Boundary values of analytic functions

Let $D$ be a connected bounded domain in $\R^2$, $S$ be its boundary which is closed, connected and smooth. Let $\Phi(z)=\frac 1 {2\pi i}\int_S\frac{f(s)ds}{s-z}$, $f\in L^1(S)$, $z=x+iy$. Boundary values of $\Phi(z)$ on $S$ are studied. The function $\Phi(t)$, $t\in S$, is defined in a new way. Necessary and sufficient conditions are given for $f\in L^1(S)$ to be boundary value of an analytic in $D$ function. The Sokhotsky-Plemelj formulas are derived for $f\in L^1(S)$.


Introduction
Let D be a connected bounded domain on the complex plane, S be its boundary, which is closed and C 1,a -smooth, 0 < a ≤ 1. Consider an analytic function in D, defined as We assume that f ∈ L 1 (S). This is the basic new assumption: in the literature it was assumed that f is Hölder-continuous or f ∈ L p (S), p > 1, see [2], [7]. In [1] there is a result for f ∈ L 1 (S), which we mention in Remark 3. It is of great interest to have a proof of the Sokhotsky-Plemelj formulas for f ∈ L 1 (S) and of the the relation (9) for f ∈ L 1 (S), see below. The contents of Chapter 2 in [2] is based on the relation (9), which is proved in [2] under the assumption that f is Hölder-continuous, f ∈ H µ (S), µ ∈ (0, 1]. We prove that this relation holds almost everywhere (a.e.) in the sense of the Lebesgue measure on S. The Sokhotsky-Plemelj formulas for f ∈ L 1 (S) are important in applications to singular integral operators and boundary value problems, [2], [8].
By D + we denote D, by D − = D ′ we denote R 2 \D, theD is the closure of D. The function Φ is analytic in D and in D ′ , Φ(∞) = 0. Boundary value of Φ on S has been studied by many authors, [1]- [9]. In [2] and [8] it is assumed that f ∈ H µ (S). In [7] and in [6] it is assumed that f ∈ L p (S), p > 1. The case p = 1 is not discussed in these books. In [1] there are results related to p = 1 related to the Cauchy principal value definition of Φ(t). We give a new definition of the singular integral Φ(t), see (12).
Define ψ(z) by formula (2). Denote the limiting values of Φ(z) and ψ(z), when z → t ∈ S along the non-tangential to S directions, by Φ + (t) and ψ + (t) when z ∈ D, z → t, and by Φ − (t) and ψ − (t) when z ∈ D ′ , z → t. By Φ(t) and ψ(t) we denote the values of Φ(z) and ψ(z) when z = t ∈ S. Let us assume that z → t along the unit normal N t to S, where the normal is directed out of D.
We are interested in the following question: How to characterize functions f ∈ L 1 (S) which are boundary values of analytic functions in D?
We answer this question in Theorem 1 (see below). Rewrite (1) as and One has Φ + (t) = lim z→t,z∈D where ψ + (t) = lim z→t,z∈D ψ(z) and If t ∈ S, then one gets (see equation (3), the line z ∈ S) : The ψ(t) is the value of ψ(z) at z = t. The ψ(t) and Φ(t) are understood as in Remark 3, see below.
If some equation holds almost everywhere with respect to the Lebesgue measure on S, then we write that this equation holds a.e.
Any f ∈ L 1 (S) is uniquely identified with the linear functional in L 1 (S) defined as follows: It is known, see [2], p. 52, that Bφ := 1 iπ S φ(s)ds s−t ∈ H µ (S), provided that φ ∈ H µ (S) and µ ∈ (0, 1). Moreover, the range of B is equal to H µ (S) when φ runs through H µ (S), see [2], p. 178. This follows from the inversion formulas: if φ, ψ ∈ H µ (S) and Bφ = ψ, then φ = Bψ, see [8], p. 115, so B 2 = I, the identity operator. Therefore, the right side of (12) defines f ∈ L 1 (S) uniquely. The integrals on the left and right sides of (12) are equal to the integral which is absolutely convergent on S × S. We have proved formula (10) with the minus sign. Similarly one proves this formula with the plus sign. Lemma 2 is proved. ✷ One can interpret formula (11) as follows: where the first integral on the right is understood as in formula (12), see also Remark 3 below, and f (t) is defined a.e: almost everywhere on S with respect to the Lebesgue measure on S.
Usually the delta-function is defined as a linear continuous functional on the space of con- In place of f (t) one has to put f (t) a.e. because f ∈ L 1 (S) is defined a.e. with respect to the Lebesgue measure on S. By Lusin's theorem, see [10], p. 157, if f ∈ L 1 (S), then f is continuous on a subset S ǫ of S, |measS − measS ǫ | < ǫ, where ǫ > 0 can be chosen arbitrarily small and measS is the Lebesgue measure of the set S. One can consider in this setting the delta-function δ(s − t) as a kernel of the identity operator in L 1 (S).
In [4], p. 83, there is a formula 1 understood in the sense of distributions. The formula in Lemma 2 is of the similar type.
From Lemma 1 and formula (7) one derives the Sokhotsky-Plemelj formulas for f ∈ L 1 (S), see Lemma 3. These formulas are derived in [2] and [8] under the assumption that f is Höldercontinuous. Under such an assumption, these formulas hold everywhere, not almost everywhere.
Lemma 3. If f ∈ L 1 (S), then the Sokhotsky-Plemelj formulas hold: Proof of Lemma 3. Formulas (14) follow from formulas (4)- (9). Indeed, by formulas (7) and (9) The following question is of interest: When is the boundary value Φ + (t) of Φ(z) on S equal to f a.e.? From formula (4) the answer follows immediately: Equation (14) yields another necessary and sufficient condition: If equation (15)  If one wants to formulate a necessary and sufficient condition for f (s) ∈ L 1 (S) to be a boundary value of an analytic in D ′ function Φ(z), Φ(∞) = 0, then an argument, similar to the above yields the following conditions: If equation (16) holds, then Φ + (t) = 0 and, consequently, Φ(z) = 0 if z ∈ D. Remark 1. Integral equation in formula (15) has infinitely many linearly independent solutions: every boundary value of an analytic function Φ(z) in D such that Φ(z) = 0 in D ′ solves this integral equation. .e., f ∈ L 1 (S) and Φ(∞) = 0, then Bf ∈ L 1 (S). Since for some f ∈ L 1 (S) one does not have Φ(z) = 0, z ∈ D ′ , it follows that not every f ∈ L 1 (S) is a boundary value of an analytic function in D.
The singular integral equation can be solved uniquely and explicitly for any h ∈ H µ (S): and its solution belongs to H µ (S). This is proved in [2], p. 66. Consequently, any h ∈ H µ (S), µ ∈ (0, 1), can be represented by the integral (18) with φ ∈ H µ (S). This is in contrast to the integral equation (18) with φ ∈ L 1 (S): not for every such φ the h belongs to L 1 (S), see Remark 5 below.
In [1] it is proved that ds, ∀φ ∈ H µ (S). The limit of the right side, as n → 0, exists, so the limit of the left side exists and is equal to that of the right side. This leads to the formula (20) below. An alternative derivation of this formula goes as follows. Since φ(t)f (s) ∈ L 1 (S × S), the operator S×S φ(t)f (s)|s − t| −1 dsdt is a bounded operator in L 1 (S × S), see [5], p.159. Consequently, the change of the order of integration in the double integral: is legitimate. Let us summarize the results: Theorem 1. A necessary and sufficient condition for f ∈ L 1 (S) to be the boundary value of an analytic in D function Φ(z) is equation (15).
A necessary and sufficient condition for f ∈ L 1 (S) to be the boundary value of an analytic in D ′ function Φ(z), Φ(∞) = 0, is equation (16).

Proof of Lemma 1.
Write formula (2) for ψ as Using Lemma 2, one obtains: Let us explain (23). The union of the discontinuity points of a summable function f and the points where the function f is not defined has measure zero. The L 1 (S) norm of a summable function is not changed if this function is changed on the set of measure zero. This yields the conclusion that equation (23) holds.
Usually the distribution δ(s − t) is defined as a linear bounded functional on the space of continuous functions with the sup-norm. On the other hand, one can define δ(s − t) as the kernel of the identity operator I in the space L p (S), p ≥ 1. In this case, the relation S f (s)δ(s − t)ds = f (t) holds a.e. The expression f (t) for f ∈ L p (S), p ≥ 1, is not defined at every point t, but is defined almost everywhere. The set of points at which f is discontinuous, or is not defined (including the points at which f = ∞) is of measure zero. Outside of this set, that is, a.e., one has where I is the identity operator with the kernel δ(s − t). Formula (24) gives another way to see that formula (23) is valid. Since in the general theory of distributions the product of two distributions is not defined (including the case when one of the distributions is a function, belonging to L p (S), p ≥ 1), formula (24) is useful for f ∈ L p (S), p ≥ 1. Finally, recall that for a continuous function f on S equation (23) is obvious. If f ∈ L 1 (S) then the Luzin's theorem, see, for example, [10], p.157, says that for any f ∈ L 1 (S), where S is bounded, and for an arbitrary small ǫ > 0, there exists a closed subset Q of S, measQ > measS − ǫ, on which f is continuous. Since ǫ > 0 is arbitrary small, equation (23) holds.
Lemma 1 is proved. ✷ Remark 4. Let S = S 0 be the unit circle on the complex plane and For an arbitrary f ∈ L 1 (S 0 ) the B 0 f may be more singular than f . Example 1. Let see [3], p.52. The function (26) is a bounded function on the interval 0 ≤ φ ≤ 2π. One can check, that (27) see [3], p.52. The validity of the change of the order of application of B 0 and the summation with respect to n follows from the convergence of the series ∞ n=1 cos nφ n . We have used the formulas, see [7], p. 47, B 0 e inφ = e inφ , n > 0; B 0 e −inφ = −e −inφ , n > 0.
Thus, B 0 f has a singularity at φ = 0 on the interval 0 ≤ φ ≤ 2π, while f is a bounded function.
Remark 5. One can check that if f ∈ L 1 (R) and Sf := 1 2πi R f (s)ds s−t , then for some f the Sf is more singular than L 1 (R). Indeed, consider Sf as a convolution in the sense of distributions.
Take the Fourier transform of Sf and use the formula F (f * 1 s ) = F f (iπsignξ), where F f = R e iξs f (s)ds, * denotes the convolution: f * h := R f (s)h(t − s)ds, F f * h = F (f )F (h). One has F 1 s = iπsignξ, see [4]. Since the set F f is the set of continuous functions of ξ when f runs through L 1 (R), the set of functions {F f × signξ} is not the set of continuous functions of ξ. Therefore, Sf does not belong to L 1 (R) for some f ∈ L 1 (R).

Conclusion
Let D be a connected bounded domain in R 2 , S be its boundary, which is closed, connected and smooth. Let Φ = Φ(z), z = x + iy, be an analytic function in D.
A necessary and sufficient condition for a function f = f (s) ∈ L 1 (S) to be a boundary value of an analytic in D function Φ(z) is equation (15).
A necessary and sufficient condition for a function f = f (s) ∈ L 1 (S) to be a boundary value of an analytic in D ′ function Φ(z), Φ(∞) = 0, is equation (16).