On boundedness of Bergman projection operators in Banach spaces of holomorphic functions in half-plane and harmonic functions in half-space
Let \(A^p(\mathbb{D})\) denote the Bergman space on the unit disk \(\mathbb{D}\); the space of holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^p dA(z) \] is finite; here \(dA(z)=dxdy\) is the usual area measure. It is well known that for a finite \(p>1\), the Bergman projection from \(L^p(\mathbb{D})\) to \(A^p(\mathbb{D})\) is bounded. There has been a lot of interest to extend this result to regions other than the unit disk, as well as to the variable exponent Bergman space. This latter space consists of holomorphic functions in a given domain \(\Omega\) for which the integral \[ \int_{\Omega}|f(z)|^{p(z)} dxdy \] is finite. In the paper under review, the authors are mainly concerned with the upper half-space \(\mathbb{R}^n_+\) for \(n\geq 2\), and the upper half-plane when \(n=2\). For \(f\in L^1(\mathbb{R}^2_+)\), the Bergman projection is given by \[ Bf(z)=-\frac{1}{\pi}\int_{\mathbb{R}^2_+}\frac{f(w)}{(z-\overline w)^2}dA(w),\quad z\in \mathbb{R}^2_+, \] where \[ K(z,w)=\frac{-1}{\pi(z-\overline w)^2} \] is the Bergman kernel function for the upper half-plane. It is also known that the Bergman projection operator \(B:L^p(\mathbb{R}^2_+) \to A^p(\mathbb{R}^2_+)\) is bounded if \(p\in (1, \infty)\). In a similar fashion, one can define the harmonic Bergman space on the upper half-space. For \(n\geq 2\), this is the space \(h^p(\mathbb{R}^n_+)\) consisting of all harmonic functions in \(\mathbb{R}^n_+\) that belong to \(L^p(\mathbb{R}^n_+, dA(x))\) where \(dA(x)\) is the usual volume measure in \(\mathbb{R}^n_+\). It is known that the Bergman projection \[ bf(x)=\int_{\mathbb{R}^n_+}R(x,y)f(y)dA(y),\quad x\in \mathbb{R}^n_+, \] from \(L^p(\mathbb{R}^n_+)\) to \(h^p(\mathbb{R}^n_+)\) is bounded; here \(R(x,y)\) stands for the corresponding kernel function. The main purpose of the current paper is to extend the aforementioned results both to the variable exponent Bergman space on the upper half-plane, and to the variable exponent harmonic Bergman space on the upper half-space. A variable exponent is a measurable function \(p(\cdot):\mathbb{R}^n_+\to [1,\infty)\) such that \(1\leq p^-\leq p^+<\infty\) where the minus and plus superscripts denote the essential infimum and the essential supremum of \(p(\cdot)\) on \(\mathbb{R}^n_+\), respectively. We recall that \(L^{p(\cdot)}(\mathbb{R}^n_+)\) is the set of measurable functions \(f\) on \(\mathbb{R}^n_+\) such that \(m_{p(\cdot)}(f/\lambda)<\infty\) for some \(\lambda>0\), where \[ m_{p(\cdot)}(f)=\int_{\mathbb{R}^n_+}|f(x)|^{p(x)}dA(x). \] Subsequently, we define the norm on \(L^{p(\cdot)}(\mathbb{R}^n_+)\) by \[ \| f\|_{L^{p(\cdot)}(\mathbb{R}^n_+)}=\inf \left\{ \lambda >0 : m_{p(\cdot)}\Big(\frac{f}{\lambda}\Big)\leq 1\right \}. \] In the paper under review, it is proved that (Theorem 3.1), if the variable exponent \(p(\cdot)\) satisfies some conditions, then the Bergman projection from \(L^{p(\cdot)}(\mathbb{R}^2_+)\) to \(A^{p(\cdot)}(\mathbb{R}^2_+)\) is bounded. Moreover, some growth estimates for the functions in \(A^{p(\cdot)}(\mathbb{R}^2_+)\) are obtained (Theorem 4.1). Under similar conditions, it is shown that the harmonic Bergman projection from \(L^{p(\cdot)}(\mathbb{R}^n_+)\) to \(h^{p(\cdot)}(\mathbb{R}^n_+)\) is bounded as well. Similar estimates for functions in \(A^{p(\cdot)}(\mathbb{R}^n_+)\) are obtained. In a parallel fashion, the same results are established for the Orlicz space, as well as for the variable exponent generalized Morrey space.
scientific article; zbMATH DE number 6822963
scientific article