Local Hardy Spaces and the Tb Theorem

This paper presents a theoretical exploration of local Hardy spaces associated with special para-accretive functions, denoted as \(h_b^p(X)\) $h_b^p(X)$ , where X is a space of homogeneous type. In pursuit of this goal, we establish inhomogeneous Plancherel–Pôlya inequality and subsequently derive the atomic and block decomposition characterizations for \(h_b^p(X)\) $h_b^p(X)$ . Furthermore, we culminate our study by demonstrating the boundedness of \((\delta ,\sigma )\) $(\delta ,\sigma )$ -type inhomogeneous Calderón–Zygmund operators on \(h_b^p(X)\) $h_b^p(X)$ with \(\max \{\frac{1}{1+\delta },\frac{1}{1+\sigma }\}<p\le 1\) $max\{\frac{1}{1+\delta },\frac{1}{1+\sigma }\}<p\le 1$ when \(T_b^*(1)\in Lip_b(\varepsilon )\) $T_b^{\ast }(1)\in Li{p}_b(\epsilon )$ , where \(\varepsilon \) $\epsilon$ represents the regularity exponent of the approximation to the identity.