Let \(\Pi =\{z=x+iy\in \mathbb {C}:\;y>0\} \) be the upper half-plane and the interval [a, b] be a subset of \( \partial \Pi =\mathbb {R}\). We derive a Carleman integral representation formula for all holomorphic functions \(f\in {\mathcal H}(\Pi )\) that have angular boundary values on [a, b] and which belong to the class \(\mathcal { N H}^1_{[a,b]}(\Pi )\). The class \(\mathcal {NH}^1_{[a,b]}(\Pi )\) is the class of holomorphic functions in \(\Pi \) which belong to the Hardy class \({\mathcal H}^1\)near the interval [a, b] (“The Class of Functions Representable by Carleman Integral Representation Formula” section). As an application of the above characterization, our main result is an extension theorem for a function \(f\in L^1([a,b])\) to a function \(f\in \mathcal {NH}^1_{[c,d]}(\Pi )\), for almost all intervals \([c,d]\subset (a,b)\). Similar results can be proved for a function f holomorphic in a slanted disc, with integrable boundary values on the horizontal part of the boundary.

中文翻译：

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用Carleman公式表示的上半平面的全纯函数

令\（\ Pi = \ {z = x + iy \ in \ mathbb {C}：\; y> 0 \} \）是上半平面，间隔[ a，  b ]是\（ \ partial \ Pi = \ mathbb {R} \）。我们为所有全纯函数\（f \ in {\ mathcal H}（\ Pi）\）导出一个Carleman积分表示公式，这些函数在[ a，b ]上具有角边界值， 并且属于类\（\ mathcal {NH } ^ 1 _ {[a，b]}（\ Pi} \）。类\（\ mathcal {NH} ^ 1 _ {[A，B]}（\ PI）\）是在类的全纯函数\（\裨\），其属于哈迪类\（{\ mathcal H}区间[ a附近的^ 1 \），  b ]（“由Carleman积分表示公式表示的函数类别”一节）。作为上述表征的应用，我们的主要结果是函数\（f \ in L ^ 1（[a，b]）\）到函数\（f \ in \ math {NH} ^ 1_的扩展定理{[c，d]}（\ Pi）\），几乎在所有间隔\（[c，d] \ subset（a，b）\）中。类似的结果可以证明为函数˚F在倾斜圆盘全纯，与在边界的水平部分积的边界值。