭} ∈ C^{λ+2}S_{1, \frac{1}{λ+1}}^{−1− \frac{1}{λ+1}} . Then using the improved estimates

(3) on a^{♯}, we have

K ∘ a = K ∘ a^{♯} + K ∘ a^{♭}
= Kã^{♯} + ∑_{1≤|α|≤N} c_{α}∂_{ξ}^{α}K∂_{x}^{α}ã^{♯} + \overline{O}_{(−1,λ+2)}^{−1− \frac{1}{λ+1}}
= Kã − Kã^{♭} + \overline{O}_{(−1,λ+2)}^{−1}
= Kã + \overline{O}_{(−1,λ+2)}^{−1}
= (Kã |D_{x}|) ∘ |D_{x}|^{−1} + \overline{O}_{(−1,λ+2)}^{−1}.

Now Kã |D_{x}| has the same mapping properties as T, and so the assertions regarding K∘a∘φ follow.

Now we turn our attention to K ∘ Da ∘ φ. We have

K ∘ Da = K ∘ Da^{♯} + K ∘ Da^{♭}
= K (Dã^{♯}) + \overline{O}_{(−1,λ+1)}^{−1}
= K (Dã) − K (Dã^{♭}) + \overline{O}_{(−1,λ+1)}^{−1}
= K (Dã) + \overline{O}_{(−1,λ+1)}^{−1}.

So by Proposition 18, K ∘ Da ∈ \overline{O}_{(0,λ+1)}^{− \frac{1}{4} −ϵ(λ)}. Since Λ^{s} ⊂ H_{p}^{s− \frac{ϵ}{2}} for all 1 < p < ∞, we have

K ∘ Da : Λ^{s} → Λ^{s+ \frac{1}{4}} for 0 < s < λ + \frac{3}{4}, and it remains only to show that K ∘ Da : Λ^{λ+ \frac{3}{4}} → Λ^{λ+1}. For this it suffices to show

∂_{t} ∘ K ∘ Da : Λ^{λ+ \frac{3}{4}} → Λ^{λ}, ∂_{x} ∘ K ∘ Da : Λ^{λ+ \frac{3}{4}} → Λ^{λ}.

Now ∂_{t} ∘ K ∘ Da = −KaQ ∘ Da + Da and by Corollary 24,

\begin{matrix}K ∘ a ∘ φ : Λ^{s} → Λ^{s}, 0 < s ≤ λ + 1, \\ Da : Λ^{s+1} → Λ^{s}, 0 < s ≤ λ + 1, \end{matrix}

which shows that ∂_{t} ∘ K ∘ Da : Λ^{λ+ \frac{3}{4}} → Λ^{λ}. As for ∂_{x} ∘ K ∘ Da, we have

\begin{aligned}∂_{x} ∘ K ∘ Da & = K ∘ ∂_{x} ∘ Da + KÃ_{x} ∘ Da \\ & = (K ∘ Da) ∘ ∂_{x} + K ∘ ∂_{x} (Da) + KÃ_{x} ∘ Da. \end{aligned}

We have ∂_{x} (Da) : Λ^{s} → Λ^{s}, 0 < s ≤ λ and KÃ_{x} ∘ φ : Λ^{s+ \frac{1}{2}} → Λ^{s}, 0 < s ≤ λ + 1 and so

∂_{x} ∘ K ∘ Da ∘ φ : Λ^{λ+ \frac{3}{4}} → Λ^{λ}, i.e., K ∘ Da ∘ φ ∈ \overline{O}_{(0,λ+2)}^{− \frac{1}{4}}.

For R, we have by (19) that R ∘ φ ∈ \overline{O}_{(0,λ+2)}^{−μ_{∞}}, since as before, Λ^{s} ⊂ H_{p}^{s− \frac{ϵ}{2}} for all 1 < p < ∞ and we can take p very large. Thus R ∘ φ : Λ^{s} → Λ^{s+μ_{∞}}, 0 < s < λ + 2 − μ_{∞}. To show that R ∘ φ : Λ^{λ+2−μ_{∞}} → Λ^{λ+2}, it suffices to show

∂_{t} ∘ R, ∂_{x} ∘ R : Λ^{λ+2−μ_{∞}} → Λ^{λ+1}.

For this we compute

∂_{t} ∘ R = KaQ ∘ ãQ̃ − KaQãQ̃ = (KaQ) ∘ ã ∘ Q̃ − KaQãQ̃

= (KaQ ∘ ã^{♯}) ∘ Q̃ − KaQã^{♯}Q̃ + O_{(0,λ+2)}^{\frac{1}{λ+3} −1− \frac{1}{λ+1} +1}

= ∑_{|α|≤N} c_{α}∂_{ξ}^{α} (KaQ) ∂_{x}^{α} \binom{}{a^{♯}} ∘ Q̃ − KaQã^{♯}Q̃ + O_{(0,λ+2)}^{\frac{1}{λ+3} − \frac{1}{λ+1}}

= (KaQ ∘ ã^{♯}) ∘ Q̃ − KaQã^{♯}Q̃ + O(0,λ+2)^{\frac{1}{λ+3}}.

Now Q̃ = Q (x, t′, ξ, ã). Letting Q^{♯} = Q (x, t′, ξ, ã^{♯}), we obtain

\begin{aligned}∂_{t} ∘ R & = (KaQã^{♯}) ∘ Q^{♯} − KaQã^{♯}Q^{♯} + O(0,λ+2)^{\frac{1}{λ+3}}. \\ & = (KaQã) ∘ Q^{♯} − KaQãQ^{♯} + O(0,λ+2)^{\frac{1}{λ+3}} = O(0,λ+2)^{\frac{1}{λ+3}}, \end{aligned}

by the sharp estimates (3) for ã^{♯}. Also, using Proposition 23,

∂_{x} ∘ R = ∂_{x} ∘ K ∘ ãQ̃ − ∂_{x} ∘ KãQ̃

= K ∘ ∂_{x} ∘ ãQ̃ − KãQ̃ ∘ ∂_{x} + KÃ_{x}ãQ̃ − K (ãQ̃)_{x} − KÃ_{x} ∘ ãQ̃

= K ∘ ãQ̃ ∘ ∂_{x} − KãQ̃ ∘ ∂_{x} + K ∘ (ãQ̃)_{x} − K (ãQ̃)_{x} + KÃ_{x}ãQ̃_{x} − KÃ_{x} ∘ ãQ̃

= R ∘ ∂_{x} + K ∘ (ãQ̃)_{x} − K (ãQ̃)_{x} + KÃ_{x}ãQ̃ − KÃ_{x} ∘ ã^{♯}Q̃ − KÃ_{x} ∘ ã^{♭} ∘ Q̃

= R ∘ ∂_{x} + K ∘ (ã_{x}Q̃) − Kã_{x}Q̃ + K ∘ ã ∘ Q̃_{x} − KãQ̃_{x} + KÃ_{x}ãQ̃ − KÃ_{x}ã^{♯} ∘ Q̃ + \overline{O}(0,λ+1)^{\frac{1}{2}}

= R ∘ ∂_{x} + K ∘ (ã_{x}Q̃) − Kã_{x}Q̃ + K ∘ ã ∘ Q̃_{x} − KãQ̃_{x} + KÃ_{x}ãQ̃ − KÃ_{x}ã ∘ Q̃ + \overline{O}(0,λ+1)^{\frac{1}{2}}.

Now

K ∘ ã_{x} = K ∘ \binom{}{a^{♯}}_{x} + K ∘ \binom{}{a^{♭}}_{x} = Kã_{x}^{♯} + \overline{O}_{(0,λ+1)}^{−δ(λ+2)} = Kã_{x} + \overline{O}_{(0,λ+1)}^{−δ(λ+2)},

and Kã_{x} ∈ \overline{O}_{(0,λ+1)}^{− \frac{1}{4}} and KÃ_{x}ã : Λ^{s} → Λ^{s+ \frac{1}{2}} , 0 < s ≤ λ + \frac{1}{2} and KÃ_{x}ãQ̃ : Λ^{s} → Λ^{s− \frac{1}{2}} , 0 < s ≤ λ + \frac{3}{2} by Proposition 23. Also Kã, K ∘ ã : Λ^{s−1} → Λ^{s}, 0 < s ≤ λ + 2 by the first part of the proof. Also

Q̃_{x} = \frac{∂}{∂x}Q (x, t′, ξ, a (x, t′)) = ã_{x} · Q_{z} + C^{λ+2}s_{1,δ}^{1+δ},

and so we conclude that ∂_{x} ∘ R ∘ φ : Λ^{λ+2−μ_{∞}} → Λ^{λ+1}, and this completes the proof that R ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{−μ_{p}}.

1.5 Reduction of the operators K, K, and T

In the above subsections, we discussed the behaviour of the operators K, K, and T in the nonelliptic cone {|τ| < C |ξ|}. Now in general, T is not bounded on H_{p}^{s} or B_{p}^{s,p} and so Proposition 18 needn’t hold with \frac{1}{λ+3} replaced by 0, and neither K nor K need have any gain. In order to establish sharp mapping properties for these operators in the next section, we will show here that the boundedness of T, K, or K into H_{p}^{s} (or B_{p}^{s,p}) for some p, s satisfying 1 < p < ∞ and −1 < s < λ + 2, is equivalent to the boundedness of the corresponding operators φ ∘ T ∘ φ, φ ∘ K ∘ φ, or φ ∘ K into L^{p} ( or B_{p}^{0,p}). We continue to denote by φ and ψ zero order ψdo’s with support in the nonelliptic cone {|τ| < C |ξ|} and the elliptic cone {|ξ| < C |τ|} respectively.

Define K_{ν} = K ∘ |D_{x}|^{ν} and K_{ν} = K ∘ |D_{x}|^{ν} for 0 ≤ ν ≤ \frac{1}{2}.

Lemma 26 We have

\begin{aligned}T ∘ ∇_{x} ∘ φ & = & ∇_{x} ∘ T ∘ φ + B_{T} , \\ K_{ν} ∘ ∇_{x} ∘ φ & = & ∇_{x} ∘ K_{ν} ∘ φ + B_{ν}, \\ K_{ν} ∘ ∇_{x} & = & ∇_{x} ∘ K_{ν} + B_{ν}, \\ K ∘ a ∘ ∇_{x} ∘ φ & = & ∇_{x} ∘ K ∘ a ∘ φ + B^{∗}, \end{aligned}  (41)

where B_{T} ∈ O_{(−1,λ+1)}^{\frac{1}{λ+3} +δ}, B^{∗} ∈ O_{(−1,λ+1)}^{\frac{1}{λ+3} −(1−δ)} and B_{ν}, B_{ν} ∈ O_{(−1,λ+1)}^{ν+δ}.

Proof. This follows from T ∈ C^{λ+2}S1,δ^{\frac{1}{λ+3}} , K_{ν} ∈ C^{λ+2}S_{1,δ}^{ν}, K_{ν} ∈ C^{λ+2}S_{1,δ}^{ν}, and K ∘ a ∈

C^{λ+2}S_{1,δ}^{\frac{1}{λ+3} −1}.

Lemma 27 We have

ψ ∘ T ∘ φ, ψ ∘ K ∘ a ∘ φ, ψ ∘ K_{ν} ∘ φ, ψ ∘ K_{ν} ∈ \overline{O}_{(−1,λ+2)}^{0},

If Z denotes either T ∘ φ, K_{ν} ∘ φ or K_{ν}, then Z is bounded from H_{p}^{s+ν− \frac{1}{p}} → H_{p}^{s} or from

B_{p}^{s+ν− \frac{1}{p} ,p} → B_{p}^{s,p} for some γ ≥ 0, −1 < s < λ + 2 and 1 < p < ∞ if and only if φ ∘ Z ∘ φ is bounded from H_{p}^{γ} → L^{p} or from B_{p}^{γ,p} → B_{p}^{0,p}.

Proof. We have

\begin{matrix}ψ ∘ K_{ν} ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ ∂_{t} ∘ K_{ν} ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ \{|D_{x}|^{ν} − KaQ ∘ |D_{x}|^{ν}\} ∈ \overline{O}_{(−1,λ+2)}^{−1+ \frac{1}{λ+3} +ν}, \\ ψ ∘ K_{ν} = ∂_{t}^{−1} ∘ ψ ∘ ∂_{t} ∘ K_{ν} ∘ φ = −∂_{t}^{−1} ∘ ψ ∘ KaQ ∘ |D_{x}|^{ν} ∘ φ ∈ \overline{O}_{(−1,λ+2)}^{−1+ \frac{1}{2} +ν}. \end{matrix}

Since \frac{1}{λ+3} + ν < \frac{1}{2} + ν ≤ 1, we have ψ ∘ K_{ν} ∘ φ and ψ ∘ K_{ν} ∈ \overline{O}_{(−1,λ+2)}^{0}. Now

ψ ∘ K ∘ a ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ ∂_{t} ∘ K ∘ a ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ {a − Ka ∘ a} ∘ φ
= ∂_{t}^{−1} ∘ ψ ∘ {a − KaãQ̃} ∘ φ + O_{(0,λ)}^{−1},
ψ ∘ T ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ ∂_{t} ∘ T ∘ φ = ∂_{t}^{−1} ∘ ψ ∘ {aQ − KaQãQ̃} ∘ φ,

and KaQãQ̃ is bounded from H_{p}^{1} to L^{p}. Indeed, the last assertion follows from the argument at the top of page 62 in [GS] as follows. It suffices to show that KaãQ is bounded on L^{p}, but this is a consequence of the argument used to prove Theorem 28 in the next section below, since after the change of variables s = A_{x} (t) = ∫_{t_{0}}^{t} a (x, θ) dθ as in (65), matters reduce to

the validity of the Hardy inequality (66) with weights w = ^{\frac{a(A_{x}^{−1}(s))^{p−1}}{s^{p}}} and v = \frac{1}{a(A_{x}^{−1}(s))}.

But w ≤ s^{−p} and v ≥ 1, and so (66) holds trivially by (67). Thus ψ ∘ T ∘ φ : H_{p}^{1} → H_{p}^{1}. The first two identities in Lemma 26 together with Proposition 25 show that ψ ∘ T is bounded on H_{p}^{s} for −1 < s < λ + 1 and 1 < p < ∞. By interpolation, ψ ∘ T ∘ φ is bounded on B_{p}^{s,p} for −1 < s < λ and 1 < p < ∞. Finally, the rest of the lemma follows from Proposition 25 and Lemma 26. This completes the proof of Lemma 27.

2 Boundedness properties of K, K and T.

In the previous paper [GSI], we reduced matters regarding the gain from f in the oblique derivative problem to the boundedness of the operator

T f(x, t) = ∫_{R^{n}} e^{ix·ξ} ∫_{0}^{t} a(x, t′)Q(x, t′, ξ)e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}f^{∼}(ξ, t′)dt′dξ

from a Sobolev space into L^{p} (or B_{p}^{0,p}). Earlier, in the previous section, we observed that

aK ∈ C^{λ}S_{1, \frac{1}{2}}^{− \frac{λ}{λ+3}} , and it follows that T is in C^{λ}S1, \frac{1}{2}^{\frac{1}{λ+3}} , and so of order \frac{1}{λ+3} on L^{p} Sobolev spaces (of the appropriate index). This is best possible for all 1 < p < ∞, but if we fix attention on a particular p, then we can do better, namely T ∈ O^{\frac{1}{p(λ+3)}} . To see this, we apply some techniques from harmonic analysis to characterize the boundedness on L^{p} of the operators T_{α} = T Q^{−α}:

T_{α}f(x, t) = ∫_{R^{n}} e^{ix·ξ} ∫_{0}^{t} a(x, t′) (Q(x, t′, ξ))^{1−α} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}f^{∼}(ξ, t′)dt′dξ. (42)

In [GS], we showed that for the case α = 0, T_{0} = T is bounded on L^{p}(R^{n} × (0, 1)) if and only if a satisfies the A_{p}^{−} condition,

[\frac{1}{∫_{σ}^{β} a(x, t)dt} ∫_{σ}^{β} a(x, t)^{p′}dt]^{p−1} ≤ C \frac{1}{γ − β} ∫_{β}^{γ} a(x, t)dt,  (43)

for all x in R^{n} and 0 < σ < β < γ < 1 such that ∫_{σ}^{β} a(x, t)dt = ∫_{β}^{γ} a(x, t)dt. This was accomplished by using the change of variables s = A_{x}(t) = ∫_{0}^{t} a(x, θ)dθ (for each fixed x) and the Calderón reproducing formula to reduce the boundedness of T on L^{p}(R^{n} × (0, 1)) to the family of vector-valued weighted norm inequalities,

∫_{0}^{A_{x}(1)} (∑_{k=0}^{∞} |M^{−}h_{k}(s)|^{q})^{p/q} w_{x}(s)ds ≤ C ∫_{0}^{A_{x}(1)} (∑_{k=0}^{∞} |h_{k}(s)|^{q})^{p/q} w_{x}(s)ds,  (44)

for all sequences {h_{k}}_{k=0}^{∞} of nonnegative functions on (0, A_{x}(1)), where M^{−} denotes the one-sided Hardy-Littlewood maximal operator,

M^{−}f(s) = sup_{0<δ<s} \frac{1}{δ} ∫_{s−δ}^{s} |f(t)| dt,

and w_{x}(s) = \frac{d}{ds}A_{x}^{−1}(s) = \frac{1}{a(A_{x}^{−1}(s))}. Using [Sa2], these inequalities were in turn shown to be equivalent to the family of A_{p}^{−} conditions,

[\frac{1}{δ} ∫_{b}^{b+δ} w_{x}(s)ds] [\frac{1}{δ} ∫_{b−δ}^{b} w_{x}(s)^{1−p′}ds]^{p−1} ≤ C, 0 < δ < b < A_{x}(1) − δ,

which become (43) upon reversing the change of variable s = A_{x}(t).

However, a simpler approach is to exploit the fact that for functions f which have been microlocalized to have support f̂ ⊂ {|ξ| > |τ|}, the Besov space norm ∥f∥_{B_{p}^{0,p}} is comparable to the mixed norm

∥f∥_{L^{p}(B_{p}^{0,p})} = \{∫ ∥f(·, t)∥_{B_{p}^{0,p}(dx)}^{p} dt\}^{\frac{1}{p}} .  (45)

¿From this, it is not hard to reduce matters to the boundedness of T_{α} on the mixed Lebesgue-Besov space L^{p} (B_{p}^{0,p}) normed by (45).

In the next subsection, we characterize the boundedness of T_{α} on L^{p} (B_{p}^{0,p}) in terms of a simple modification of the the A_{p}^{−} condition (43). The proofs are modelled after the arguments in section 6 of [GS].

2.1 Boundedness of T_{α}

Our main theorem here is

Theorem 28 Suppose 1 < p < ∞ and a(x, t) is a bounded nonnegative function on R^{n} × (0, 1). Then T_{α} is bounded on L_{(0,1)}^{p} (B_{p}^{0,p} (R^{n})), i.e.

(∫_{0}^{1} ∥T_{α}f(·, t)∥_{B_{p}^{0,p}}^{p} dt)^{\frac{1}{p}} ≤ C (∫_{0}^{1} ∥f(·, t)∥_{B_{p}^{0,p}}^{p} dt)^{\frac{1}{p}} ,

if and only if a satisfies the A_{p,α}^{−} condition:

[\frac{1}{∫_{σ}^{β} a(x, t)dt} ∫_{σ}^{β} a(x, t)^{p′}dt]^{p−1} ≤ C \frac{1}{γ − β} [∫_{β}^{γ} a(x, t)dt]^{1−pα} ,  (46)

for all x in R^{n} and 0 < σ < β < γ < 1 with ∫_{σ}^{β} a(x, t)dt = ∫_{β}^{γ} a(x, t)dt > 0.

Proof. To show the necessity of the A_{p,α}^{−} condition, we first prove the apparently weaker condition

[\frac{1}{∫_{σ}^{β} a(x, t)dt} ∫_{σ}^{β} a(x, t)^{p′}dt]^{p−1} ≤ C \frac{1}{δ − γ} [∫_{γ}^{δ} a(x, t)dt]^{1−pα} ,  (47)

for all x in R^{n} and 0 < σ < β < γ < δ < 1 such that ∫_{σ}^{β} a(x, t)dt = ∫_{β}^{γ} a(x, t)dt = ∫_{γ}^{δ} a(x, t)dt > 0. For this, fix w in R^{n} and let r = ∫_{σ}^{β} a(w, θ)dθ = ∫_{β}^{γ} a(w, θ)dθ = ∫_{γ}^{δ} a(w, θ)dθ > 0. As in [GS], let ϕ̂ ∈ C^{∞}(R^{n}) have support contained in \{ξ : \frac{1}{2} ≤ |ξ| ≤ 4\} and satisfy ϕ̂(ξ) = 1 for 1 ≤ |ξ| ≤ 2. Set ϕ_{r}(x) = r^{−n}ϕ(\frac{x}{r}) so that ϕ̂_{r}(ξ) = ϕ̂(rξ). Letting ϕ_{r} denote the operation of convolution in the x-variable (in R^{n} × (0, 1) = {(x, t) : x ∈ R^{n}, 0 < t < 1}), then just as in ?, the real part of the kernel of T_{α}ϕ_{r} satisfies the following estimate for |x − w|, |x′ − w| < cr, t′ ∈ (α, β) and t ∈ (γ, δ):

Re(T_{α}ϕ_{r})(x, t, x′t′) ≥ ca(x, t′)r^{α−n−1}.  (48)

As in [GS], define f by

f(x′, t′) = χ_{B(w,cr)}(x′)χ_{(σ,β)}(t′)a(w, t′)^{p′−1}.  (49)

Combining (48) and (49) we obtain for |x − w| < cr, t ∈ (γ, δ),

Re(T_{α}ϕ_{r}f)(x, t) ≥ cr^{α−1} ∫_{σ}^{β} a(x, t′)a(w, t′)^{p′−1}dt′.  (50)

where the last integral in (50) satisfies

∫_{σ}^{β} a(w, θ)^{p′}dθ ≤ 2 ∫_{σ}^{β} a(x, θ)a(w, θ)^{p′−1}dθ  (51)

for |x − w| < cr.

Now choose η ∈ C_{c}^{∞}(R^{n}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n}} η(x)dx = 1. With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows easily from (50) and (51) that

Re(η_{r} ∗ T_{α}ϕ_{r}f)(x, t) ≥ cr^{α−1} ∫_{σ}^{β} a(w, t′)^{p′}dt′  (52)

for |x − w| < cr, t ∈ (γ, δ), with a perhaps smaller constant c - note that r ≤ (δ − γ)/3 if, as we may assume, a(x, t) ≤ 1/3. We will also need the following inequality, valid for any g ∈ B_{p}^{0,p}(R^{n}), 1 < p < ∞:

\begin{aligned}∥η_{r} ∗ g∥_{L^{p}} & = ∥∑_{k=0}^{∞} φ_{k} ∗ φ_{k} ∗ η_{r} ∗ g∥_{L^{p}} \\ & ≤ ∑_{k=0}^{∞} ∥φ_{k} ∗ η_{r}∥_{L^{1}} ∥φ_{k} ∗ g∥_{L^{p}} \\ & ≤ (∑_{k=0}^{∞} ∥φ_{k} ∗ η_{r}∥_{L^{1}}^{p′})^{\frac{1}{p′}} (∑_{k=0}^{∞} ∥φ_{k} ∗ g∥_{L^{p}}^{p})^{\frac{1}{p}} \\ & ≤ ∥η_{r}∥_{B^{·}}{0,p′ \atop 1} ∥g∥_{B_{p}^{0,p}} = ∥η∥_{B^{·}}{0,p′ \atop 1} ∥g∥_{B_{p}^{0,p}} ≤ C ∥g∥_{B_{p}^{0,p}} . \end{aligned}  (53)

Combining (52) and (53) with the boundedness of T_{α} on L^{p} (B_{p}^{0,p}), we obtain

(cr^{α−1} ∫_{σ}^{β} a(w, t′)^{p′}dt′)^{p} cr^{n}(δ − γ) ≤ ∫_{γ}^{δ} ∫_{B(w,cr)} |η_{r} ∗ T_{α}ϕ_{r}f(x, t)|^{p}dxdt
≤ ∫_{0}^{1} ∥η_{r} ∗ T_{α}ϕ_{r}f∥_{L^{p}(dx)}^{p} dt
≤ C ∫_{0}^{1} ∥T_{α}ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p} dt ≤ C ∫_{0}^{1} ∥ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p} dt

= C ∫_{0}^{1} (∑_{k=0}^{∞} ∥φ_{k} ∗ ϕ_{r}f∥_{L^{p}(dx)}^{p}) dt ≤ C ∫_{0}^{1} ∥f∥_{L^{p}(dx)}^{p} dt

= C ∫_{0}^{1} ∫_{R^{n}} |f(x′, t′)|^{p}dx′dt′ = cr^{n} ∫_{σ}^{β} a(w, t′)^{p′}dt′.

(54)

If we rewrite (54) as

(r^{α−1} ∫_{σ}^{β} a(w, t′)^{p′}dt′)^{p−1} ≤ C \frac{r^{1−α}}{δ − γ} ,

and use r = ∫_{σ}^{β} a(w, t′)dt′ = ∫_{γ}^{δ} a(w, t′)dt′, we obtain (47). It remains to show that (47) implies the A_{p,α}^{−} condition (46), and this follows using the argument beginning on page 43 of [GS]. This completes the proof that A_{p,α}^{−} is necessary for the boundedness of T_{α} on L^{p} (B_{p}^{0,p}).

Conversely, to show that the A_{p,α}^{−} condition (46) implies the boundedness of T_{α} on L^{p} (B_{p}^{0,p}), we begin by using the Calderón reproducing formula to reduce matters to a Littlewood-Paley decomposition of T_{α}. Choose ϕ̂_{0} and ϕ̂_{1} nonnegative and infinitely differentiable on R^{n} with supports in \{ξ : |ξ| ≤ 1\} and \{ξ : \frac{1}{2} ≤ |ξ| ≤ 2\} respectively so that

∑_{k=0}^{∞} ϕ̂_{k}(ξ)^{2} = 1, for all ξ ∈ R^{n},

where ϕ̂_{k}(ξ) = ϕ̂_{1}(2^{−k}ξ) for k ≥ 2. Also choose ψ̂_{1} ≥ 0, C^{∞} with support in \{ξ : \frac{1}{4} ≤ |ξ| ≤ 4\} such that ψ̂_{1} = 1 on the support of ϕ̂_{1}. Set ψ̂_{k}(ξ) = ψ̂_{1}(2^{−k}ξ). Letting ϕ_{k} and ψ_{k} denote the operation of convolution in the x-variable (in R^{n} × (0, 1) = {(x, t) : x ∈ R^{n}, 0 < t < 1}) with (ϕ̂_{k})^{∨} and (ψ̂_{k})^{∨}, the inverse Fourier transforms of ϕ̂_{k} and ψ̂_{k} respectively, we have for any f, g ϵ C_{c}^{∞}(R^{n} × (0, 1)),

\begin{aligned}∫_{0}^{1} ∫_{R^{n}} T_{α}f(x, t)g(x, t)dxdt & = ∑_{k=0}^{∞} ∫_{0}^{1} ∫_{R^{n}} T_{α}ϕ_{k}^{2}(x, t)g(x, t)dxdt, \\ & = ∫_{0}^{1} ∫_{R^{n}}(T_{α}ϕ_{0}^{2})f(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}[(1 − ψ_{k})T_{α}ϕ_{k}](ϕ_{k}f)(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}(T_{α}ϕ_{k})(ϕ_{k}f)(x, t)(ψ_{k}g)(x, t)dxdt \\ & = I + II + III. \end{aligned} (55)

Term I in (55) is estimated as in [GS]. Note first that by the argument on page 45 of [GS], the kernel of T_{α}ϕ_{0}^{2} satisfies

|T_{α}ϕ_{0}^{2}(x, t, x′, t′)| ≤ C(1 + |x − x′|)^{−(n+ \frac{1}{2} )}

uniformly in t′ and t. Thus T_{α}ϕ_{0}^{2} is bounded on L^{p} (B_{p}^{0,p}) and

|I| ≤ ∥T_{α}ϕ_{0}^{2}f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} ≤ C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})}  (56)

for f, g ∈ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate term II in (55), we apply Propositions 4 and 5 to the composition (I − ψ_{k})∘ T_{α}ϕ_{k}, with M = 1, to obtain that (I − ψ_{k}) T_{α}ϕ_{k} maps B_{p}^{−μ,p}(dx) into B_{p}^{0,p}(dx) with norm independent of k, t and t′. Thus

\begin{aligned}|II| & ≤ ∑_{k=1}^{∞} ∥(I − ψ_{k})T_{α}ϕ_{k}(ϕ_{k}f)∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C∥ϕ_{k}f∥_{L^{p}(B_{p}^{−μ,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C2^{−kμ}∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & = C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \end{aligned}  (57)

for f, g ϵ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate the main term III in (55), we need the following inequality, uniformly in k:

∫_{0}^{1} ∫_{R^{n}} |T_{α}ϕ_{k}h(x, t)|^{p}dxdt ≤ C ∫_{0}^{1} ∫_{R^{n}} |h(x, t)|^{p}dxdt, k ≥ 1,  (58)

for all h ∈ S(R^{n} × (0, 1)), the subspace of S(R^{n+1}) whose elements are supported in R^{n} × (0, 1). Assuming (58), we have for f, g ϵ C_{c}^{∞}(R^{n} × (0, 1)),

|III| ≤ ∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |T_{α}ϕ_{k}(ϕ_{k}f)(x, t)|^{p})^{\frac{1}{p}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′})^{\frac{1}{p′}} dxdt

≤ \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |T_{α}ϕ_{k}(ϕ_{k}f)(x, t)|^{p}) dxdt\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}

≤ C \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ϕ_{k}f(x, t)|^{p}) dxdt\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}
by (58) applied with h = ϕ_{k}f ∈ S(R^{n} × (0, 1)),
≤ C (∫_{0}^{1} ∥f(·, t)∥_{B_{p}^{0,p}}^{p} dt)^{\frac{1}{p}} (∫_{0}^{1} ∥g(·, t)∥_{B_{p′}^{0,p′}}^{p′} dt)^{\frac{1}{p′}} = C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})}.

(59)

Combining (55), (56), (57) and (59) shows that T_{α} is bounded on L^{p} (B_{p}^{0,p}), and thus it remains only to establish (58).

To estimate the kernel T_{α}ϕ_{k}(x, t, x′, t′) of T_{α}ϕ_{k}, we write, using (42),

|T_{α}ϕ_{k}(x, t, x′, t′)| = χ_{(0,t)}(t′) |∫_{R^{n}} e^{i(x−x′)·ξ}a(x, t′) (Q(x, t′, ξ))^{1−α} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|
= χ_{(0,t)}(t′) |∫_{R^{n}} ((\frac{2^{−2k}−△_{ξ}}{2^{−2k}+|x−x′|^{2}} )^{N} e^{i(x−x′)·ξ}) ×
a(x, t′) (Q(x, t′, ξ))^{1−α} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|

≤ χ_{(0,t)}(t′) (2^{−2k} + |x − x′|^{2})^{−N}2^{nk}2^{−2kN}a(x, t′)2^{k(1−α)}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ}

(60)

since | (−△_{ξ})^{ℓ} ϕ̂_{k}(ξ)| ≤ C_{ℓ}2^{−2kℓ}, 2^{1−m}|ξ| ≤ Q(x, θ, ξ) ≤ 2^{m−1}|ξ| for some fixed m, and ϕ̂_{k} is supported in {ξ : 2^{k−1} ≤ |ξ| ≤ 2^{k+1}}. Denote by M the Hardy-Littlewood maximal operator in the x-variable,

Mh(x, t) = sup_{r>0} r^{−n} ∫_{|x−x′|≤r} |h(x′, t)|dx′.

Then ([St]; p. 63)

∫_{R^{n}} 2^{kn}[1 + (2^{k}|x − x′|^{2})]^{−}^{N} |h(x′, t)| dx′ ≤ C_{N}Mh(x, t),  (61)

for N > \frac{n}{2} . For k ∈ Z_{+} and a(t) ≥ 0 for 0 < t < 1, define T_{a,k}^{α} by

T_{a,k}^{α}g(t) = ∫_{0}^{t} a(t′)2^{k(1−α)}e^{−2 ∫_{t′}^{t} a(θ)dθ}g(t′)dt′, 0 < t < 1,  (62)

for any g integrable on (0, 1). Using (60), (61) and (62) we then have

|T_{α}ϕ_{k}h(x, t)| ≤ C_{N} ∫_{0}^{t} a(x, t′)2^{k(1−α)}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ} ∫_{R^{n}} \frac{2^{kn}}{[1+(2^{k}|x−x′|)^{2}]^{N}} |h(x′, t′)|dx′dt′
≤ C_{N} ∫_{0}^{t} a(x, t′)2^{k(1−α)}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ}Mh(x, t′)dt′
≤ C_{N}T_{a_{x},k}^{α}(Mh)(x, t),

(63)

where a_{x}(t) = a(x, t).

We now claim that

∫_{0}^{1} |T_{a_{x},k}^{α}g(t)|^{p}dt ≤ C ∫_{0}^{1} |g(t)|^{p}dt  (64)

for all sequences of functions {f_{k}}. To see this, let A_{x}(t) = ∫_{0}^{t} a(x, θ)dθ and make the change of variable s = A_{x}(t) = ∫_{0}^{t} a(x, θ)dθ and s′ = A_{x}(t′) in (64) and (62). Then with g̃(s) = g(A_{x}^{−1}(s)) and (T_{a_{x},k}^{α}g)^{∼}(s) = T_{a_{x},k}^{α}g(A_{x}^{−1}(s)), we have

\begin{aligned}(T_{a_{x},k}^{α}g)^{∼}(s) & = ∫_{0}^{A_{x}^{−1}(s)} a(x, t′)2^{k(1−α)}e^{−2^{k}(A_{x}(t)−A_{x}(t′))}g(t′)dt′ \\ & = ∫_{0}^{s} 2^{k(1−α)}e^{−2^{k}(s−s′)}g̃(s′)ds′, \end{aligned}

and so with w_{x}(s) = \frac{d}{ds}A_{x}^{−1}(s) = \frac{1}{a(A_{x}^{−1}(s))} so that dt = w_{x}(s)ds, (64) is equivalent to

\begin{matrix}∫_{0}^{A_{x}(1)} |∫_{0}^{s} 2^{k(1−α)}e^{−2^{k}(s−s′)}g(s′)ds′|^{p} w_{x}(s)ds \\ ≤ C ∫_{0}^{A_{x}(1)} |g(s)|^{p}w_{x}(s)ds \end{matrix}  (65)

for all nonnegative functions g on (0, A_{x}(1)). We now claim that (65) follows from the A_{p,α}^{−} condition (46) and the weighted norm inequality for the Hardy operator ([Mu]), namely

∫_{0}^{A} |∫_{0}^{s} h(s′)ds′|^{p} w(s)ds ≤ C ∫_{0}^{A} |h(s)|^{p}v(s)ds,  (66)

for all h ≥ 0 if and only if

sup_{0<b<A} (∫_{b}^{A} w(s)ds) (∫_{0}^{b} v(s)^{1−p′}ds)^{p−1} < ∞.  (67)

To see that (65) follows, replace g(s′) by e^{−2^{k}s′}g(s′) in (65) to obtain (66) with A = A_{x}(1), w(s) = 2^{pk(1−α)}e^{−p2^{k}s}w_{x}(s) and v = e^{−p2^{k}s}w_{x}(s). Thus we must show that (67) holds for these weights, i.e.,

sup_{0<b<A_{x}(1)} (∫_{b}^{A_{x}(1)} 2^{pk(1−α)}e^{−p2^{k}s}w_{x}(s)ds) (∫_{0}^{b} e^{p′2^{k}s}w_{x}(s)^{1−p′}ds)^{p−1} < ∞.  (68)

We now rewrite the A_{p,α}^{−} condition (46) using the change of variable s = A_{x}(t) = ∫_{0}^{t} a(x, θ)dθ to get

δ^{pα} [\frac{1}{δ} ∫_{b}^{b+δ} w_{x}(s)ds] [\frac{1}{δ} ∫_{b−δ}^{b} w_{x}(s)^{1−p′}ds]^{p−1} ≤ C,  (69)

for 0 < δ < b < A_{x}(1) − δ. Set r = 2^{−k}. If 1 < p ≤ 2, then we estimate the product in (68) by

r^{p(α−1)} (∑_{i=0}^{∞} ∫_{b+ir}^{b+(i+1)r} e^{−ps/r}w_{x}(s)ds) (∑_{j=0}^{∞} ∫_{b−(j+1)r}^{b−jr} e^{p′s/r}w_{x}(s)^{1−p′}ds)^{p−1}
≤ r^{p(α−1)} ∑_{i=0}^{∞} (∫_{b+ir}^{b+(i+1)r} e^{−ps/r}w_{x}(s)ds) ∑_{j=0}^{∞} (∫_{b−(j+1)r}^{b−jr} e^{p′s/r}w_{x}(s)^{1−p′}ds)^{p−1}
≤ Cr^{p(α−1)} ∑_{i,j≥0} e^{−p(i+j)} (∫_{b+ir}^{b+(i+1)r} w_{x}(s)ds) (∫_{b−(j+1)r}^{b−jr} w_{x}(s)^{1−p′}ds)^{p−1}
≤ Cr^{p(α−1)} ∑_{i,j≥0} e^{−p(i+j)} [(i + j) r]^{p(1−α)} , by (69) with δ = (i + j) r,
≤ C ∑_{i,j≥0} e^{−p(i+j)} (i + j)^{p(1−α)} ≤ C.

On the other hand, if 2 ≤ p < ∞, then we raise the product in (68) to the power p′ − 1 and estimate

r^{p′(α−1)} (∑_{i=0}^{∞} ∫_{b+ir}^{b+(i+1)r} e^{−ps/r}w_{x}(s)ds)^{p′−1} (∑_{j=0}^{∞} ∫_{b−(j+1)r}^{b−jr} e^{p′s/r}w_{x}(s)^{1−p′}ds)

by a constant C as above. This establishes (64) with C independent of k.

We thus have

∫_{0}^{1} ∫_{R^{n}} |T_{α}ϕ_{k}h(x, t)|^{p}dxdt ≤ C ∫_{0}^{1} ∫_{R^{n}} |T_{a_{x},k−m}(Mh)(x, t)|^{p}dtdx, by (63),
≤ C ∫_{0}^{1} ∫_{R^{n}} |Mh(x, t)|^{p}dtdx, by (64) ,
= C ∫_{0}^{1} ∫_{R^{n}} |Mh(x, t)|^{p}dxdt ≤ C ∫_{0}^{1} ∫_{R^{n}} |h(x, t)|^{p}dxdt

since M is bounded on L^{p}. This establishes (58) and completes the proof of Theorem 28.

We now turn to the local estimates for T_{α} on B_{p}^{s,p}(R^{n+1}) that we need in [GSI]. In this setting, both Γ and T⃗ have been straightened out and the flow for T⃗ =\frac{∂}{∂t} through (x, 0) in Γ is given by \overrightarrow{γ} ((x, 0) , t) = (x, t). Thus the A_{p,α}^{∓} condition becomes

Definition 29 The function a satisfies the A_{p,α}^{∓} condition at the fibre F_{(y,0)}, (y, 0) ∈ Γ, if there are constants r > 0, R^{−} < 0 < R^{+}, such that a (x, R^{−}) ≠ 0 and a (x, R^{+}) ≠ 0 for x ∈ R^{n}, |x − y| < r and both of the following conditions hold:

[\frac{1}{∫_{σ}^{β} a (x, t) dt} ∫_{σ}^{β} a (x, t)^{p′} dt]^{p−1} ≤ C \frac{1}{γ − β} [∫_{β}^{γ} a (x, t) dt]^{1−pα}  (70)

for all x ∈ Γ, |x − y| < r and all 0 < σ < β < γ < R^{+} with ∫_{σ}^{β} a (x, t) dt = ∫_{β}^{γ} a (x, t) dt > 0, and also

[\frac{1}{∫_{β}^{γ} |a (x, t)| dt} ∫_{β}^{γ} |a (x, t)|^{p′} dt]^{p−1} ≤ C \frac{1}{β − σ} [∫_{σ}^{β} |a (x, t)| dt]^{1−pα}  (71)

for all x ∈ Γ, |x−y| < r and all R^{−} < σ < β < γ < 0 with ∫_{σ}^{β} |a (x, t)| dt = ∫_{β}^{γ} |a (x, t)| dt > 0.

We shall need to know that for functions f (x, t) with frequencies in the cone {|ξ| > |τ|}, the mixed norm ∥·∥_{L^{p}(B_{p}^{0,p})} is equivalent to the Besov norm ∥·∥_{B_{p}^{0,p}}.

Lemma 30 Suppose supp f̂(ξ, τ) ⊂ {(ξ, τ) ∈ R^{n} × R : |ξ| > |τ|}. Then

∥f∥_{B_{p}^{0,p}} ≅ ∥f∥_{L^{p}(B_{p}^{0,p})} .

Proof. Let {Φ_{k}}_{k=0}^{∞} (respectively {ϕ_{k}}_{k=0}^{∞}) satisfy the usual conditions for the Calderon reproducing formula in R^{n+1} (respectively R^{n}). Then by the condition on the support of f̂, we have Φ_{k}f ≅ ϕ_{k}f, and so

\begin{aligned}∥f∥_{B_{p}^{0,p}}^{p} & = ∑_{k=0}^{∞} ∥Φ_{k}f∥_{L^{p}(R^{n+1})}^{p} ≅ ∑_{k=0}^{∞} ∥ϕ_{k}f∥_{L^{p}(R^{n+1})}^{p} \\ & = ∑_{k=0}^{∞} ∫_{R} ∥ϕ_{k}f (·, t)∥_{L^{p}(R^{n})}^{p} dt = ∫_{R} ∑_{k=0}^{∞} ∥ϕ_{k}f (·, t)∥_{L^{p}(R^{n})}^{p} dt = ∥f∥_{L^{p}(B_{p}^{0,p})}^{p} . \end{aligned}

Now we can state our characterization of the local boundedness of T_{α} on B_{p}^{s,p} in Case II.

Theorem 31 Let a(x, t) be a C^{λ+2} function with a(x, t) ≥ 0 when t ≥ 0 and a(x, t) ≤ 0 when t ≤ 0, let y ∈ R^{n} and suppose that a ∈ A ¯+ ; p,α at the fibre F_{(y,0)}. Let U = {(x, y) : |x − y| < r, R^{−} < t < R^{+}} where r, R^{−} and R^{+} are as in (70) and (70). Let φ, ψ be C^{∞} functions supported in U, and suppose that P denotes a multiplier with symbol supported in the cone {|ξ| > |τ|}. Then if M_{φ}, M_{ψ} are the operators of multiplication by φ and ψ respectively, the operator P M_{φ}T_{α}M_{ψ}P is bounded on B_{p}^{s,p} for −\frac{1}{2} < s < λ + 2. Conversely, if φ = ψ = 1 on U, and M_{φ}T_{α}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then a ∈ A ¯+ ; p,α at the fibre F_{(y,0)}. In particular, P M_{φ}T_{\frac{1}{p(λ+3)}} M_{ψ}P is bounded on B_{p}^{s,p} for

−\frac{1}{2} < s < λ + 2.

Proof. Choose ρ ∈ C_{c}^{∞}(R^{n}) with supp ρ ⊂ {x : |x−y| < r} and ρ(x) = 1 on the support of ψ. Then ρ(x)a(x, t) satisfies A ¯+ ; p,α since for any fixed x, the inequality in A ¯+ ; p,α is unaffected by multiplying a(x, t) by a positive constant. Thus the operator T_{α}^{∗}, obtained from T_{α} by replacing a with ρa, is bounded on L^{p}(B_{p}^{0,p}) by Theorem 28 together with the same result scaled to R^{n} × (−R, 0). Now if supp φ, supp ψ ⊂ U, then P M_{φ}T_{α}M_{ψ}P = P M_{φ}T_{α}^{∗}M_{ψ}P is bounded on L^{p}(B_{p}^{0,p}), and so also on B_{p}^{0,p} by Lemma 30. By Lemma 27, we now conclude that P M_{φ}T_{α}M_{ψ}P is bounded on B_{p}^{s,p} for −\frac{1}{2} < s < λ + 2.

Conversely, if φ = ψ = 1 on U and M_{φ}T_{α}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then by Lemma 27, it is bounded on B_{p}^{0,p} . So suppose T̃_{α} = M_{φ}T_{α}M_{ψ} is bounded on B_{p}^{0,p}. The proof of necessity in Theorem 28 carries over here with just a few changes, as follows. With notation as in the proof of Theorem 28, we have from (50) and (51) that

Re(T̃_{α}ϕ_{r}f)(x, t) = Re(M_{φ}T_{α}ϕ_{r}f)(x, t) ≥ cr^{α−1} ∫_{α}^{β} a(w, t′)^{p′}dt′  (72)

for |x−w| < cr, t ∈ (γ, δ). Now choose η ∈ C_{c}^{∞}(R^{n+1}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n+1}} η(x)dx = 1 (note the use of R^{n+1} in place of R^{n} here). With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows easily from (72) that

Re(η_{r} ∗ T̃_{α}ϕ_{r}f)(x, t) ≥ cr^{α−1} ∫_{α}^{β} a(w, t′)^{p′}dt′  (73)

for |x − w| < cr, t ∈ (γ, δ), with a perhaps smaller constant c - note that r ≤ (δ − γ)/3 if, as we may assume, a(x, t) ≤ 1/3. If we use the inequality ∥η_{r} ∗ g∥_{L^{p}} ≤ C ∥g∥_{B_{p}^{0,p}}, valid for

any g ∈ B_{p}^{0,p}(R^{n+1}), 1 < p < ∞, (this is (53) with R^{n+1} in place of R^{n}), the A ¯+ ; p,α condition in the open set \overrightarrow{γ}(U_{R}) can be derived as in the proof of Theorem 28.

Finally, a ∈ C^{λ+2} implies a ∈ A ¯+ ; p, \frac{1}{p(λ+3)} by the following elementary computation using

Lemma 13:

[_{\frac{1}{∫_{σ}^{β} a(x,t)dt}} ∫_{σ}^{β} a(x, t)^{p′}dt]^{p−1} ≤ sup_{σ≤t≤γ} a (x, t)
≤ C (\frac{1}{γ−σ} ∫_{σ}^{γ} a (x, t) dt)^{1− \frac{1}{λ+3}} , by (17),
≤ C \frac{1}{γ−β} [∫_{β}^{γ} a(x, t)dt]^{1−pα} ,

for 0 < σ < β < γ since ∫_{σ}^{γ} a (x, t) dt = 2 ∫_{β}^{γ} a(x, t)dt, \frac{1}{λ+3} = pα and γ − β ≤ γ − σ ≤ 1. Similarly for A_{p,α}^{=} and A_{p,α}^{‡}. Simple examples show this is sharp. This of course shows that P M_{φ}T_{\frac{1}{p(λ+3)}} M_{ψ}P is bounded on B_{p}^{s,p} for −\frac{1}{2} < s < λ + 2 as required.

2.2 Boundedness of K_{ν}

In [GSI] we reduced matters regarding the gain from g in the oblique derivative problem to the boundedness of the operator

Kf(x, t) = ∫_{R^{n}} e^{ix·ξ} ∫_{0}^{t} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}f^{∼}(ξ, t′)dt′dξ

from a Besov space into L^{p}. It is known that if a is C^{∞} and has type k, then K gains \frac{1}{k+1}

derivatives, i.e. K ∈ O^{\frac{1}{k+1}} . In this subsection, we extend this result to rough a by replacing the type condition on a with the condition

\begin{aligned}(T_{ν}) & β − α ≤ C (∫_{α}^{β} |a(x, t)| dt)^{ν} \\ & for α < β, and x ∈ R^{n}. \end{aligned}  (74)

We begin by proving the analogue of Theorem 28 for the operators K_{ν} = KQ^{ν} given by

K_{ν}f(x, t) = ∫_{R^{n}} e^{ix·ξ} ∫_{0}^{t} Q(x, t′, ξ)^{ν}e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}f^{∼}(ξ, t′)dt′dξ.  (75)

Theorem 32 Suppose 1 < p < ∞ and a(x, t) is a bounded nonnegative function on R^{n} × (0, 1). Then K_{ν} is bounded on L_{(0,1)}^{p} (B_{p}^{0,p} (R^{n})) if and only if a satisfies the (T_{ν}) condition.

Proof. As in the proof of Theorem 28, fix w in R^{n} and let r = ∫_{α}^{β} a(w, θ)dθ. Then with ϕ_{r} as before, the real part of the kernel of K_{ν}ϕ_{r} satisfies the following estimate for |x − w|, |x′ − w| < cr, and t, t′ ∈ (α, β):

\begin{aligned}Re(K_{ν}ϕ_{r})(x, t, x′t′) & = ∫_{R^{n}} cos((x − x′) · ξ) (Q(x, t′, ξ))^{ν} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂(rξ)dξ \\ & ≥ ∫_{\frac{1}{2r} ≤|ξ|≤ \frac{4}{r}} cr^{−ν}e^{− ∫_{t′}^{t} a(x,θ)Cr^{−1}dθ}dξ \\ & ≥ cr^{−ν−n}. \end{aligned} (76)

Now define f by

f(x′, t′) = χ_{B(w,cr)}(x′)χ_{(α,β)}(t′).  (77)

Combining (76) and (77) we obtain for |x − w| < cr, t ∈ (α, β),

\begin{aligned}Re(K_{ν}ϕ_{r}f)(x, t) & ≥ c ∫_{B(w,cr)} r^{−ν−n}(β − α) \\ & ≥ cr^{−ν}(β − α). \end{aligned}  (78)

As in the previous subsection, choose η ∈ C_{c}^{∞}(R^{n}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n}} η(x)dx = 1. With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows immediately from (78) that

Re(η_{r} ∗ K_{ν}ϕ_{r}f)(x, t) ≥ cr^{−ν}(β − α).  (79)

for |x − w| < cr, t ∈ (α, β), with a perhaps smaller constant c - note that r ≤ (β − α)/3 if, as we may assume, a(x, t) ≤ 1/3. From (77), (79), (53) and the boundedness of K_{ν}ϕ_{r} on L^{p} (B_{p}^{0,p}), we now obtain

(cr^{−ν}(β − α))^{p} cr^{n}(β − α) ≤ ∫_{B(w,cr)} ∫_{α}^{β} |η_{r} ∗ K_{ν}ϕ_{r}f(x, t)|^{p}dxdt
≤ ∫_{0}^{1} ∥η_{r} ∗ K_{ν}ϕ_{r}f∥_{L^{p}(dx)}^{p} dt
≤ C ∫_{0}^{1} ∥K_{ν}ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p} dt ≤ C ∫_{0}^{1} ∥ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p} dt
= C ∫_{0}^{1} (∑_{k=0}^{∞} ∥φ_{k} ∗ ϕ_{r}f∥_{L^{p}(dx)}^{p}) dt ≤ C ∫_{0}^{1} ∥f∥_{L^{p}(dx)}^{p} dt
= C ∫_{0}^{1} ∫_{R^{n}} |f(x′, t′)|^{p}dx′dt′ = Cr^{n}(β − α),

which yields the T_{ν} condition since r = ∫_{α}^{β} a(w, t′)dt′.

To show that the T_{ν} condition implies the boundedness of K_{ν} on L^{p} (B_{p}^{0,p}), we proceed as before by using the Calderón reproducing formula on a Littlewood- Paley decomposition of K_{ν}. Let ϕ_{k} and ψ_{k} be as in subsection 5.1. We have for any f, g ∈ C_{c}^{∞}(R^{n} × (0, 1)),

\begin{aligned}∫_{0}^{1} ∫_{R^{n}} K_{ν}f(x, t)g(x, t)dxdt & = ∑_{k=0}^{∞} ∫_{0}^{1} ∫_{R^{n}} K_{ν}ϕ_{k}^{2}(x, t)g(x, t)dxdt, \\ & = ∫_{0}^{1} ∫_{R^{n}}(K_{ν}ϕ_{0}^{2})f(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}[(1 − ψ_{k})K_{ν}ϕ_{k}](ϕ_{k}f)(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}(K_{ν}ϕ_{k})(ϕ_{k}f)(x, t)(ψ_{k}g)(x, t)dxdt \\ & = I + II + III. \end{aligned} (80)

Term I in (80) is handled just as in the previous subsection, obtaining

|I| ≤ C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})}  (81)

for f, g ∈ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate term II in (80), we apply Propositions 4 and 5 to the composition (I − ψ_{k})∘ K_{ν}ϕ_{k}, with M = 1, to obtain that (I − ψ_{k}) K_{ν}ϕ_{k} maps B_{p}^{−μ,p}(dx) into B_{p}^{0,p}(dx) with norm independent of k, t and t′. Thus

\begin{aligned}|II| & ≤ ∑_{k=1}^{∞} ∥ (I − ψ_{k}) K_{ν}ϕ_{k}(ϕ_{k}f)∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C∥ϕ_{k}f∥_{L^{p}(B_{p}^{−μ,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C2^{−kμ}∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & = C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \end{aligned}  (82)

for f, g ϵ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate the main term III in (80), we need the following inequality, uniformly in k:

∫_{0}^{1} ∫_{R^{n}} |K_{ν}ϕ_{k}h(x, t)|^{p}dxdt ≤ C ∫_{0}^{1} ∫_{R^{n}} |h(x, t)|^{p}dxdt, k ≥ 1,  (83)

for all h ∈ S(R^{n} × (0, 1)), the subspace of S(R^{n+1}) whose elements are supported in R^{n} × (0, 1). Assuming (83), we have for f, g ϵ C_{c}^{∞}(R^{n} × (0, 1)),

|III| ≤ ∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |K_{ν}ϕ_{k}(ϕ_{k}f)(x, t)|^{p})^{\frac{1}{p}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′})^{\frac{1}{p′}} dxdt

≤ \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |K_{ν}ϕ_{k}(ϕ_{k}f)(x, t)|^{p}) dxdt\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}

≤ C \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ϕ_{k}f(x, t)|^{p}) dxdt\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}
by (83) applied with h = ϕ_{k}f ∈ S(R^{n} × (0, 1)),
≤ C (∫_{0}^{1} ∥f(·, t)∥_{B_{p}^{0,p}}^{p} dt)^{\frac{1}{p}} (∫_{0}^{1} ∥g(·, t)∥_{B_{p′}^{0,p′}}^{p′} dt)^{\frac{1}{p′}} = C∥f∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})}.

(84)

Combining (80), (81), (82) and (84) shows that K_{ν} is bounded on L^{p} (B_{p}^{0,p}), and thus it remains only to establish (83).

To estimate the kernel K_{ν}ϕ_{k}(x, t, x′, t′) of K_{ν}ϕ_{k}, we write, using (75),

|K_{ν}ϕ_{k}(x, t, x′, t′)| = χ_{(0,t)}(t′) |∫_{R^{n}} e^{i(x−x′)·ξ} (Q(x, t′, ξ))^{ν} e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|
= χ_{(0,t)}(t′) |∫_{R^{n}} ((\frac{2^{−2k}−△_{ξ}}{2^{−2k}+|x−x′|^{2}} )^{N} e^{i(x−x′)·ξ}) (Q(x, t′, ξ))^{ν} ×
e^{− ∫_{t′}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|
≤ χ_{(0,t)}(t′) (2^{−2k} + |x − x′|^{2})^{−N}2^{nk}2^{−2kN}2^{kν}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ}

(85)

since | (−△_{ξ})^{ℓ} ϕ̂_{k}(ξ)| ≤ C_{ℓ}2^{−2kℓ}, 2^{1−m}|ξ| ≤ Q(x, θ, ξ) ≤ 2^{m−1}|ξ| for some fixed m, and ϕ̂_{k} is supported in {ξ : 2^{k−1} ≤ |ξ| ≤ 2^{k+1}}. Denote by M the Hardy-Littlewood maximal operator in the x-variable, so that

∫_{R^{n}} 2^{kn}[1 + (2^{k}|x − x′|^{2})]^{−}^{N} |h(x′, t′)| dx′ ≤ C_{N}Mh(x, t′),  (86)

for N > \frac{n}{2} . Since a satisfies the T_{ν} condition (74), we have

2^{kν}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ} ≤ 2^{kν}e^{−2^{k−m}C^{−1/ν}|t−t′|^{1/ν}}.  (87)

Now the right side of (87) is an even integrable function of t − t′ satisfying

∫_{0}^{1} 2^{kν}e^{−2^{k−m}C^{−1/ν}|t−t′|^{1/ν}}dt (or dt′) ≤ C_{m,ν}

uniformly in k, and so we conclude that

∫_{0}^{1} 2^{kν}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ}Mh(x, t′)dt′ ≤ C_{m,ν}M̃ (Mh) (x, t),  (88)

where M̃ denotes the Hardy-Littlewood maximal operator in the t-variable. Combining (85), (86) and (88) yields

\begin{aligned}|K_{ν}ϕ_{k}h(x, t)| & ≤ C_{N} ∫_{0}^{t} 2^{kν}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ} ∫_{R^{n}} \frac{2^{kn}}{[1+(2^{k}|x−x′|)^{2}]^{N}} |h(x′, t′)|dx′dt′ \\ & ≤ C_{N} ∫_{0}^{t} 2^{kν}e^{−2^{k−m} ∫_{t′}^{t} a(x,θ)dθ}Mh(x, t′)dt′ \\ & ≤ C_{N} M̃(Mh)(x, t). \end{aligned}  (89)

We thus have

\begin{aligned}∫_{0}^{1} ∫_{R^{n}} |K_{ν}ϕ_{k}h(x, t)|^{p}dxdt & ≤ C ∫_{0}^{1} ∫_{R^{n}} |M̃(Mh)(x, t)|^{p}dtdx, by (89), \\ & ≤ C ∫_{0}^{1} ∫_{R^{n}} |Mh(x, t)|^{p}dtdx ≤ C ∫_{0}^{1} ∫_{R^{n}} |h(x, t)|^{p}dxdt \end{aligned}

since both M̃ and M are bounded on L^{p}. This establishes (83) and completes the proof of Theorem 32.

In order to state a local version of this result, we recast the definition of the T_{ν} condition in terms of open sets. Let U_{R} = \{(x, t) ϵR^{n} × R : |x| , |t| < R\}, and note that \overrightarrow{γ} ((x, s) , t) = (x, s + t) is the flow for T⃗ =\frac{∂}{∂t} through (x, s).

Definition 33 The function a satisfies the T_{ν} condition in the open set U_{R} if

\begin{matrix}β − α ≤ C (∫_{α}^{β} |a (x, t)| dt)^{ν} \\ for all − R < α < β < R, and |x| < R. \end{matrix}

Now we can state our characterization of the local boundedness of K_{ν} on B_{p}^{s,p}.

Theorem 34 Let a(x, t) be a C^{λ+2} function, and suppose that a ∈ T_{ν} in the open set U_{R}. Let φ, ψ be C^{∞} functions supported in U_{R}, and suppose that P denotes a multiplier with symbol supported in the cone {|ξ| > |τ|}. Then if M_{φ}, M_{ψ} are the operators of multiplication by φ and ψ respectively, the operator P M_{φ}K_{ν}M_{ψ}P is bounded B_{p}^{s,p} on for −\frac{1}{2} < s < λ + 2. Conversely, if φ = ψ = 1 on U_{R}, and M_{φ}K_{ν}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then a ∈ T_{ν} in the open set U_{R′} for some R′ > 0.

Proof. The operator K_{ν} is bounded on L^{p}(B_{p}^{0,p}) by Theorem 32 together with the same result scaled to R^{n} × (−R, 0). Now if supp φ, supp ψ ⊂ U_{R}, then P M_{φ}K_{ν}M_{ψ}P is bounded on L^{p}(B_{p}^{0,p}), and so also on B_{p}^{0,p} by Lemma 30. By Lemma 27, we now conclude that P M_{φ}K_{ν}M_{ψ}P is bounded on B_{p}^{s,p} for −\frac{1}{2} < s < λ + 2.

Conversely, if φ = ψ = 1 on U_{1} and M_{φ}K_{ν}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then by Lemma 27, it is bounded on B_{p}^{0,p} . So suppose K̃_{ν} = M_{φ}K_{ν}M_{ψ} is bounded on B_{p}^{0,p}. The proof of necessity in Theorem 32 carries over here with just a few changes, as follows. With notation as in the proof of Theorem 32, we have from (78) that

Re(K̃_{ν}ϕ_{r}f)(x, t) = Re(M_{φ}K_{ν}ϕ_{r}f)(x, t) ≥ cr^{−ν}(β − α)  (90)

for |x−w| < cr, t ∈ (α, β). Now choose η ∈ C_{c}^{∞}(R^{n+1}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n+1}} η(x)dx = 1 (note the use of R^{n+1} in place of R^{n} here). With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows easily from (90) that

Re(η_{r} ∗ K̃_{ν}ϕ_{r}f)(x, t) ≥ cr^{−ν} ∫_{α}^{β} a(w, t′)^{p′}dt′

for |x − w| < cr, t ∈ (α, β), with a perhaps smaller constant c - note that r ≤ (β − α)/3 if, as we may assume, a(x, t) ≤ 1/3. If we use the inequality ∥η_{r} ∗ g∥_{L^{p}} ≤ C ∥g∥_{B_{p}^{0,p}}, valid for any g ∈ B_{p}^{0,p}(R^{n+1}), 1 < p < ∞, (this is (53) with R^{n+1} in place of R^{n}), the T_{ν} condition in the open set U_{R′} can be derived as in the proof of Theorem 32.

2.3 Boundedness of K

In [GSI], we reduced matters regarding the gain from h in the oblique derivative problem, to the boundedness of the operator

Kf(x, t) = ∫_{R^{n}} e^{ix·ξ}e^{− ∫_{0}^{t} a(x,θ)Q(x,θ,ξ)dθ}f̂(ξ) dξ

from a Besov space into B_{p}^{0,p}. In this subsection, we show that if a has type k, then K gains \frac{1}{p(k+1)} derivatives from h in the L^{p} scale of smoothness spaces (note that since a vanishes at the origin, k ≥ 1; for k = 0, this would agree with the classical result for the Poisson integral in the half space). More generally, we extend this result to fractional γ replacing the type condition on a with a type conditon at the origin,

\begin{aligned}(P_{γ}) & |β| ≤ C |∫_{0}^{β} a(x, t)dt|^{γ} \\ & for all β ∈ R, x ∈ R^{n}. \end{aligned}  (91)

We begin by proving the analogue of Theorems 28 and 32 for the operators K_{γ} = KQ^{γ} given by

K_{γ}f(x, t) = ∫_{R^{n}} e^{ix·ξ}Q(x, t′, ξ)^{γ}e^{− ∫_{0}^{t} a(x,θ)Q(x,θ,ξ)dθ}f̂(ξ) dξ.  (92)

Theorem 35 Suppose 1 < p < ∞ and a(x, t) is a bounded nonnegative function on R^{n} × (0, 1). Then K_{γ} is bounded from B_{p}^{0,p} to L_{(0,1)}^{p} (B_{p}^{0,p} (R^{n})) if and only if a satisfies the P_{pγ} condition.

Proof. As in the proof of Theorem 32, fix w in R^{n} and let r = ∫_{α}^{β} a(w, θ)dθ. Then with ϕ_{r} as before, the real part of the kernel of K_{γ}ϕ_{r} satisfies the following estimate for |x − w|, |x′ − w| < cr, and t ∈ (0, β):

\begin{aligned}Re(K_{γ}ϕ_{r})(x, t, x′) & = ∫_{R^{n}} cos((x − x′) · ξ) (Q(x, t′, ξ))^{γ} e^{− ∫_{0}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂(rξ)dξ \\ & ≥ ∫_{\frac{1}{2r} ≤|ξ|≤ \frac{4}{r}} cr^{−γ}e^{− ∫_{0}^{t} a(x,θ)Cr^{−1}dθ}dξ \\ & ≥ cr^{−γ−n} \end{aligned} (93)

Now define f by

f(x′) = χ_{B(w,cr)}(x′).  (94)

Combining (93) and (94) we obtain for |x − w| < cr, t ∈ (0, β),

\begin{aligned}Re(K_{γ}ϕ_{r}f)(x, t) & ≥ c ∫_{B(w,cr)} r^{−γ−n} \\ & ≥ cr^{−γ}. \end{aligned}  (95)

As in the previous subsection, choose η ∈ C_{c}^{∞}(R^{n}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n}} η(x)dx = 1. With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows immediately from (95) that

Re(η_{r} ∗ K_{γ}ϕ_{r}f)(x, t) ≥ cr^{−γ}.  (96)

for |x − w| < cr, t ∈ (0, β), with a perhaps smaller constant c - note that r ≤ β/3 if, as we may assume, a(x, t) ≤ 1/3. From (94), (96), (53) and the boundedness of K_{γ}ϕ_{r} from B_{p}^{0,p} to L^{p} (B_{p}^{0,p}), we now obtain

(cr^{−γ})^{p} cr^{n}β ≤ ∫_{B(w,cr)} ∫_{0}^{β} |η_{r} ∗ K_{γ}ϕ_{r}f(x, t)|^{p}dxdt
≤ ∫_{0}^{1} ∥η_{r} ∗ K_{γ}ϕ_{r}f∥_{L^{p}(dx)}^{p} dt ≤ C ∫_{0}^{1} ∥K_{γ}ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p} dt ≤ C ∥ϕ_{r}f∥_{B_{p}^{0,p}(dx)}^{p}
= C ∑_{k=0}^{∞} ∥φ_{k} ∗ ϕ_{r}f∥_{L^{p}(dx)}^{p} ≤ C ∥f∥_{L^{p}(dx)}^{p}
= C ∫_{R^{n}} |f(x′, t′)|^{p}dx′ = Cr^{n},

which yields the P_{pγ} condition since r = ∫_{0}^{β} a(w, t′)dt′.

To show that the P_{pγ} condition implies the boundedness of K_{γ} from B_{p}^{0,p} to L^{p} (B_{p}^{0,p}), we proceed as before by using the Calderón reproducing formula on a Littlewood- Paley decomposition of K_{γ}. Let ϕ_{k} and ψ_{k} be as in subsection 5.1. We have for any f ∈ C_{c}^{∞}(R^{n}) and g ∈ C_{c}^{∞}(R^{n} × (0, 1)),

\begin{aligned}∫_{0}^{1} ∫_{R^{n}} K_{γ}f(x, t)g(x, t)dxdt & = ∑_{k=0}^{∞} ∫_{0}^{1} ∫_{R^{n}} K_{γ}ϕ_{k}^{2}f(x, t)g(x, t)dxdt, \\ & = ∫_{0}^{1} ∫_{R^{n}}(K_{γ}ϕ_{0}^{2})f(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}[(1 − ψ_{k})K_{γ}ϕ_{k}](ϕ_{k}f)(x, t)g(x, t)dxdt \\ & + ∑_{k=1}^{∞} ∫_{0}^{1} ∫_{R^{n}}(K_{γ}ϕ_{k})(ϕ_{k}f)(x, t)(ψ_{k}g)(x, t)dxdt \\ & = I + II + III. \end{aligned} (97)

To estimate term I in (97), we proceed as in the previous section to obtain

|I| ≤ C∥f∥_{B_{p}^{0,p}}∥g∥_{L^{p′}(B_{p′}^{0,p′})}  (98)

for f ∈ C_{c}^{∞}(R^{n}) and g ∈ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate term II in (97), we apply Propositions 4 and 5 to the composition (I − ψ_{k})∘ K_{γ}ϕ_{k}, with M = 1, to obtain that (I − ψ_{k}) K_{γ}ϕ_{k} maps B_{p}^{−μ,p}(dx) into B_{p}^{0,p}(dx) with norm independent of k and t. Thus

\begin{aligned}|II| & ≤ ∑_{k=1}^{∞} ∥ (I − ψ_{k}) K_{γ}ϕ_{k}(ϕ_{k}f)∥_{L^{p}(B_{p}^{0,p})}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C∥ϕ_{k}f∥_{B_{p}^{−μ,p}}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & ≤ ∑_{k=1}^{∞} C2^{−kμ}∥f∥_{B_{p}^{0,p}}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \\ & = C∥f∥_{B_{p}^{0,p}}∥g∥_{L^{p′}(B_{p′}^{0,p′})} \end{aligned}  (99)

for f ∈ C_{c}^{∞}(R^{n}) and g ∈ C_{c}^{∞}(R^{n} × (0, 1)).

To estimate the main term III in (97), we need the following inequality, uniformly in k:

∫_{0}^{1} ∫_{R^{n}} |K_{γ}ϕ_{k}h(x, t)|^{p}dxdt ≤ C ∫_{R^{n}} |h(x)|^{p}dx, k ≥ 1,  (100)

for all h ∈ S(R^{n}). Assuming (100), we have for f ∈ C_{c}^{∞}(R^{n}) and g ∈ C_{c}^{∞}(R^{n} × (0, 1)),

|III| ≤ ∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |K_{γ}ϕ_{k}(ϕ_{k}f)(x, t)|^{p})^{\frac{1}{p}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′})^{\frac{1}{p′}} dxdt

≤ \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |K_{γ}ϕ_{k}(ϕ_{k}f)(x, t)|^{p}) dxdt\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}

≤ C \{∫_{R^{n}} (∑_{k=1}^{∞} |ϕ_{k}f(x)|^{p}) dx\}^{\frac{1}{p}} \{∫_{0}^{1} ∫_{R^{n}} (∑_{k=1}^{∞} |ψ_{k}g(x, t)|^{p′}) dxdt\}^{\frac{1}{p′}}
by (100) applied with h = ϕ_{k}f ∈ S(R^{n}),
≤ C ∥f∥_{B_{p}^{0,p}} (∫_{0}^{1} ∥g(·, t)∥_{B_{p′}^{0,p′}}^{p′} dt)^{\frac{1}{p′}} = C∥f∥_{B_{p}^{0,p}}∥g∥_{L^{p′}(B_{p′}^{0,p′})}.

(101)

Combining (97), (98), (99) and (101) shows that K_{γ} is bounded from B_{p}^{0,p} to L^{p} (B_{p}^{0,p}), and thus it remains only to establish (100).

To estimate the kernel K_{γ}ϕ_{k}(x, t, x′) of K_{γ}ϕ_{k}, we write, using (92),

|K_{γ}ϕ_{k}(x, t, x′)| = |∫_{R^{n}} e^{i(x−x′)·ξ} (Q(x, t, ξ))^{γ} e^{− ∫_{0}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|

= |∫_{R^{n}} ((\frac{2^{−2k}−△_{ξ}}{2^{−2k}+|x−x′|^{2}} )^{N} e^{i(x−x′)·ξ}) (Q(x, t, ξ))^{γ} e^{− ∫_{0}^{t} a(x,θ)Q(x,θ,ξ)dθ}ϕ̂_{k}(ξ)dξ|

≤ (2^{−2k} + |x − x′|^{2})^{−N}2^{nk}2^{−2kN}2^{kγ}e^{−2^{k−m} ∫_{0}^{t} a(x,θ)dθ}

(102)

since | (−△_{ξ})^{ℓ} ϕ̂_{k}(ξ)| ≤ C_{ℓ}2^{−2kℓ}, 2^{1−m}|ξ| ≤ Q(x, t, ξ) ≤ 2^{m−1}|ξ| for some fixed m, and ϕ̂_{k} is supported in {ξ : 2^{k−1} ≤ |ξ| ≤ 2^{k+1}}. As in subsection 5.1, denote by M the Hardy-Littlewood maximal operator in the x-variable, so that

∫_{R^{n}} 2^{kn}[1 + (2^{k}|x − x′|^{2})]^{−}^{N} |h(x′)| dx′ ≤ C_{N}Mh(x),  (103)

for N > \frac{n}{2} . Since a satisfies the P_{pγ} condition (see (91) with γ replaced by pγ), we have

2^{kγ}e^{−2^{k−m} ∫_{0}^{t} a(x,θ)dθ} ≤ 2^{kγ}e^{−2^{k−m}ct^{\frac{1}{pγ}}} .  (104)

Combining (102), (103) and (104) yields

\begin{aligned}|K_{γ}ϕ_{k}h(x, t)| & ≤ C_{N}2^{kγ}e^{−2^{k−m}ct^{\frac{1}{pγ}}} ∫_{R^{n}} \frac{2^{kn}}{[1+(2^{k}|x−x′|)^{2}]^{N}} |h(x′)|dx′ \\ & ≤ C_{N}2^{kγ}e^{−2^{k−m}ct^{\frac{1}{pγ}}} Mh(x). \end{aligned}  (105)

We thus have

\begin{aligned}∫_{0}^{1} ∫_{R^{n}} |K_{γ}ϕ_{k}h(x, t)|^{p}dxdt & ≤ C (∫_{0}^{1} |2^{kγ}e^{−2^{k−m}ct^{\frac{1}{pγ}}} |^{p} dt) ∫_{R^{n}} |Mh(x)|^{p} dx, \\ & ≤ C ∫_{R^{n}} |Mh(x)|^{p} dx ≤ C ∫_{R^{n}} |h(x)|^{p}dx, \end{aligned}

since

∫_{0}^{1} |2^{kγ}e^{−2^{k−m}ct^{\frac{1}{pγ}}} |^{p} dt = 2^{kpγ} ∫_{0}^{2^{k}} e^{−2^{−m}cs}d (\frac{s}{2^{k}} )^{pγ} ≤ C_{p,γ},

and M is bounded on L^{p}. This establishes (100) and completes the proof of Theorem 35.

Now, as at the end of the previous subsection, we recast the definition of the P_{γ} condition in terms of open sets. Recall that U_{R} = {(x, t) ∈ R^{n} × R : |x| , |t| < R}, and that \overrightarrow{γ} ((x, s) , t) = (x, s + t) is the flow for T⃗ = \frac{∂}{∂t} through (x, s).

Definition 36 The function a satisfies the P_{γ} condition in the open set U_{R} if

\begin{matrix}|β| ≤ C |∫_{0}^{β} a (x, t) dt|^{γ} \\ for all − R < β < R, and |x| < R. \end{matrix}

Now we can state our characterization of the local boundedness of K_{γ} on B_{p}^{s,p}.

Theorem 37 Let a(x, t) be a C^{λ+2} function, and suppose that a ∈ P_{pγ} in the open set U_{R}. Let φ, ψ be C^{∞} functions supported in U_{R}, and suppose that P denotes a multiplier with symbol supported in the cone {|ξ| > |τ|}. Then if M_{φ}, M_{ψ} are the operators of multiplication by φ and ψ respectively, the operator P M_{φ}K_{γ}M_{ψ} is bounded B_{p}^{s,p} on for −\frac{1}{2} < s < λ + 2. Conversely, if φ = ψ = 1 on U_{R}, and M_{φ}K_{γ}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then a ∈ P_{pγ} in the open set U_{R′} for some R′ > 0.

Proof. The operator K_{γ} is bounded on L^{p}(B_{p}^{0,p}) by Theorem 35 together with the same result scaled to R^{n} × (−R, 0). Now if supp φ, supp ψ ⊂ U_{R}, then P M_{φ}K_{γ}M_{ψ} is bounded on L^{p}(B_{p}^{0,p}), and so also on B_{p}^{0,p} by Lemma 30. By Lemma 27, we now conclude that P M_{φ}K_{γ}M_{ψ} is bounded on B_{p}^{s,p} for −\frac{1}{2} < s < λ + 2.

Conversely, if φ = ψ = 1 on U_{R} and M_{φ}K_{γ}M_{ψ} is bounded on B_{p}^{s,p} for some s ∈ (−\frac{1}{2}, λ + 2), then by Lemma 27, it is bounded on B_{p}^{0,p} . So suppose K′_{γ} = M_{φ}K_{γ}M_{ψ} is bounded on B_{p}^{0,p}. The proof of necessity in Theorem 35 carries over here with just a few changes, as follows. With notation as in the proof of Theorem 35, we have from (95) that

Re(K′_{γ}ϕ_{r}f)(x, t) = Re(M_{φ}K_{γ}ϕ_{r}f)(x, t) ≥ cr^{−γ}β  (106)

for |x−w| < cr, t ∈ (0, β). Now choose η ∈ C_{c}^{∞}(R^{n+1}) such that η ≥ 0, support η ⊂ B(0, \frac{1}{2}) and ∫_{R^{n+1}} η(x)dx = 1 (note the use of R^{n+1} in place of R^{n} here). With η_{r}(x) = r^{−n}η (\frac{x}{r}), it follows easily from (106) that

Re(η_{r} ∗ K′_{γ}ϕ_{r}f)(x, t) ≥ cr^{−γ} ∫_{α}^{β} a(w, t′)^{p′}dt′

for |x − w| < cr, t ∈ (0, β), with a perhaps smaller constant c - note that r ≤ β/3 if, as we may assume, a(x, t) ≤ 1/3. If we use the inequality ∥η_{r} ∗ g∥_{L^{p}} ≤ C ∥g∥_{B_{p}^{0,p}}, valid for any g ∈ B_{p}^{0,p}(R^{n+1}), 1 < p < ∞, (this is (53) with R^{n+1} in place of R^{n}), the P_{pγ} condition in the open set U_{R′} can be derived as in the proof of Theorem 35.

2.4 Boundedness of K in Hölder spaces

We study K when a satisfies the T_{ν} condition:

|t − t′| ≤ C |∫_{t′}^{t} a(x, θ)dθ|^{ν}  (107)

for all x ∈ R^{n} and all t, t′ ∈ R with tt′ ≥ 0. Denote the unit ball in R^{n} by B. Using the argument in [Gu] we have the following lemma.

Lemma 38 Suppose ν > 0 and a satisfies the T_{ν} condition (107). Then there is a constant C_{ν} such that for every ϵ > 0,

∫_{0}^{t} ∫_{|x−x′|<ϵ} |K(x, x′, t, t′)| dx′dt′ ≤ C_{ν}ϵ^{ν},  (108)
∫_{0}^{t} ∫_{B} |K(x + h, x′, t, t′) − K(x, x′, t, t′)| dx′dt′ ≤ C_{ν} |h|^{ν} ,  (109)

and if in addition a(x, t)a(x, t + η) ≥ 0, then

∫_{t}^{t+η} ∫_{B} |K(x, x′, t, t′)| dx′dt′ ≤ C_{ν} |η| ln |η|^{−1} ,  (110)
∫_{0}^{t} ∫_{B} |K(x, x′, t + η, t′) − K(x, x′, t, t′)| dx′dt′ ≤ C_{ν} |η| ln |η|^{−1} .  (111)

If we apply the argument used in the proof of Corollary 24, we now obtain

Theorem 39 If λ > 0, and in addition a satisfies the T_{ν} condition (107), then K maps Λ^{s} to Λ^{s+ν} for all s > 0 with s + ν ≤ λ + 2.

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