Preduals of Quadratic Campanato Spaces Associated to Operators with Heat Kernel Bounds

Abstract

Let L be a nonnegative, self-adjoint operator on $L^{2}(\mathbb {R}^{n})$ with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space $\mathcal {L}^{2,\lambda }_{-\Delta }(\mathbb R^{n})$ , in Duong et al. (J. Fourier Anal. Appl. 13:87–111, 2007) the quadratic Campanato space $\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})$ is defined by a variant of the maximal function associated with the semigroup {e −t L } t≥0. On the basis of Dafni and Xiao (J. Funct. Anal. 208:377–422, 2004) and Yang and Yuan (J. Funct. Anal. 255:2760–2809, 2008) this paper addresses the preduality of $\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})$ through an induced atom (or molecular) decomposition. Even in the case L = −Δ the discovered predual result is new and natural.