Let \(\Omega \) be a \(\sigma \)-finite measurable space. Suppose that (X, Y) is a couple of quasi-Banach lattices of measurable functions on \({\mathbb {T}} \times \Omega \) satisfying some additional assumptions. The Hardy-type spaces \(X_A\) consist of functions on \({\mathbb {D}} \times \Omega \) belonging to the Smirnov class \(\mathrm {N}^+\) in the first variable such that their boundary values are in X. Here \({\mathbb {T}}\) is the unit circle and \({\mathbb {D}}\) is the open unit disc of the complex plane. Couple \((X_A, Y_A)\) is said to be K-closed in (X, Y) with constant C if for any \(f \in X\), \(g \in Y\) such that \(H = f + g \in X_A + Y_A\) there exist some \(F \in X_A\), \(G \in Y_A\) satisfying \(H = F + G\), \(\Vert F\Vert _X \leqslant C \Vert f\Vert _X\) and \(\Vert G\Vert _Y \leqslant C \Vert g\Vert _Y\). This property is shown to be equivalent to the stability of the real interpolation \((X_A, Y_A)_{\theta , p} = (X_A + Y_A) \cap (X, Y)_{\theta , p}\) and to the \(\mathrm {BMO}\)-regularity of the associated lattices \(\left( \mathrm {L}_{1}, \left( X^r\right) ' Y^r\right) _{\delta , q}\) under fairly broad assumptions. The inclusion \(\left( X^{1 - \theta } Y^\theta \right) _A \subset \left( X_A, Y_A\right) _{\theta , \infty }\) is also characterized in these therms. New examples of couples \((X_A, Y_A)\) with this stability are given, proving that this property is strictly weaker than the usual \(\mathrm {BMO}\)-regularity of (X, Y).

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Hardy型空间和BMO正则性的实插值

令 \（\ Omega \）为\（\ sigma \）有限的可测量空间。假设（X，  Y）是\（{\ mathbb {T}} \ times \ Omega \）上的几个可测量函数的准Banach格， 满足一些附加假设。Hardy类型的空间 \（X_A \）由属于Smirnov类 \（\ mathrm {N} ^ + \）的\（{\ mathbb {D}} \ times \ Omega \）上的函数组成， 例如第一个变量，例如它们的边界值在 X中。这里 \（{\ mathbb {T}} \）是单位圆，  \（{\ mathbb {D}} \）是复平面的开放单位圆盘。情侣 \（（X_A，Y_A）\）被说成是ķ在（ -封闭X，  ÿ）具有恒定 Ç如果由于任何 \（F \在X \） ，\（克\在Y \） ，使得 在X_A + Y_A \（H = F + G \ \）存在一些 \（F \ X_A \），\（G \ Y_A \）满足 \（H = F + G \），\（\ Vert F \ Vert _X \ leqslant C \ Vert f \ Vert _X \）和 \（\ Vert G \ Vert _Y \ leqslant C \ Vert g \ Vert _Y \）。已显示此属性等效于实插值 \（（X_A，Y_A）_ {\ theta，p} =（X_A + Y_A）\ cap（X，Y）_ {\ theta，p} \）的稳定性并移至\（\ mathrm {BMO} \）相当宽泛的假设下，关联格 \（\ left（\ mathrm {L} _ {1}，\ left（X ^ r \ right）'Y ^ r \ right）_ {\ delta，q} \）的正则性。列入 \（\左（X ^ {1 - \ THETA} Y 1 \ THETA \右）_A \子集\左（X_A，Y_A \右）_ {\ THETA，\ infty} \）的特征还在于这些克卡。给出了具有这种稳定性的偶对（（（X_A，Y_A）\））的新示例，证明了此属性严格比（X，  Y）的常规\（\ mathrm {BMO} \） -正则性弱。