Georgian Mathematical Journal Volume 10 (2003), Number 3, 573–593

A DIRECT METHOD FOR BOUNDARY INTEGRAL EQUATIONS ON A CONTOUR WITH A PEAK

V. MAZ’YA AND A. SOLOVIEV

In memoriam V.D. Kupradze

Abstract. Boundary integral equations in the logarithmic potential theory are studied by the direct method under the assumption that the contour has a peak.

2000 Mathematics Subject Classification: 31A10, 45A05.

Key words and phrases: Boundary integral equation, logarithmic potential, direct method.

1. Introduction

In this paper we apply the so-called direct variant of the method of boundary integral equations to solve the Dirichlet problem

Δu = 0 in Ω, u = φ on Γ \ {O}  (1)

and the Neumann problem

Δu = 0 in Ω, ∂u/∂n = ψ on Γ \ {O}  (2)

with boundary data ψ satisfying ∫_{Γ} ψds = 0, where Ω is a plane simply connected domain having compact closure and the boundary Γ with a peak at z = 0. Here and elsewhere we assume that the normal n is directed outwards.

The classical method for solving boundary value problems is their reduction to boundary integral equations by using potentials. In the case of the Dirichlet and Neumann problems for the Laplace equation the solutions of these problems are represented in the form of simple and double layer potentials whose densities satisfy boundary integral equations. However, there exists another way of reduction when solutions of integral equations are represented explicitly by the solutions of the boundary value problems (1) and (2). In this case the integral equations can be obtained from the integral representation for a harmonic function:

u(z) = \frac{1}{2π} (V \frac{∂u}{∂n})(z) − \frac{1}{2π} (W u)(z), z ∈ Ω,

where V is the simple layer potential

(V σ)(z) = ∫_{Γ} σ(q) log \frac{|z|}{|z − q|}ds_{q}

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and W is the double layer potential

(W σ)(z) = ∫_{Γ} σ(q)\frac{∂}{∂n} log \frac{1}{|z − q|} ds_{q} .

By making use of the continuity of the simple layer potential and the limit relation for the double layer potential we obtain

πu(z) = (V \frac{∂u}{∂n})(z) − (W u)(z), z ∈ Γ \ \{O\},  (3)

where (W u)(z) is the direct value of W u at the point z on Γ\{O}. By inserting the known values φ of u on Γ \ {O} into (1) we obtain that ∂u/∂n satisfies the integral equation of first kind

V γ = πφ + W φ on Γ \ {O}.

The normal derivative on Γ \ {O} of the solution u of (2) is defined by the boundary data ψ. From (3) it follows that u on Γ \ {O} is a solution of the integral equation

πσ + W σ = V ψ .

Using the results of [1]–[3] we study the equations obtained by the direct reduction of problems (1) and (2) for a domain with a peak to integral equations. For every integral equation we choose a pair of function spaces and prove the solvability of the equation in one of these spaces with a right-hand side from another. We also describe solutions of the corresponding homogeneous equation.

We mention recent articles [4], [5] dealing with boundary integral equations in weighted L_{p}-spaces on contours with peaks. The solvability in Smirnov’s classes of boundary value problems in domains with piecewise smooth boundaries was studied in [6], [7] where the reduction to the Riemann-Hilbert problem for analytic functions in the unit disc was used.

Let Ω be a plane simply connected domain with boundary Γ which has a peak at the origin O. We assume that Γ\{O} belongs to the class C^{2}. We say that O is an outward (inward) peak if Ω (the exterior domain Ω′) is given near O by the inequalities κ_{−}(x) < y < κ_{+}(x), 0 < x < δ, where

x^{−μ−1}κ_{±}(x) ∈ C^{2}[0, δ], lim_{x→+0} x^{−μ−1}κ_{±}(x) = α_{±}

with μ > 0 and α_{+} > α_{−}.

By Γ_{±} we denote the arcs {(x, κ_{±}(x)) : x ∈ [0, δ]}. Points on Γ_{+} and Γ_{−} with equal abscissas are denoted by q_{+} and q_{−}.

We say that φ belongs to L_{p,β}(Γ) if |q|^{β}φ ∈ L_{p}(Γ). The norm in this space is given by

∥ φ ∥_{L_{p,β}(Γ)}=∥ |q|^{β}φ ∥_{L_{p}(Γ)} .

Let L_{p,β}^{1}(Γ) be the space of absolutely continuous functions on Γ \ {O} with the finite norm

∥ φ ∥_{L_{p,β}^{1}(Γ)}=∥ (∂/∂s)φ ∥_{L_{p,β}(Γ)} + ∥ φ ∥_{L_{p,β−1}(Γ)} .

We introduce the space 𝔑_{p,β}(Γ) of absolutely continuous functions φ on Γ\{O} with the finite norm

∥ φ ∥_{𝔑_{p,β}(Γ)}= ( ∫_{Γ_{+}∪Γ_{−}} |φ(q_{+}) − φ(q_{−})|^{p}|q|^{p(β−μ)}ds_{q} )^{1/p}+ ∥ φ ∥_{L_{p,β+1}^{1}(Γ)} .

By 𝔑_{p,β}^{−1}(Γ) we denote the space of functions on Γ\{O} represented in the form φ = (d/ds)ψ, where ψ ∈ 𝔑_{p,β}(Γ) and ψ(z_{0}) = 0 for a fixed point z_{0} ∈ Γ \ {O}. A norm on 𝔑_{p,β}^{−1}(Γ) is defined by

∥ φ ∥_{𝔑_{p,β}^{−1}(Γ)}=∥ ψ ∥_{𝔑_{p,β}(Γ)} .

Furthermore, we introduce the space 𝔑_{p,β}^{(+)}(Γ) of absolutely continuous functions φ on Γ \ {O} supplied with the norms

∥ φ ∥_{𝔑_{p,β}^{(+)}(Γ)}= ( ∫_{Γ_{+}∪Γ_{−}} |φ(q_{+}) + φ(q_{−})|^{p}|q|^{p(β−μ)}ds_{q} )^{1/p}+ ∥ φ ∥_{L_{p,β+1}^{1}(Γ)} .

Let 𝔓(Γ) denote the space of restrictions to Γ \ {O} of functions of the form p(z) = ∑_{k=0}^{m} t^{(k)}Re z^{k}, where m = [μ − β − p^{−1} + 2^{−1}] and t^{(k)} are real numbers. We endow 𝔓(Γ) with the norm

∥ p ∥_{𝔓(Γ)}= ∑_{k=0}^{m} | t^{(k)} | .

The space 𝔐_{p,β}(Γ) is defined as the direct sum of 𝔑_{p,β}^{(+)}(Γ) and 𝔓(Γ).

Now, we can describe our results. We assume that p > 1 and 0 < β + p^{−1} < min{μ, 1}.

Let Ω have an outward peak. We introduce the double layer potential in Ω′ by setting

(W^{ext}σ)(z) = ∫_{Γ} σ(q)(\frac{∂}{∂n_{q}} log \frac{1}{|z − q|} + 1)ds_{q}, z ∈ Ω′ .

The value of this potential at the point z ∈ Γ \ {O} will be also denoted by (W^{ext}σ)(z). In Theorem 1 we prove that the integral equation

πσ + W^{ext}σ = V ψ  (4)

with the function ψ ∈ 𝔑_{p,β}^{−1}(Γ) on the right-hand side has a unique solution σ in 𝔑_{p,β}(Γ) satisfying ∫_{Γ} σds = 0.

As is shown in Theorem 3, the integral equation of first kind

V γ = πφ + W φ  (5)

with the function φ ∈ 𝔑_{p,β}(Γ) on the right-hand side has a unique solution γ in 𝔑_{p,β}^{−1}(Γ).

For Ω with an inward peak, we show in Theorem 2 that the integral equation

πσ + W^{ext}σ = V ψ′ ,  (6)

with the function ψ ∈ L_{p,β+1}^{1}(Γ) is solvable in L_{p,β+1}^{1}(Γ) and that the solution σ satisfies ∫_{Γ} σds = 0. In Theorem 4 we prove that the integral equation

V γ = πφ + W φ  (7)

with the function φ ∈ L_{p,β+1}^{1}(Γ) has a solution γ ∈ L_{p,β+1}(Γ).

In Theorems 2 and 4 we prove that the homogeneous equations (6) and (7) have only trivial solutions for 0 < β + p^{−1} < 1/2 and a one-dimensional space of solutions for 1/2 < β + p^{−1} < 1.

We shall use the following statements proved in [1]–[3].

Theorem A (see [1]). Let Ω have either an outward or an inward peak and let 0 < β + p^{−1} < min{μ, 1}. Then the operator

L_{p,β+1}(Γ) × ℝ ∋ (σ, t)⟼^{V} V σ + c ∈ 𝔑_{p,β}(Γ)

is continuous and, if β + p^{−1} ≠ 2^{−1}, it is surjective. The kernel of V is trivial for 0 < β + p^{−1} < 1/2 and one-dimensional for 1/2 < β + p^{−1} < 1. If Ω has an outward peak and 1/2 < β + p^{−1} < 1, then

ker V = \{\frac{t}{π} \frac{∂}{∂n} Im \frac{1}{γ^{(out)}} , t Im \frac{1}{γ^{(out)}(∞)}\} ,

where t ∈ ℝ and γ^{(out)} is the conformal mapping of Ω′ onto ℝ_{+}^{2} = {z : Im z > 0} subject to γ^{(out)}(0) = 0 and γ^{(out)}(∞) = i.

If Ω has an inward peak and 1/2 < β + p^{−1} < 1 then

ker V = \{\frac{t}{π} \frac{∂}{∂n} Im \frac{1}{γ^{(in)}} , 0\} ,

where t ∈ ℝ and γ^{(in)} is the conformal mapping of Ω onto ℝ_{+}^{2} subject to the conditions γ^{(in)}(0) = 0 and γ^{(in)}(z_{0}) = i with a fixed point z_{0} ∈ Ω.

Theorem B (see [2]). Let Ω have an inward peak and let 0 < β + p^{−1} < min{μ, 1}. Then the operator

L_{p,β+1}^{1}(Γ) ∋ σ ⟼^{W_{in}} (πI + W^{ext})σ ∈ 𝔑_{p,β}(Γ)

is continuous and, provided β + p^{−1} ≠ 2^{−1}, it is surjective. The operator W has a trivial kernel for 0 < β + 1/p < 1/2 and a one-dimensional kernel for 1/2 < β + 1/p < 1. In the latter case

Ker W = \{t Re\frac{1}{γ_{0}} \} ,

where t ∈ ℝ and γ_{0} is the conformal mapping of Ω onto ℝ_{+}^{2} subject to

γ_{0}(0) = 0, ∫_{Γ} Re \frac{1}{γ_{0}} ds = 0 and Im γ_{0}(z_{0}) = 1 ,

with a fixed point z_{0} in Ω.

We introduce the functions I_{k}^{ext}, k = 1, 2, …, by setting

I_{k}^{ext}(z) = Im(\frac{zz_{0}}{z_{0} − z} )^{k−1/2}, z ∈ Ω′,

where z_{0} is a fixed point in Ω.

Theorem C (see [3]). Let Ω have an outward peak and let 0 < β + p^{−1} < min{μ, 1} and m = [μ − β − p^{−1} + 2^{−1}]. Then the operator

L_{p,β+1}^{1}(Γ) × ℝ^{m} ∋ (σ, t) ⟼^{W_{out}} πσ + W^{ext}σ + ∑_{k=1}^{m} t^{(k)}I_{k}^{ext} ∈ 𝔐_{p,β}(Γ)

is continuous and injective. In case μ − β − p^{−1} + 2^{−1} ∉ N and β + p^{−1} ≠ 2^{−1} the operator W_{μ} is bijective.

In the proof of the surjectivity of W_{out} we use the next proposition.

Proporision 1. Let Ω have an inward peak and let φ ∈ 𝔑_{p,β}^{(+)}(Γ), where

0 < β + p^{−1} < min{μ, 1}, β + p^{−1} ≠ 2^{−1} and μ − β − p^{−1} + 2^{−1} ∉ ℕ.

Then there exists a harmonic extension of φ onto Ω′ with the normal derivative in the space L_{p,β+1}(Γ) such that the conjugate function g with g(z_{0}) = 0 for a fixed point z_{0} ∈ Γ \ {O} can be written as

∑_{k=1}^{m} c_{k}(φ)Re z^{k−1/2} + g^{#}(z),

where c_{k}(φ) are linear continuous functionals in 𝔑_{p,β}^{(+)}(Γ) and g^{#} satisfies

∥g^{#}∥_{𝔑_{p,β}^{(−)}(Γ)} ≤ c ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)}.

with a constant c independent of φ.

The proof of Proposition 1 in [3] was not complete. In particular, the condition β + p^{−1} ≠ 2^{−1} was omitted. In Appendix we give a complete proof of this proposition.

2. The Dirichlet and Neumann problems for domains with peaks

2.1. Let the operator T be defined by

T f(x) = ∫_{ℝ} K(x, y)f(y)dy,

where

|K(x, y)| ≤ c \frac{1}{|x − y|} \frac{1}{(1 + |x − y|^{J})} , J ≥ 0.

Here and elsewhere by c we denote different positive constants.

We introduce the space L_{p,α}(ℝ) of functions on ℝ with the norm

∥φ∥_{L_{p,α}(ℝ)} = ∥(1 + x^{2})^{α/2}φ∥_{L_{p}(ℝ)}.

The following lemma can be proved in the same way as Stein’s theorem on the boundedness of a singular integral operator in a weighted L_{p}-space [8].

Lemma 1. If T : L_{p}(ℝ) → L_{p}(ℝ), 1 < p < ∞, is bounded and −J < α + p^{−1} < J + 1, then T is a continuous operator in L_{p,α}(ℝ).

2.2. Let G ⊂ ℝ^{2} be a domain with C^{2}-boundary such that the set {(τ, ν) ∈ G : τ ≤ 0} has compact closure and {(τ, ν) ∈ G : τ > 0} = {(τ, ν) : τ > 0, |ν| < 1}.

As usual, by C_{0}^{∞}(G) we mean the space of infinitely differentiable functions with compact supports in G. By W_{p}^{k}(G), k = 0, 1, 2, p ∈ (1, ∞), we denote the Sobolev space of functions in L_{p}(G) with derivatives up to order k in L_{p}(G). The notation W̊_{p}^{k}(G) stands for the completion of C_{0}^{∞}(G) in W_{p}^{k}(G). Let W_{p}^{k−1/p}(∂G) be the space of traces on ∂G of functions in W_{p}^{k}(G). We introduce also the space W_{p}^{−1}(G) of distributions on G with the finite norm

∥ φ ∥_{W_{p}^{−1}(G)}= inf ∑_{j=0}^{2} ∥ φ_{j} ∥_{L_{p}(G)},

where the infimum is taken over all representations φ = φ_{0} + (∂/∂τ)φ_{1} + (∂/∂ν)φ_{2} with φ_{j} ∈ L_{p}(G), j = 0, 1, 2.

Let α ∈ ℝ. We say that φ ∈ W_{p,α}^{k}(G) if (1+τ^{2})^{α/2}φ ∈ W_{p}^{k}(G), k = −1, 0, 1 …, and define the norm

∥ φ ∥_{W_{p,α}^{k}(G)}=∥ (1 + τ^{2})^{α/2}φ ∥_{W_{p}^{k}(G)} .

The spaces W_{p,α}^{k}(∂G) and W_{p,α}^{k−1/p}(∂G) are introduced in the same way. By L_{p,α}(∂G) we denote the space of functions with the finite norm

∥ φ ∥_{L_{p,α}(∂G)}=∥ (1 + τ^{2})^{α/2}φ ∥_{L_{p}(∂G)} .

We shall make use of the same definitions for the strip Π = {(τ, ν) : τ ∈ ℝ, |ν| < 1}.

The following lemma is contained in more general results of [9].

Lemma 2. The operator

W_{p,α}^{k}(G) ∋ u → {Δu, u|_{∂G}} ∈ W_{p,α}^{k−2}(G) × W_{p,α}^{k−1/p}(∂G)

performs an isomorphism for every real α and k = 1, 2. The same is true for k = 1, 2 if G is replaced by the strip Π.

2.3. In this section we consider the Dirichlet problem in Ω with an outward peak. The following proposition is an improvement of Proposition 1 from [1].

Proporision 2. Let φ belong to space 𝔑_{p,β}(Γ), where 0 < β+p^{−1} < min{μ, 1}. Then there exists a harmonic extension u of φ onto Ω with the normal derivative ∂u/∂n in 𝔑_{p,β}^{−1}(Γ) satisfying

∥ ∂u/∂n ∥_{𝔑_{p,β}^{−1}(Γ)}≤ c ∥ φ ∥_{𝔑_{p,β}(Γ)} .  (8)

Proof. It is sufficient to obtain (8) under the assumption φ = 0 in a neighborhood of the peak.

We start with the case where φ ∈ 𝔑_{p,β}(Γ) vanishes on Γ ∩ {|q| < δ/2}. Let θ be a conformal mapping of the unit disk D onto Ω. We introduce the harmonic extension F of the continuous function φ ∘ θ onto D. The normal derivative ∂F/∂n on ∂D has the form

\frac{∂F}{∂n} (ζ) = \frac{1}{2π} ∫_{−π}^{π} \frac{d}{dt}φ(θ(e^{it})) cotan \frac{s − t}{2} dt, ζ = e^{is}.

Hence u = F ∘ θ^{−1} satisfies

∥ ∂u/∂n ∥_{L_{p,γ+1}(Γ)}≤ c_{1} ∥ ∂F/∂n ∥_{L_{p}(∂D)}≤ c_{2} ∥ (φ ∘ θ)′ ∥_{L_{p}(∂D)}≤ c_{3} ∥ φ ∥_{𝔑_{p,γ}(Γ)}

for every real γ. Since ∫_{Γ}(∂u/∂n)ds = 0, ∂u/∂n is represented in the form (∂/∂s)v, where the function v satisfies

∥ v ∥_{L_{p,β−μ}(Γ)} + ∥ v′ ∥_{L_{p,β+1}(Γ)}≤ c ∥ φ ∥_{𝔑_{p,β}(Γ)} .

It remains to prove (8) for φ ∈ 𝔑_{p,β}(Γ) vanishing outside Γ_{+} ∪ Γ_{−}. Let G be the domain defined in 2.2.

By z = ω(τ + iν) we denote a conformal mapping of G onto Ω such that ω(∞) = 0 and ω(∂G ∩ {(x, y) : x > 1}) ⊃ Γ_{+} ∪ Γ_{−}. It is well-known and follows essentially from Warschawski’s asymptotic formula for conformal mappings [10] that

ω(τ + iν) = c (τ + iν)^{−1/μ}(1 + o(1)), |ω′(τ + iν)| ≤ c |τ + iν|^{−1−1/μ} (9)

as τ → +∞, τ + iν ∈ G, and that

ω^{−1}(z) = c z^{−μ}(1 + o(1)), |(ω^{−1})′(z)| ≤ c |z|^{−1−μ}  (10)

as x → 0, z = x + iy ∈ Ω. We introduce the functions Φ_{±} on ℝ by

Φ_{±}(τ) = φ(ω(τ ± i)) for τ ≥ 0 and Φ_{±}(τ) = 0 for τ < 0.

The following estimates hold:

c_{1} ∥φ∥_{𝔑_{p,β}(Γ)} ≤ ∥ Φ_{+} − Φ_{−} ∥_{L_{p,1−α}(ℝ)}
+ ∑_{±} (∥Φ_{±}∥_{L_{p,−α}(ℝ)} + ∥Φ′_{±}∥_{L_{p,1−α}(ℝ)}) ≤ c_{2} ∥φ∥_{𝔑_{p,β}(Γ)}  (11)

with α defined by β + p^{−1} = μ(α − p^{−1}).

Let r be a measurable function on (0, ∞) subject to |r(τ)| ≤ 1. We have

∫_{0}^{∞}|Φ_{±}(τ) − Φ_{±}(τ + r(τ))|^{p} τ^{(1−α)p}dτ

≤ c_{1} ∫_{0}^{∞}( ∫_{τ−1}^{τ+1}|Φ′_{±}(ν)|dν)^{p}τ^{(1−α)p}dτ ≤ c_{2} ∫_{0}^{∞}|Φ′_{±}(τ)|^{p}τ^{(1−α)p}dτ. (12)

For z, z_{1} ∈ Ω with |z_{1} − z| ≤ x^{μ+1} the distance between ω^{−1}(z_{1}) and ω^{−1}(z) is bounded by (10). Hence and by (12) we obtain the left inequality in (11).

Let h be a measurable function on [0, δ] such that |h(x)| ≤ x^{μ+1}. By the theorem on the boundedness of the Hardy–Littlewood maximal operator in weighted L^{p}-spaces (see [11]) we have

∫_{0}^{δ} |φ(x) − φ(x + h(x))|^{p}x^{(β−μ)p}dx

≤ c ∫_{0}^{δ} (\frac{1}{x^{μ+1}} ∫_{x−x^{μ+1}}^{x+x^{μ+1}}|t φ′(t)|dt)^{p}x^{βp}dx ≤ c ∫_{0}^{δ} |φ′(t)|^{p}t^{(β+1)p}dt.  (13)

By using (9) we obtain that for ζ, ζ_{1} ∈ G with |ζ − ζ_{1}| < 1 the distance between ω(ζ) and ω(ζ_{1}) does not exceed c x^{μ+1}. Hence and by (13) the right inequality in (11) follows.

Let Π denote the same strip as in 2.2. We introduce the bounded harmonic function Φ^{(+)} on Π taking the same value (Φ_{+}(τ)+Φ_{−}(τ))/2 at the points (τ, 1) and (τ, −1) of ∂Π. The Fourier transform of Φ^{(+)}(τ, ν) with respect to τ is given by

c (Φ̂_{+}(ξ) + Φ̂_{−}(ξ)) cosh (νξ) (cosh ξ)^{−1} ,

where Φ̂_{±} denote the Fourier transform of Φ_{±}. Therefore the Fourier transform of (∂/∂n)Φ^{(+)}(τ, ±1) is equal to

c i (Φ̂′_{+}(ξ) + Φ̂′_{−}(ξ)) tanh ξ .

Hence

\frac{∂Φ^{(+)}}{∂n} (τ, ±1) = c ∫_{ℝ} \frac{d}{dt} (Φ_{+}(t) + Φ_{−}(t)) (sinh (\frac{π}{2} (τ − t)))^{−1} dt .

By Lemma 1,

∥ ∂Φ^{(+)}/∂n ∥_{L_{p,1−α}(∂Π)}≤ c ∥ (Φ_{+} + Φ_{−})′ ∥_{L_{p,1−α}(∂Π)} .

We rewrite the Fourier transform of (∂/∂n)Φ^{(+)}(τ, ν) with respect to τ in the form

c iξ (Φ̂′_{+}(ξ) + Φ̂′_{−}(ξ)) \frac{sinh ξ}{ξ cosh ξ} .

Clearly, the function (∂/∂n)Φ^{(+)}(τ, ±1) can be represented as (∂/∂s)Y^{(−)}(τ, ±1), where Y^{(−)}(τ, +1) = −Y^{(−)}(τ, −1). By a theorem on the Fourier multipliers in weighted L_{p}-spaces (see [12]) it follows that

∥ Y^{(−)} ∥_{L_{p,1−α}(∂Π)} + ∥ (∂/∂τ)Y^{(−)} ∥_{L_{p,1−α}(∂Π)}≤ c ∥ (∂/∂τ)(Φ_{+} + Φ_{−}) ∥_{L_{p,1−α}(∂Π)} .

Let Φ^{(−)} be the bounded harmonic function on Π taking the opposite values (Φ_{+}(τ) − Φ_{−}(τ))/2 and (Φ_{−}(τ) − Φ_{+}(τ))/2 at the points (τ, 1) and (τ, −1) of ∂Π. The Fourier transform of Φ^{(−)}(τ, ν) with respect to τ has the form

c (Φ̂_{+}(ξ) − Φ̂_{−}(ξ)) sinh(νξ) (sinh ξ)^{−1} .

Therefore the Fourier transform of (∂/∂n)Φ^{(−)}(τ, ±1) is equal to

±c ξ (Φ̂_{+}(ξ) − Φ̂_{−}(ξ)) cosh ξ (sinh ξ)^{−1}

= ±c\{ (Φ̂_{+}(ξ) − Φ̂_{−}(ξ)) \frac{ξ}{sinh ξ} + i ({ \atop \frac{d}{dt}Φ_{+}}(ξ) − { \atop \frac{d}{dt}Φ_{−}}(ξ)) tanh ξ\}.

and hence

\frac{∂Φ^{(−)}}{∂n} (τ, ±1) = ±c_{1} ∫_{ℝ} (Φ_{+}(t) − Φ_{−}(t)) (cosh \frac{π}{2} (τ − t))^{−2} dt

±c_{2} ∫_{ℝ} (\frac{dΦ_{+}}{dt} (t) − \frac{dΦ_{−}}{dt} (t)) (sinh π(τ − t))^{−1} dt .

By Lemma 1,

∥ ∂Φ^{(−)}/∂n ∥_{L_{p,1−α}(∂Π)}≤ c ∥ Φ_{+} − Φ_{−} ∥_{W_{p,1−α}^{1}(∂Π)} .

It is clear that (∂/∂n)Φ^{(−)} can be written as (∂/∂s)Y^{(+)}(τ, ±1), where Y^{(+)}(τ, +1) = Y^{(+)}(τ, −1) and according to the theorem on the Fourier multipliers in weighted L_{p}-spaces (see [12]) the following estimate is valid:

∥ Y^{(+)} ∥_{L_{p,−α}(∂Π)} + ∥ (∂/∂τ)Y^{(+)} ∥_{L_{p,1−α}(∂Π)}≤ c ∥ (Φ_{+} − Φ_{−}) ∥_{W_{p,1−α}^{1}(∂Π)} .

Let χ ∈ C^{∞}(ℝ) be equal to 1 for t > 1 and vanish for t < 0, and let Ψ = Δ(χ(Φ^{(−)} + Φ^{(+)})). Using Lemma 2 we have

∥ Ψ ∥_{L_{p}(Π)}≤ c(∥ Φ^{(−)} ∥_{W_{p,−α}^{1−1/p}(∂Π)} + ∥ Φ^{(+)} ∥_{W_{p,−α}^{1−1/p}(∂Π)}).

By the inclusion W_{p}^{1}(∂Π) ⊂ W_{p}^{1−1/p}(∂Π) the right-hand side has the majorant

c (∥ Φ^{(−)} ∥_{W_{p,1−α}^{1}(∂Π)} + ∥ Φ^{(+)} ∥_{L_{p,−α}(∂Π)} + ∥ dΦ^{(+)}/dt ∥_{L_{p,1−α}(∂Π)}).

Applying Lemma 2 with k = 2, we obtain that the Dirichlet problem

ΔZ = −Ψ in G, Z = 0 on ∂G

has a solution satisfying

∥ ∂Z/∂n ∥_{L_{p,γ}(∂G)}≤ c_{1} ∥ Z ∥_{W_{p,γ}^{2}(G)}≤ c_{2} ∥ Ψ ∥_{L_{p}(G)}

for every real γ. Since ∫_{∂G}(∂Z/∂n)ds = 0, the function ∂Z/∂n can be represented in the form ∂X/∂s, where X satisfies

∥ X ∥_{L_{p,1−α}(Γ)} + ∥ X′ ∥_{L_{p,1−α}(Γ)}≤ c ∥ Φ_{+} − Φ_{−} ∥_{L_{p,1−α}(ℝ)}

+ ∑_{±} (∥ Φ_{±} ∥_{L_{p,−α}(ℝ)} + ∥ Φ′_{±} ∥_{L_{p,1−α}(ℝ)}) .

We set F = Z + χ(Φ^{(−)} + Φ^{(+)}). By (12) the function u = F ∘ ω^{−1} is the required harmonic extension of φ onto Ω. □

Since 1 ∈ 𝔑_{p,β}(Γ), the next proposition follows from the Cauchy–Riemann conditions.

Proporision 3. Let ψ belong to 𝔑_{p,β}^{−1}(Γ), where 0 < β + p^{−1} < 1. Then the Neumann problem in Ω with boundary data ψ has a solution v in 𝔑_{p,β}(Γ) satisfying

∥ v ∥_{𝔑_{p,β}(Γ)}≤ c ∥ ψ ∥_{𝔑_{p,β}^{−1}(Γ)} .

3. An Integral Equation of the Neumann Problem

Theorem 1. Let Ω have an outward peak, and let 0 < β + p^{−1} < min{μ, 1}. Then, for any ψ ∈ 𝔑_{p,β}^{−1}(Γ), the integral equation

πσ + W^{ext}σ = V ψ  (14)

has a unique solution σ in 𝔑_{p,β}(Γ), satisfying ∫_{Γ} σ ds = 0.

Proof. Let ψ belong to C_{0}^{∞}(Γ\{O}). By h we denote a solution of the Neumann problem in Ω with boundary data ψ as in Proposition 3. From the integral representation of the harmonic function h in Ω and the limit relation for the double layer potential we obtain

h(z) + \frac{1}{π} ∫_{Γ} h(q)\frac{∂}{∂n_{q}} log \frac{1}{|z − q|}ds_{q} = \frac{1}{π} ∫_{Γ} log \frac{|z|}{|z − q|}ψ(q)ds_{q}  (15)

We choose h so that ∫_{Γ} h(q)ds_{q} = 0. According to Proposition 3, h belongs to 𝔑_{p,β}(Γ). From (15) it follows that σ = h ∈ 𝔑_{p,β}(Γ) is a solution of (14).

Now let ψ be an arbitrary function in 𝔑_{p,β}^{−1}(Γ). For β > −p^{−1} there exists a sequence {ψ_{n}}_{n≥1} of smooth functions on Γ \ {O} vanishing near the peak and approaching ψ in 𝔑_{p,β}^{−1}(Γ). By σ_{n} we denote the constructed solution of (14) with ψ_{n} on the right-hand side which is unique by Theorem C. Since the operator

L_{p,β+1}(Γ) ∋ ψ ⟼ V ψ ∈ 𝔑_{p,β}(Γ)

is continuous (see Theorem A), it follows that {V ψ_{n}} converges in L_{p,β+1}^{1}(Γ) to the limit V ψ. According to Proposition 3, the sequence {σ_{n}} converges in L_{p,β+1}^{1}(Γ) to the limit σ. Since the operator

L_{p,β+1}^{1}(Γ) ∋ σ ⟼^{W} (πI + W^{ext})σ ∈ 𝔐_{p,β}(Γ) ⊂ L_{p,β+1}^{1}(Γ)

is continuous (see Theorem C), we obtain that σ is a solution of (14) by passing to the limit.

The kernel of W in L_{p,β+1}^{1}(Γ) is trivial. Therefore equation (14) is uniquely solvable in 𝔑_{p,β}(Γ). □

Remark 1. Under the assumptions of Theorem 1 and provided μ − β − p^{−1} + 2^{−1} ∉ N, β + p^{−1} ≠ 2^{−1}, ψ ∈ 𝔑_{p,β}^{−1}(Γ) we conclude that V ψ belongs to 𝔑_{p,β}(Γ) ∩ 𝔐_{p,β}(Γ) and satisfies the “orthogonality” conditions

∫_{Γ} \frac{∂}{∂s} (V ψ) Re \frac{1}{ζ^{2k−1}} ds = 0, k = 1, …, m ,

where ζ is the conformal mapping of Ω′ onto ℝ_{+}^{2} subject to

ζ(0) = 0, Re ζ(∞) = 0 and Re (1/ζ(z)) = ±x^{−1/2} + O(1).

Proof. By Theorem 1 we have V ψ ∈ 𝔑_{p,β}(Γ), since ψ ∈ L_{p,β+1}(Γ), and V ψ ∈ 𝔐_{p,β}(Γ). Therefore V ψ = φ + ∑_{k=0}^{m} d^{(k)}Re z^{k}, where φ ∈ 𝔑_{p,β}^{(+)}(Γ). According to Proposition 1, there exists a harmonic extension of φ onto Ω′ such that the conjugate function g satisfying g(∞) = 0 has the representation

∑_{k=1}^{m} c^{(k)}(φ)Re z^{k−1/2} + g^{#}(z),

where c^{(k)}(φ) are linear continuous functionals in 𝔑_{p,β}^{(+)}(Γ) and g^{#} ∈ 𝔑_{p,β}(Γ). We apply the Green formula to the functions g and Re ζ^{1−2k} in Ω′ ∩ {|z| < ε}. Passing to the limit as ε → 0, we obtain

c^{(k)}(φ) = c_{k} ∫_{Γ} \frac{∂g}{∂n} Re \frac{1}{ζ^{2k−1}} ds = c_{k} ∫_{Γ} φ′_{s}Re \frac{1}{ζ^{2k−1}} ds.

Let (σ, t) ∈ L_{p,β+1}^{1}(Γ) × ℝ^{m} be the solution of the equation

(πI + W^{ext})σ + ∑_{k=1}^{m} t^{(k)}I_{k} = V ψ

with the right-hand side from 𝔐_{p,β}(Γ). As is shown in Theorem 2 [3], the components t^{(k)}, are equal to c^{(k)}(φ), k = 1, …, m. Since the operator W is surjective and since

∫_{Γ} \frac{∂}{∂n} Im z^{k} Re \frac{1}{ζ^{2k−1}} ds = 0 ,

it follows that V ψ ∈ 𝔑_{p,β}(Γ) ∩ 𝔐_{p,β}(Γ) and c^{(k)}(V ψ) = 0, k = 1, …, m. □

Theorem 2. Let Ω have an inward peak, and let 0 < β + p^{−1} < min{μ, 1} and β +p^{−1} ≠ 1/2. Then, for any ψ ∈ L_{p,β+1}^{1}(Γ), the boundary integral equation

πσ + W^{ext}σ = V ψ′  (16)

has a solution σ ∈ L_{p,β+1}^{1}(Γ) satisfying ∫_{Γ} σ ds = 0. The homogeneous equation (16) has only a trivial solution for 0 < β + p^{−1} < 1/2 and a one-dimensional space of solutions for 1/2 < β + p^{−1} < 1, given by

\{t Re\frac{1}{γ_{0}} \},

where t ∈ ℝ and γ_{0} is the conformal mapping of Ω onto ℝ_{+}^{2} subject to

γ_{0}(0) = 0, ∫_{Γ} Re \frac{1}{γ_{0}} ds = 0 and Imγ_{0}(z_{0}) = 1

with a fixed point z_{0} ∈ Ω.

Proof. Let ψ ∈ L_{p,β+1}^{1}(Γ). Then V ψ′ ∈ 𝔑_{p,β}(Γ) (see Theorem A). According to Theorem B, equation (16) is solvable in L_{p,β+1}^{1}(Γ). Since the harmonic extension of V ψ′ vanishes at infinity, we have ∫_{Γ} σ ds = 0. By Theorem B, the set of solutions to the homogeneous equation (16) in L_{p,β+1}^{1}(Γ) is one-dimensional for 1/2 < β + p^{−1} < 1 and trivial for 0 < β + p^{−1} < 1/2. The set of solutions to the homogeneous equation (16) is described in Theorem B. □

4. An Integral Equation of the Dirichlet Problem

Theorem 3. Let Ω have an outward peak and let 0 < β + p^{−1} < min{μ, 1}. Then the boundary integral equation

V γ = πφ + W φ  (17)

has a solution γ ∈ 𝔑_{p,β}^{−1}(Γ) for every φ ∈ 𝔑_{p,β}(Γ). This solution is unique for β + p^{−1} ≠ 2^{−1}.

Proof. Let φ ∈ C_{0}^{∞}(Γ \ {O}). By u we denote the bounded harmonic extension of φ onto Ω constructed in Proposition 2. By the integral representation of the harmonic function u on Ω and by the limit relation for the simple layer potential we obtain for z ∈ Γ \ {O}

∫_{Γ} log \frac{|z|}{|z − q|} \frac{∂u}{∂n}(q) ds_{q} = πφ(z) + ∫_{Γ} φ(q)\frac{∂}{∂n_{q}} log \frac{1}{|z − q|} ds_{q} . (18)

Since ∂u/∂n belongs to 𝔑_{p,β}^{−1}(Γ), from (18) it follows that γ = ∂u/∂n satisfies (17).

Now let φ be an arbitrary function in 𝔑_{p,β}(Γ). There exists a sequence {φ_{n}}_{n≥1} of smooth functions on Γ \ {O} vanishing near the peak and converging to φ in 𝔑_{p,β}(Γ). By γ_{n} we denote the constructed solution of (17) with φ_{n} instead of φ on the right-hand side. Since the operator (πI + W) : L_{p,β+1}^{1}(Γ) → 𝔐_{p,β}(Γ) is continuous (see Theorem C), we obtain by taking the limit that {πφ_{n} + W φ_{n}}_{n≥1} converges in L_{p,β+1}^{1}(Γ). In view of Proposition 2, the sequence {∂u_{n}/∂n}, where u_{n} is the bounded extension of φ_{n} onto Ω, converges in L_{p,β+1}(Γ). According to Theorem A,

L_{p,β+1}(Γ) ∋ γ ⟼^{V} V γ ∈ 𝔑_{p,β}(Γ)

is continuous. Then by passing to the limit in the equation V γ_{n} = πφ_{n} + W φ_{n} we obtain that γ is a solution of (17). For Ω with an outward peak Theorem 1 implies that Ker V is trivial provided β +p^{−1} ≠ 2^{−1}. Therefore the just obtained solution of (17) is unique. □

Theorem 4. Let Ω have an inward peak, and let 0 < β + p^{−1} < min{μ, 1}, β + p^{−1} ≠ 1/2. Then the integral equation

V γ = πφ + W φ  (19)

has a solution γ ∈ L_{p,β+1}(Γ) for every φ ∈ L_{p,β+1}^{1}(Γ). This solution is unique for 0 < β+p^{−1} < 1/2 and the homogeneous equation (19) has a one-dimensional space of solutions for 1/2 < β + p^{−1} < 1 given by

\{t \frac{∂}{∂n} Im \frac{1}{γ^{(in)}} \},

where t ∈ ℝ and γ^{(in)} is the conformal mapping of Ω onto ℝ_{+}^{2} subject to the conditions γ^{(in)}(0) = 0 and γ^{(in)}(z_{0}) = i with a fixed point z_{0} ∈ Ω.

Proof. Let φ ∈ L_{p,β+1}^{1}(Γ). According to Theorem B, (πI + W)φ belongs to 𝔑_{p,β}(Γ) and its harmonic extension onto Ω′

(W σ)(z) = ∫_{Γ} σ(q)\frac{∂}{∂n_{q}} log \frac{1}{|z − q|} ds_{q}, z ∈ Ω′,

vanishes at infinity. The range of the operator

L_{p,β+1}(Γ) ∋ γ ⟼ V γ ∈ 𝔑_{p,β}(Γ)

consists of the elements of 𝔑_{p,β}(Γ) whose harmonic extensions to Ω′ vanish at infinity (see Theorem A). Therefore equation (19) has a solution in L_{p,β+1}(Γ). The homogeneous equation (19) has only a trivial solution in L_{p,β+1}(Γ) for 0 < β + p^{−1} < 1/2 and a one-dimensional space of solutions for 1/2 < β + p^{−1} < 1. The set of solutions to the homogeneous equation (19) is described in Theorem A. □

5. Appendix: Proof of Proposition 1

(i) We shall make use of the representation of a conformal mapping θ of ℝ_{+}^{2} = {ζ = ξ + iη : η > 0} onto Ω

θ(ξ) = \{ \begin{matrix}∑_{k=2}^{[2μ]+1} B^{(k)}ξ^{k} + B^{([2μ]+2)}ξ^{[2μ]+2} log |ξ| + B^{(±)}|ξ|^{2μ+2} \\ +O(ξ^{2μ+2+γ})  if 2μ ∈ ℕ, \\ ∑_{k=2}^{[2μ]+2} B^{(k)}ξ^{k} + B^{(±)}|ξ|^{2μ+2} + B^{([2μ]+3)}ξ^{[2μ]+3} \\ +O(ξ^{2μ+2+γ})  if 2μ ∉ ℕ, \end{matrix} (20)

as ξ → ±0, where B^{(k)}, k = 2, …, [2μ] + 2, are real coefficients and 0 < γ < min(μ, 1). Decomposition (20) can be differentiated at least once ( see [3]).

By D we denote the image of Ω under the mapping u + iv = (x + iy)^{1/2}. Let θ̃ denote a conformal mapping of ℝ_{+}^{2} onto D normalized by θ̃(0) = 0. Then

θ̃(ξ)=\begin{cases}∑_{k=1}^{[2μ]} b^{(k)}ξ^{k}+b^{([2μ]+1)}ξ^{[2μ]+1} log |ξ|+b^{(±)}|ξ|^{2μ+1}+O(ξ^{2μ+1+γ}) & if 2μ ∈ ℕ, \\ ∑_{k=1}^{[2μ]+1} b^{(k)}ξ^{k} + b^{(±)}|ξ|^{2μ+1} + b^{([2μ]+2)}ξ^{[2μ]+2} + O(ξ^{2μ+1+γ}) & if 2μ ∉ ℕ, \end{cases}

The inverse mapping θ^{−1}(z) restricted to Γ_{±} has the form

ξ = ∑_{k=1}^{[2μ]}(±1)^{k}β^{(k)}x^{k/2}+(±1)^{[2μ]+1}β^{([2μ]+1)}x^{([2μ]+1)/2} log \frac{1}{x} +β^{(±)}x^{μ+1/2}+o (x^{μ+1/2})

if 2μ ∈ ℕ, and

ξ = ∑_{k=1}^{[2μ]+1}(±1)^{k}β^{(k)}x^{k/2} + β^{(±)}x^{μ+1/2} + o (x^{μ+1/2})  (21)

if 2μ ∉ ℕ. Here β^{(k)}, k = 1, … [2μ] + 1, are real coefficients. We notice that there exists a function of the form

d_{0}(ζ) = \{ \begin{matrix}ζ + ∑_{k=2}^{[2μ]} a^{(k)}ζ^{k}  if 2μ ∈ ℕ, \\ ζ + ∑_{k=2}^{[2μ]+1} a^{(k)}ζ^{k}  if 2μ ∉ ℕ, \end{matrix}

defined on ℝ_{+}^{2} and satisfying

(θ̃∘ d_{0})(ζ) = ζ + γ(ζ) .

Here γ(ζ) is the holomorphic function on a neighborhood of ζ = 0 in ℝ_{+}^{2} taking real values in a neighborhood of ξ = 0 in ℝ and having the representation

γ(ξ) = \begin{cases}b^{[2μ]+1}ξ^{[2μ]+1} log |ξ| + O(|ξ|^{2μ+1}) & if 2μ ∈ ℕ, \\ O(|ξ|^{2μ+1}) & if 2μ ∉ ℕ . \end{cases}

Then the function

d(ζ) = \begin{cases}d_{0}(ζ) + γ(ζ) & if 2μ ∈ ℕ, \\ d_{0}(ζ) & if 2μ ∉ ℕ \end{cases}

satisfies

(θ̃∘ d)(ξ) = ξ + O(|ξ|^{2μ+1}) .

It is clear that θ_{0} = (θ̃∘ d)^{2} is a conformal mapping of a neighborhood of ζ = 0 in ℝ_{+}^{2} onto a neighborhood of the peak in Ω and admitting the representation

x = Re θ_{0}(ξ) = ξ^{2} + O(|ξ|^{2μ+2}) as ξ → ±0 .  (22)

The inverse mapping θ_{0}^{−1} has the form

ξ = Re θ_{0}^{−1}(z) = ±x^{1/2} + O(x^{μ+1/2}) on Γ_{±} .

By diminishing δ in the definition of Γ_{±}, we can assume that θ_{0} is defined on Γ_{+} ∪ Γ_{−}.

(ii) Now, let φ ∈ 𝔑_{p,β}^{(+)}(Γ) vanish outside Γ_{+} ∪ Γ_{−}. We extend the function Φ(τ) = (φ ∘ θ_{0})(τ) by zero outside a small neighborhood of O.

We first prove the estimate

c_{1} ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)} ≤ ∥Φ^{(+)}∥_{L_{p,2β−2μ+1/p}(ℝ)}
+ ∥(dΦ/dτ)∥_{L_{p,2β+1+1/p}(ℝ)} + ∥Φ∥_{L_{p,2β+1/p}(ℝ)} ≤ c_{2} ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)}, (23)

where Φ^{(+)}(ξ) = (Φ(ξ) + Φ(−ξ))/2.

Let r be a measurable function on (0, ∞) subject to |r(ξ)| ≤ |ξ|^{2μ+1}. We choose ℓ ∈ [0, 1] such that

ℓ/2 < β + p^{−1} < (ℓ + 1)/2 .

Then, from the boundedness of the Hardy–Littlewood maximal operator in a weighted L_{p}-space (see [11]), we obtain

∫_{ℝ} |Φ(ξ) − Φ(ξ + r(ξ))|^{p}|ξ|^{2βp−2μp+1}dξ

≤ c ∫_{ℝ} (\frac{1}{|ξ|^{2μ+1}} ∫_{ξ−c|ξ|^{2μ+1}}^{ξ+c|ξ|^{2μ+1}} |τ^{1+ℓ}(dΦ/dτ)(τ)|dτ)^{p}|ξ|^{2βp−ℓp+1}dξ

≤ c ∫_{ℝ} |(dΦ/dτ)(ξ)|^{p}|ξ|^{2βp+p+1}dξ . (24)

For z ∈ Γ_{+} we have

|θ_{0}^{−1}(z) + θ_{0}^{−1}(z_{−})| ≤ c |ξ|^{2μ+1} .

Hence and by (24) the left inequality in (23) follows.

Let h be a measurable function on [0, δ] such that |h(x)| ≤ x^{μ+1}. Similarly to (24) we have

∫_{0}^{δ} |φ(x) − φ(x + h(x))|^{p}x^{(β−μ)p}dx

≤ c ∫_{0}^{δ} (\frac{1}{x^{μ+1}} ∫_{x−cx^{μ+1}}^{x+cx^{μ+1}}t |(dφ/dt)(t)|dt)^{p}x^{β p}dx

≤ c ∫_{0}^{δ} |(dφ/dt)(t)|^{p}x^{(β+1)p}dx . (25)

By using (22) we obtain that for ξ in a small neighborhood of the origin the distance between θ_{0}(ξ) and θ_{0}(−ξ) does not exceed c x^{μ+1}. Hence and by (25) the right inequality in (23) follows.

We introduce a function H by

H(ζ) = \frac{1}{π} ∫_{ℝ} \frac{dΦ}{dτ}(τ) Re log \frac{ζ − τ}{ζ} dτ, ζ = ξ + iη ∈ ℝ_{+}^{2} .

From the norm inequality for the Hilbert transform of even functions in the space L_{p,2β−1+p^{−1}}(ℝ) (see [13]) it follows that the function

(∂/∂ξ)H^{(+)}(ξ) = \frac{1}{πξ^{2}} ∫_{ℝ} \frac{d}{dτ} Φ^{(−)}(τ)\frac{τ^{2}dτ}{ξ − τ}

satisfies

∥(∂/∂ξ)H^{(+)}∥_{L_{p,2β+1+1/p}(ℝ)} ≤ c ∥(d/dξ)Φ^{(−)}∥_{L_{p,2β+1+1/p}(ℝ)} ,  (26)

where Φ^{(−)}(ξ) = (Φ(ξ) − Φ(−ξ))/2 and H^{(+)}(ξ) denote the function (H(ξ) + H(−ξ))/2.

Using again the norm inequality for the Hilbert transform of even functions in the space L_{p,2β+p^{−1}}(ℝ) (see [13]), we obtain that the function

(∂/∂ξ)H^{(−)}(ξ) = \frac{1}{πξ} ∫_{ℝ} \frac{d}{dτ} Φ^{(+)}(τ)\frac{τdτ}{ξ − τ}

satisfies

∥(∂/∂ξ)H^{(−)}∥_{L_{p,2β+1+1/p}(ℝ)} ≤ c ∥(d/dξ)Φ^{(+)}∥_{L_{p,2β+1+1/p}(ℝ)}  (27)

for 0 < β + \frac{1}{p} < \frac{1}{2}. Here Φ^{(+)}(ξ) = (Φ(ξ) + Φ(−ξ))/2 and H^{(−)}(ξ) = (H(ξ) − H(−ξ))/2.

Let ν be an even C_{0}^{∞}-function on ℝ with a small support, vanishing inside a neighborhood of ξ = 0 and subjected to ∫_{ℝ} ν(u)du = 1. We remark that ∫_{ℝ} Φ^{(+)}(τ)dτ is a linear continuous functional on 𝔑_{p,β}^{(+)}(Γ). We introduce the function φ_{0} := Φ^{(+)} ∘ θ^{−1} − (ν ∘ θ^{−1}) ∫_{ℝ} Φ^{(+)}τdτ and, for \frac{1}{2} < β + \frac{1}{p} < 1, we

consider the function φ − φ_{0} ∈ 𝔑_{p,β}^{(+)}(Γ) for which we keep the same notation φ. Then the function (∂/∂ξ)H^{(−)} can be represented in the form

(∂/∂ξ)H^{(−)}(ξ) = \frac{1}{πξ^{2}} ∫_{ℝ} \frac{d}{dτ} Φ^{(+)}(τ)\frac{τ^{2}dτ}{ξ − τ} .

From the norm inequality for the Hilbert transform of odd functions in L_{p,2β−1+p^{−1}}(ℝ) (see [13]) it follows that

∥(∂/∂ξ)H^{(−)}∥_{L_{p,2β+1+1/p}(ℝ)} ≤ c ∥(d/dξ)Φ^{(+)}∥_{L_{p,2β+1+1/p}(ℝ)} .  (28)

Let n_{0} be the integer subject to the inequalities

n_{0} − 1 ≤ 2(μ − β − p^{−1}) < n_{0} .

Then m = [μ − β − p^{−1} + 2^{−1}] is the largest integer satisfying 2m ≤ n_{0}. We represent the odd function H^{(−)} on ℝ in the form

H^{(−)}(ξ) = \frac{1}{π} ∫_{ℝ} \frac{d}{dτ} Φ^{(+)}(τ) log|\frac{ξ − τ}{ξ} |dτ

= \frac{ξ^{n_{0}}}{π} ∫_{ℝ} \frac{Φ^{(+)}(τ)}{τ^{n_{0}} (ξ − τ)}dτ − ∑_{k=0}^{n_{0}−1} \frac{ξ^{k}}{π} ∫_{ℝ} \frac{Φ^{(+)}(τ)}{τ^{k+1}}dτ .

Since Φ^{(+)} ∈ L_{p,2β−2μ+1/p}(ℝ) and since

0 < 2β − 2μ + n_{0} + 2p^{−1} < 1 for even n_{0}

and

0 < 2β − 2μ + n_{0} + 2p^{−1} < 2 for odd n_{0},

it follows from the boundedness of the Hilbert transform in weighted L_{p}-spaces (see [13]) that L_{p,2β−2μ+1/p}(ℝ)–norm of

\frac{ξ^{n_{0}}}{π} ∫_{ℝ} \frac{Φ^{(+)}(τ)}{τ^{n_{0}} (ξ − τ)}dτ

does not exceed ∥Φ^{(+)}∥_{L_{p,2β−2μ+1/p}(ℝ)} provided μ − β − p^{−1} + 2^{−1} ∉ N.

Hence by (26), (27) and (28) we obtain that the function h(z) = H ∘ θ_{0}^{−1}(z) can be represented in the form

∑_{k=1}^{m} a_{k}(φ)Rez^{k−1/2} + h^{#}(z)  (29)

for z ∈ Ω lying in a small neighborhood of the peak. Here

a_{k}(φ) = ∫_{ℝ} Φ^{(+)}(τ)τ^{−2k}dτ , 1 ≤ k ≤ m ,

are linear continuous functionals in 𝔑_{p,β}^{(+)}(Γ), and h^{#} belongs to 𝔑_{p,β}(Γ) and satisfies

∥h^{#}∥_{𝔑_{p,β}(Γ_{+}∪Γ_{−})} ≤ c ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)}.

Now let æ ∈ C^{∞}(ℝ^{2}) be equal to 1 for |z| < δ and vanish for |z| > δ. We extend æh by zero outside a small neighborhood of O and set

ψ_{1}(z) = −Δ(æh)(z), z ∈ Ω′,
φ_{1}(z) = (∂/∂s)φ(z) − (∂/∂n)(æh)(z), z ∈ Γ.

Since (∂/∂η)H = (∂/∂ξ)Φ ∈ L_{p,β+1}(ℝ), it follows that ∂h/∂n belongs to L_{p,β+1}(Γ_{+} ∪ Γ_{−}) We consider the boundary value problem

ΔF(ζ) = Q(ζ), ζ ∈ ℝ_{+}^{2}, (∂/∂n)F(ξ + i0) = T (ξ), ξ ∈ ℝ,  (30)

where Q(ζ) = (ψ_{1} ∘ θ)(ζ) |θ′(ζ)|^{2} and T (ξ) = (φ_{1} ∘ θ)(ξ + i0) |θ′(ξ + i0)|. For 0 < β + 1/p < 1/2, by using the estimates

|grad H(ζ)| ≤ \frac{c}{|ζ|}|∫_{ℝ} \frac{τ (dΦ/dτ)(τ)}{ζ − τ}dτ|, |H(ζ)| ≤ |∫_{ℝ} \frac{Φ(τ)}{ζ − τ} dτ| ,  (31)

and the theorems on the boundedness of the Hardy–Littlewood maximal operator and the Hilbert transform in weighted L_{p}-spaces [11], [14], we obtain

∥T ∥_{L_{p}(ℝ)} + ∥Q∥_{L_{p}(ℝ_{+}^{2})} ≤ c (∥dΦ/dτ∥_{L_{p,2β+1+1/p}(ℝ)} + ∥Φ∥_{L_{p,2β+1/p}(ℝ)}) . (32)

In order to prove (32) for 1/2 < β + 1/p < 1, we use (31) for Φ = Φ^{(−)} and the estimates

|grad H^{(−)}(ζ)|≤ \frac{c}{|ζ|^{2}} |∫_{ℝ} \frac{τ^{2} (dΦ^{(+)}/dτ)(τ)}{ζ − τ}dτ|, |H^{(−)}(ζ)|≤ \frac{1}{|ζ|}|∫_{ℝ} \frac{τ Φ^{(+)}(τ)}{ζ − τ}dτ| .

A solution of (30) is given by

F(ζ) = ∫_{ℝ} T (u)𝔊(u, ζ) du − ∫_{ℝ_{+}^{2}} Q(w)𝔊(w, ζ) du dv, w = u + iv,

with the Green function

𝔊(w, ζ) = \frac{1}{2π} log |(1 − \frac{w}{ζ}) (1 − \frac{w}{\overline{ζ}})| .

We rewrite F on ℝ in the form

F(ξ) = t_{−1}(φ) log |ξ| + t_{0}(φ)
+\frac{1}{π} ∫_{ℝ} T (u) log|1 − \frac{ξ}{u}|du − \frac{1}{π} ∫_{ℝ_{+}^{2}} Q(w) log|1 − \frac{ξ}{w}|dudv ,  (33)

where

t_{−1}(φ) = −\frac{1}{π} ∫_{ℝ} T (u)du + \frac{1}{π} ∫_{ℝ_{+}^{2}} Q(w)dudv,

and

t_{0}(φ) = \frac{1}{π} ∫_{ℝ} T (u) log |u|du − \frac{1}{π} ∫_{ℝ_{+}^{2}} Q(w) log |w|dudv .

Hence

\frac{∂F}{∂ξ} (ξ) − \frac{t_{−1}(φ)}{ξ} = \frac{1}{π} ∫_{ℝ} \frac{T (u)}{ξ − u}du − \frac{1}{π} ∫_{ℝ_{+}^{2}} \frac{(ξ − u)Q(w)}{|ξ − w|^{2}}dudv .

By the boundedness of the Hilbert transform in L_{p}(ℝ) and the Minkowski inequality we find that

(∂F/∂ξ)(ξ) − t_{−1}(φ) ξ^{−1} ∈ L_{p}(ℝ)  (34)

and that the L_{p}-norm of this function does not exceed

∥T ∥_{L_{p}(ℝ)} + ∥Q∥_{L_{p}(ℝ_{+}^{2})}.

It is clear that in a neighborhood of infinity

(∂F/∂ξ)(ξ) = R_{∞}(ξ) ξ^{−2},  (35)

where

|R_{∞}(ξ)| ≤ c (∥T ∥_{L_{p}(ℝ)} + ∥Q∥_{L_{p}(ℝ_{+}^{2})}) for large |ξ|.

Set f = F ∘ θ. From (34) and (35) it follows that ∂f/∂s belongs to L_{p,β+1}(Γ) and satisfies

∥∂f/∂s∥_{L_{p,β+1}(Γ)} ≤ c ∥φ∥_{L_{p,β+1}^{1}(Γ)} .  (36)

By the Taylor decomposition of the integral terms in (33) we obtain

F(ξ) = t_{−1}(φ) log |ξ| + t_{0}(φ) + ∑_{k=1}^{n_{0}−1} t_{k}(φ)ξ^{k} + |ξ|^{n_{0}}R_{n_{0}}(ξ) ,  (37)

where

|t_{k}(φ)| ≤ c (∥T ∥_{L_{p}(ℝ)} + ∥Q∥_{L_{p}(ℝ_{+}^{2})}), k = −1, …, n_{0} − 1,

and

|R_{n_{0}}(ξ)| ≤ c (∥T ∥_{L_{p}(ℝ)} + ∥Q∥_{L_{p}(ℝ_{+}^{2})})

for small |ξ|. Taking into account the asymptotic representations (20), (21) of θ^{−1} and the inequality 2(μ − β − p^{−1}) < n_{0}, it follows from (36) and (37) that f is represented in the form

f(z) = ∑_{k=1}^{m} b_{k}(φ)Re z^{k−1/2} + f^{#}(z), z ∈ Ω ,  (38)

where f^{#} ∈ 𝔑_{p,β}(Γ), and b_{k}(φ), k = 1, …, m, are linear combinations of the coefficients t_{ℓ}(φ) in (37).

According to (29) and (38), the function g = æh + f is harmonic in Ω and can be written as

g(z) = ∑_{k=1}^{m} c_{k}(φ)Re z^{k−1/2} + g^{#}(z), z ∈ Ω′,

with c_{k}(φ) = a_{k}(φ) + b_{k}(φ). Moreover,

∑_{k=1}^{m} |c^{(k)}| + ∥g^{#}∥_{𝔑_{p,β}(Γ)} ≤ c ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)}

and (g ∘ θ)(∞) = 0 by the definition of g. Owing to (∂/∂s)g ∈ L_{p,β+1}(Γ), one of the functions conjugate to −g is a harmonic extension of φ onto Ω with the normal derivative in L_{p,β+1}(Γ).

(iii) Now let φ belong to 𝔑_{p,β}^{(+)}(Γ) and let φ vanish on Γ ∩ {|q| < δ/2}. We introduce the function

Φ(ξ) = (φ ∘ θ)(1/ξ), ξ ∈ ℝ,

which vanishes outside a certain interval. Set

G(ζ) = \frac{1}{π} P.V. ∫_{−∞}^{∞} Φ(τ)Re \frac{ζ}{τ(ζ − τ)}dτ, ζ ∈ ℝ_{+}^{2}.  (39)

It is clear that one of the conjugate functions G̃ is a harmonic extension of −Φ onto ℝ_{+}^{2}. It follows from the boundedness of the Hilbert transform in L_{p}-spaces that g = G(1/θ^{−1}) belongs to L_{p,β+1}^{1}(Γ) and satisfies

∥ g ∥_{L_{p,β+1}^{1}(Γ)}≤ c ∥ φ ∥_{𝔑_{p,β}^{(+)}(Γ)} .  (40)

Further, we represent G on ℝ in the form

\begin{matrix}G(ξ) = \frac{1}{π} P.V. ∫_{ℝ} & Φ(τ)τ^{−1}dτ + ∑_{k=1}^{n_{0}−1} \frac{1}{πξ^{k}} ∫_{ℝ} Φ(τ)τ^{k−1}dτ \\ + & \frac{1}{πξ^{n_{0}−1}} ∫_{ℝ} \frac{Φ(τ) τ^{n_{0}−1}}{ξ − τ} dτ = ∑_{k=0}^{n_{0}−1} t_{k}(φ)ξ^{k} + G^{#}(ξ), \end{matrix}

where

t_{k}(φ) = \frac{1}{π} ∫_{ℝ} Φ(τ)τ^{k−1}dτ and G^{#}(ξ) = \frac{1}{πξ^{n_{0}−1}} ∫_{ℝ} \frac{Φ(τ) τ^{n_{0}−1}}{ξ − τ} dτ, ξ ∈ ℝ. (41)

Since −1/p < 2μ − 2β − 3/p − n_{0} + 1 < 1 − 1/p, we have

∑_{k=0}^{n_{0}−1} |t_{k}(φ)| + ∥G^{#}∥_{L_{p,2μ−2β−3/p}(ℝ)} ≤ c ∥Φ∥_{L_{p,2μ−2β−3/p}(ℝ)} .

Hence it follows from (40) that g is represented in the form

g(z) = ∑_{k=1}^{m} c_{k}(φ)Re z^{k−1/2} + g^{#}(z), z ∈ Ω,

where g^{#} ∈ 𝔑_{p,β}(Γ), and c_{k}(φ), k = 1, …, m, are linear combinations of the coefficients t_{ℓ}(φ), ℓ = 1, …, n_{0} − 1 in (41). These coefficients and the function g^{#} satisfy

∑_{k=1}^{m} |c_{k}(φ)| + ∥g^{#}∥_{𝔑_{p,β}(Γ)} ≤ c ∥φ∥_{𝔑_{p,β}^{(+)}(Γ)}

and the conjugate function g̃ = G̃(1/θ^{−1}) is the harmonic extension of φ onto Ω with the normal derivative in L_{p,β+1}(Γ). □

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(Received 17.02.2003)

Author’s addresses:

V. Maz’ya Department of Mathematics Linköping University S-581 83 Linköping Sweden E-mail: vlmaz@math.liu.se

A. Soloviev Department of Mathematics Chelyabinsk State University 454016 Chelyabinsk Russia E-mail: alsol@csu.ru