Regularity of some nonlinear quantities on superharmonic functions in local Herz-type Hardy spaces
Let \(B_k=\{x\in \mathbb R^n; |x|\leq 2^k\}\) and \(C_k=B_k \setminus B_{k-1}\) for \(k\in \mathbb Z\). Let \(\chi_k\) denote the characteristic function of the set \(C_k\). The homogeneous Herz space \(\dot K_q^{\alpha, p}(\mathbb R^n)\) is defined in terms of \(\|f\|_{\dot K_q^{\alpha, p}(\mathbb R^n)}\) \(=\bigl\{\sum_{k=-\infty}^\infty 2^{k\alpha p}\|f\chi_{k}\|_{L^q(\mathbb R^n)}^p\bigr\}^{1/p}\) by letting \(\dot K_q^{\alpha, p}(\mathbb R^n) =\{f\in L_{\text{loc}}^q(\mathbb R^n\setminus \{0\}); \|f\|_{\dot K_q^{\alpha, p}(\mathbb R^n)}<\infty\}\). \(f\in L_{\text{loc}}^q(\mathbb R^n\setminus \{0\})\) is said to belong to the local Herz space \(\dot K_{q,\text{loc}}^{\alpha, p}(\mathbb R^n\setminus \{0\})\), if for every \(\phi\in C_c^\infty(\mathbb R^n)\), \(\phi f\in \dot K_q^{\alpha, p}(\mathbb R^n)\). Next, fix a smooth function \(\eta(x)\) satisfying \(\text{supp }\eta \subset B(0,1)\) and \(\int \eta(x) dx=1\). Set \(f^{\ast\ast}(x)= \sup_{0<r\leq 1}\bigl|r^{-n}\int_{\mathbb R^n}f(y)\eta((x-y)/r)dy\bigr|\). The local Herz-type Hardy space \(h\dot K_q^{n(1-1/q), 1}(\mathbb R^n)\) is defined as the set of all \(f\in L^1(\mathbb R^n)\) satisfying \(f^{\ast\ast}\in \dot K_q^{n(1-1/q),1}(\mathbb R^n)\). In the case \(q=1\), this is the local Hardy space \(h^1(\mathbb R^n)\) in the sense of D. Goldberg. The authors give the following: Let \(1<q<\infty\), \(u\) and its distributional first partial derivatives belong to \(\dot K_{2q,\text{loc}}^{2(1/2-1/(2q)), 2}(\mathbb R^2)\) (in the case \(q=1\), this means that \(u\) belongs to the local Sobolev space \(H_{\text{loc}}^1(\mathbb R^2))\) and \(u\) be a weak solution of the partial differential equation \(-\Delta u=w\) in \(\mathbb R^2\), where \(w\) is a nonnegative function in \(\dot K_{q,\text{loc}}^{2(1-1/q), 1}(\mathbb R^2)\). Then, it holds that \(u_{x_1}u_{x_2}\) and \(u_{x_1}^2 - u_{x_2}^2\) belong locally to \(h\dot K_{q}^{2(1-1/q), 1}(\mathbb R^2)\), i.e., for every \(\phi\in C_c^\infty(\mathbb R^n)\), \(\phi u_{x_1}u_{x_2}\) and \(\phi(u_{x_1}^2 - u_{x_2}^2)\) belong to \(h\dot K_{q}^{2(1-1/q), 1}(\mathbb R^2)\). This is a generalization of a result of \textit{L. C. Evans} and \textit{S. Müller} (the case \(q=1\)) [J. Am. Math. Soc. 7, No. 1, 199-219 (1994; Zbl 0802.35120)]. As in Evans and Müller, it is noted that in the radial case, nonnegativity of \(w\) is not required. They also point out that a sequence \(f_k\) bounded in \(h\dot K_{q}^{n(1-1/q), 1}(\mathbb R^n)\) \((1<q<\infty)\) and converging a.e. to a locally integrable function \(f\), converges distributionally to \(f\), and \(f\) is in \(h\dot K_{q}^{n(1-1/q), 1}(\mathbb R^n)\) (a local version of a result of Jones-Journé).
scientific article; zbMATH DE number 1267250
scientific article