Henkin measures, Riesz products and singular sets
The principal objects of this paper are measures defined on the complex sphere \(S=\{\zeta\in{\mathbb{C}}^n:|\zeta| =1\}, n\geq 2.\) For \((p, q)\in{\mathbb{Z}}^2_+\), \(H(p, q)\) is the vector space of all harmonic homogeneous polynomials in \(S\) of total degree \(p+q,\) of degree \(p\) in \(z_1,\dots,z_n,\) and of degree \(q\) in \(\overline z_,\dots,\overline z_n.\) The author develops the spectral function theory on \(S\) in terms of \(H(p, q).\) If \(M(S)\) is the set of all regular complex Borel measures on \(S,\) then the spectrum of \(\mu\in M(S)\) is \[ \text{spec}(\mu)=\Biggl\{(p, q)\in{\mathbb{Z}}^2_+: \mu_{pq}(z)=\int _SK_{pq}(z,\zeta)d\mu (\zeta)\neq 0,\;z\in S\Biggr\}, \] where \(K_{pq}(z, \zeta)\) is the reproducing kernel for \(H(p, q)\subset L^2(S).\) Given a set (a spectrum) \(\Lambda\subset{\mathbb{Z}}^2_+,\) the general problem is to investigate the properties of the space \(M_{\Lambda}(S)=\{\mu\in M(S):\text{spec}(\mu)\subset\Lambda\}.\) For \(d\in{\mathbb{Z}}^2_+,\) the author defines \(\Lambda (d)=\{(p, q)\in{\mathbb{Z}}^2_+: (p-d)(q-d)=0\), \(p\geq d\), \(q\geq d\}.\) In complex analysis, \(\text{spec}(\mu)\subset\Lambda =\Lambda (0)\) iff the Poisson integral \(P[\mu ]\) is a pluriharmonic function. By analogy, if \(\text{spec}(\mu)\subset\Lambda (d)\) or \(\text{spec}(\mu)\subset\Lambda (d)\cup\{(0, 0)\},\) he calls \(\mu \) \(d\)-pluriharmonic. He proved in [Zap. Nauchn. Semin. POMI 217, 54-58 (Russian) (1994); English transl. in J. Math. Sci., New York 85, No. 2, 1790-1793 (1997; Zbl 0907.31007)] that \(\Lambda (0)\) is singular. In the paper under review, he establishes (Theorem A) that also ``the set \(\Lambda (d)\) is singular \(\forall d\in{\mathbb{Z}}^2_+\)''. The classical Riesz product on the unit circle \({\mathcal T}\) is \[ \mu:=\prod_{k=1}^{\infty} \Biggl({{\overline a_k\overline z^{jk}}\over{2}}+1+{{a_kz^{jk}}\over{2}} \Biggr), \quad z\in{\mathcal T},\quad | a_k|\leq 1,\quad {{j_{k+1}}\over{j_k}}\geq 3. \] The author introduces a corresponding \(d\)-pluriharmonic Riesz product representing also an example of probability singular \(d\)-pluriharmonic measure. This Riesz product idea yields the following important existence result: (Theorem B) ``There exists a probability singular \(d\)-pluriharmonic measure \(\mu \) such that the slice \(\mu _{\zeta}\) lives on sets of Hausdorff dimension \(1\;\forall\xi\in{\mathbb{C}}P^{n-1}\), where \({\mathbb{C}}P^{n-1}\) is the projective space.'' Analogously to the classical case, he introduces the following sets: ``If \(X\subset C(S)\) is a closed subspace and \(K\subset S\) is compact, then \(K\) is an \((N)\)-set (null set) for \(X\) if \(|\mu| (K)=0\;\forall\mu\in X^{\perp}=\{\mu\in M(S): \int _Sfd\mu =0\) \(\forall f\in X\}.\) \(K\) is a \((PI)\)-set (peak interpolation set) if given a \(g\in C(K)\), \(g\neq 0\), there exists \(f\in X(B)\) with \(f_{| K}=g\) and \(| f(z)| <| f(z)| <\| g\| _K\) for \(z\in\overline B\backslash K.\) \(K\) is a \((P)\)-set (peak set) if the preceding property holds for \(g\equiv 1.\) \(K\) is an \((I)\)-set (interpolation set) if \(X_{| K}=C(K)\).'' In the classical case of the ball algebra \(A(S)=\{ f\in C(S): P[f]\) is a holomorphic function\}, the above properties of \(K\) are known to be equivalent. The author obtains for the spaces \(A_{k\ell}(S)=\{f\in C(S): \text{spec}(f)\subset\{(p, q)\in{\mathbb{Z}}^2_+: k\geq q\geq \ell\}, k, \ell\in{\mathbb{Z}_+}, k\geq\ell\}\) the equivalence between \((N)\) and the strong peak set \((SP),\) where \(K\subset (PS)\) if there exists a sequence \(\{ f_j\}_{j=1}^{\infty}\subset X(B)\) such that \(f_{j| K}=1, | f| <1\) on \(B\) and \(1>| f_j| \rightarrow 0\) on \(S\backslash K,\) while in the particular case \(A_{k0}(S)\), \((N) \Leftrightarrow (PI) \Leftrightarrow (I) \Leftrightarrow (SP).\) He observes also that if \(k\in{\mathbb{Z}}_+\) and \(M\subset S\) is a real submanifold of class \(C^{k+2}\), then every compact set \(K\subset M\) is an \((N)\)-set [\((PI), (I),\) or \((SP)\)-set] for \(A_{k0}\) iff \(M\) is complex tangential, i.e. the real tangent space to \(M\) at \(\zeta \) is in the complex tangential space to \(S\) at \(\zeta\) \(\forall\zeta\in M\).
scientific article; zbMATH DE number 1199201
scientific article