The spaces \(\mathcal{H}^{(q,p)}\) were introduced by the authors in [J. Math. Anal. Appl. 455, No. 2, 1899–1936 (2017; Zbl 1371.42024)] as analogues of the real Hardy spaces \(H^p\), replacing the condition that the maximal function \(\mathcal{M}_\varphi(f)=\sup_{t>0}\vert f\ast\varphi_t\vert \) (for suitable smooth \(\varphi\) supported in the unit ball of \(\mathbb{R}^d\)) with the condition that the same maximal function lies in the amalgam space \((L^q,\ell^p)\) with \(\Vert f\Vert_{q,p}=\Vert \{\Vert f\chi_{Q_k}\Vert_q\}\Vert_{\ell^p}\) where \(Q_k=k+[0,1)^d\). Corresponding local Hardy-type spaces are defined in terms of \(\mathcal{M}_\varphi(f)=\sup_{0<t<1}\vert f\ast\varphi_t\vert \).
This work addresses two questions. The first is whether the inclusions of \(\mathcal{H}^{(1,p)}\) in \((L^1,\ell^p)\) and of \(\mathcal{H}^{(q,p)}\) in \(\mathcal{H}^{(q,p)}_{\text{ loc}}\) (\(0<q\leq \min\{p,1\}\), \(p<\infty\)) are strict. The second is whether certain known bounds on Calderón-Zygmund operators on Hardy spaces and their duals extend to the amalgam setting. Duals of the spaces \(\mathcal{H}^{(q,p)}\) were identified by the authors in [Acta Math. Sin., Engl. Ser. 38, No. 3, 519–546 (2022; Zbl 1491.42032)]. Up to isomorphism they are Campanato-type spaces \(\mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}\) (\(r^\prime=r/(r-1)\), \(\delta\geq \lfloor d(1/q-1)\rfloor\)) with \(\phi_1: \, {\text {cubes}}\to (0,\infty)\) defined by \(\phi_1(Q)=\frac{\Vert \chi_Q\Vert_{q,p}}{\vert Q\vert}\). The infimum of constants \(C\) such that \[\sum_{j\in\mathbb{Z}} 2^j O(g,\Omega^j,r)\leq C\Vert \sum_{j\in\mathbb{Z}} 2^{j\eta}\chi_{\Omega^j}\Vert_{\frac{q}{\eta},\frac{p}{\eta}}^{\frac{1}{\eta}}\] defines a norm on \(\mathcal{L}^{(q,p,\eta)}_{r',\phi_1,\delta}\) where \(\{\Omega^j\}\) ranges over families of open sets such that \(\Vert \sum_{j\in\mathbb{Z}}2^{j\eta}\chi_{\Omega_j}\Vert_{\frac{q}{\eta},\frac{p}{\eta}}<\infty\), and \[O(g,\Omega^j,r)=\sup\sum_{n\geq 0}\vert Q^n\vert ^{1/r'}\Vert (g-P^\delta_{\tilde{Q}^n}(g))\chi_{\tilde{Q}^n}\Vert_r \] with \(P^\delta_Q(g)\) the unique polynomial of degree at most \(\delta\) such that \(g-P^\delta_{Q}(g)\) is orthogonal on \(Q\) to all polynomials of degree at most \(\delta\). The supremum is taken over all families of cubes whose unions are contained in \(\Omega\) and such that \(\sum_n \chi_{Q^n}\leq K\) for some constant \(K\) depending only on dimension \(d\). A corresponding description applies to the dual of the local Hardy-type space.
It is demonstrated that the inclusion \(\mathcal{H}^{1,p}\subset (L^1,\ell^p)\) is strict (in the same way that the real Hardy space is strictly contained inside of \(L^1\)). It is also proven that the inclusion \(\mathcal{H}^{q,p}\subset \mathcal{H}^{q,p}_{\text{loc}}\) is strict. These facts are proven, but not stated as theorems.
Let \(X\) stand for the amalgam space \((L^q,\ell^p)\). Define \(\mathcal{L}_{X,r,\delta,\eta}\) in terms of \[ \vert f\Vert_{X,r,\delta,\eta}=\sup\frac{\sum_{j=1}^m \frac{\lambda_i}{\Vert \chi_{B_i}\Vert_X}\Bigl(\frac{1}{\vert B_j\vert }\int_{B_j}\vert f(x)-P_{B_j}^\delta(f)(x)\vert ^r\Bigr)^{1/r} }{\Vert \{\sum_{i=1}^m\frac{\lambda_i}{\Vert \chi_{B_i}\Vert_X^\eta}\chi_{B_i}\}^{1/\eta}\Vert_X}\] where the supremum is taken over all finite collections of balls \(\{B_j\}\) and \(\{\lambda_j\}\subset [0,R)\) for some \(R>0\).
It is shown that \(\mathcal{L}_{X,r,\delta,\eta}\) is continuously embedded in \(\mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}\). It is also stated as unknown whether a reverse inclusion of \(\mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}\) inside \(\mathcal{L}_{X,r,\delta,\eta}\) holds. Boundedness properties of certain Calderón-Zygmund operators are also proven (Theorems 4.4 and 4.7). Let \(1<p<\infty\), \(1\leq r<p^\prime\), \(0<\eta<1\) (allowing \(\eta=1\) if \(r=1\)) and let \(K\) be a tempered distribution such that \(\vert \widehat{K}\vert \leq A\) and \(\vert \partial^\beta K(x)\vert \leq \frac{B}{\vert x\vert ^{d+\beta}}\) for all multi-indices \(\vert \beta\vert \leq \delta\). It is proven that \(T(f)=K\ast f\) (\(f\in\mathcal{S}\)) is extendable to \((L^\infty,\ell^{p^\prime})\) and that there is a \(C>0\) such that for all \(f\in (L^\infty,\ell^{p^\prime})\), \[\Vert T(f)\Vert_{\mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}}\leq C\Vert f\Vert_{\infty,p^\prime}\, .\] Supposing further that \(\frac{d}{d+\delta}q\leq 1<p<\infty\), and otherwise \(p\), \(r\), \(\eta\), \(q\) are as above, one further has for all \(f\in \mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}\) that \[\Vert T(f)\Vert_{\mathcal{L}^{(q,p,\eta)}_{r^\prime,\phi_1,\delta}}\leq C\Vert f\Vert_{\mathcal{L}^{(q,p,\eta)}_{r',\phi_1,\delta}}\, .\]