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 \(h_b^p(X)\), 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 \(h_b^p(X)\). Furthermore, we culminate our study by demonstrating the boundedness of \((\delta ,\sigma )\)-type inhomogeneous Calderón–Zygmund operators on \(h_b^p(X)\) with \(\max \{\frac{1}{1+\delta },\frac{1}{1+\sigma }\}<p\le 1\) when \(T_b^*(1)\in Lip_b(\varepsilon )\), where \(\varepsilon \) represents the regularity exponent of the approximation to the identity.