Local Hardy spaces and summability of Fourier transforms

Characterization of inclusion relations between Wiener amalgam and some classical spaces

In our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space h dis p ⁢ ( ℝ n 1 × ℝ n 2 ) {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

The inclusion relations between α-modulation spaces and Lp-Sobolev spaces or local Hardy spaces

Let \(L^{q ( x ) } (\mathbb{R} ) \) be variable exponent Lebesgue space and \(\ell^{ \{ q_{n} \} }\) be discrete analog of this space. In this work we define the amalgam spaces W(L p(x),L q(x)) and \(W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } ) \), and discuss some basic properties of these spaces. Since the global components \(L^{q ( x ) } (\mathbb{R} ) \) and \(\ell^{ \{ q_{n} \} }\) are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space \(\ell^{ \{ q_{n} \}}\) is the discrete space of \(L^{q ( x ) } (\mathbb{R} )\). By using this result we prove that \(W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} ) \). We also study the frame expansion in \(L^{p ( x ) } ( \mathbb{R} )\). At the end of this work we prove that the Hardy–Littlewood maximal operator from \(W ( L^{s ( x ) },\ell^{ \{t_{n} \} } ) \) into \(W ( L^{u ( x ) },\ell^{ \{v_{n} \} } ) \) is bounded under some assumptions.

We prove that any distribution q satisfying the grad-div system \({\nabla q={\rm div}\,{\bf f}}\) for some tensor \({{\bf f}=(f^i_j), \,f^i_j\in h^r(U)\,(1\leq r < \infty}\)) -the local Hardy space; q is in hr and q is locally represented by the sum of singular integrals of \({f^i_j}\) with Calderon-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incompressible elastic energy-minimizing deformation u satisfying \({|\nabla{\bf u}|^2,\,|{\rm cof}\,\nabla{\bf u}|^2\in h^1}\). We also derive the system of Euler–Lagrange equations for volume preserving local minimizers u that are in the space \({K^{1,3}_{\rm loc}}\) [defined in (1.2)]—partially resolving a long standing problem. In two dimensions we prove partial C1,α regularity of weak solutions provided their gradient is in L3 and p is Holder continuous.

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 &lt; p &lt; ∞ {1&lt;p&lt;\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 &lt; p ≤ 1 {0&lt;p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 &lt; p &lt; ∞ {1&lt;p&lt;\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 &lt; p ≤ 1 {0&lt;p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 &lt; p ≤ 1 {0&lt;p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 &lt; p ≤ 1 {0&lt;p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

The aim of this paper is to establish the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ generated by the bilinear pseudo-differential operator $$T_{\sigma }$$ with smooth symbols and $$b_{1},\ b_{2}\in \mathrm {BMO}({\mathbb {R}}^{n})$$ on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator $$T_{\sigma }$$ is bounded from the Lebesgue space $$L^{p}({\mathbb {R}}^{n})$$ into $$h^{p_{1}(\cdot )}({\mathbb {R}}^{n})\times h^{p_{2}(\cdot )}({\mathbb {R}}^{n})$$ . Moreover, the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ on product of local Hardy spaces with variable exponents is also obtained.

Let $$({{\mathcal {X}}},d,\mu )$$ be an RD-space, $$H^1_{\rho }({{\mathcal {X}}})$$ , and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$ be, respectively, the local Hardy space and the local BMO space associated with an admissible function $$\rho $$ . Under an additional assumption that there exists a specific generalized approximation of the identity, the authors prove that the product $$f\times g$$ of $$f\in H^1_{\rho }({{\mathcal {X}}})$$ and $$g\in {\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$ , viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $$H^1_{\rho }({{\mathcal {X}}})\times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})$$ into $$L^1({{\mathcal {X}}})$$ and from $$H^1_{\rho }({{\mathcal {X}}}) \times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})$$ into $$H^{\log }({{\mathcal {X}}})$$ , which is of wide generality. The authors also give out four applications of this result to Schrodinger operators, respectively, over different underlying spaces, where three of these applications are new.

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

Remarks on Fourier multipliers and applications to the wave equation

Inclusion relations between [formula omitted]-Sobolev and Wiener amalgam spaces

Invertibility of the Gabor frame operator on the Wiener amalgam space

A general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebraW(C,ℓ1)($\mathbb{R}$d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from theHpHardy space toLp(orHp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.

Tauberian conditions for the recovery of convergence out of summability, depending on a parameter cease to be Tauberian conditions for some limiting values of the parameter. This situation motivates the following question. Are there some more general summability methods for which the limiting case would become a Tauberian condition for the behavior of {Sn(a)} ? A general summability method is designed whose special cases lead to generalized Abel’s summability methods. The Tauberian theorems for those Abel’s summability methods contain as a particular case the answer to the above question.

Div–curl type estimates for elliptic systems of complex vector fields