Abstract
We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.
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Notes
Grafakos in [21, p.63] uses the Fourier transform of P, but an exact formula still seems essential.
where \(\vec {q}\) is the conjugate of \(\vec {p}\) i.e \(\vec {q}=(q_1,\dots ,q_n)\) such that \(q_j=p_j',\;j=1,\dots ,n.\)
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Supported by the Danish Council for Independent Research | Natural Sciences, Grant 12-124675, “Mathematical and Statistical Analysis of Spatial Data”.
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Appendix
Appendix
1.1 Proof of Lemma 4.2
Proof
Let \(\phi \in {\mathcal {S}}\) such that
We set \(\varphi (x):=\phi (x)-2^{-\nu }\phi (2^{-\vec {a}}x)\) and \(\psi _k(x)=\varphi _{(k)}(x)\) for every \(k\in {\mathbb {N}}\). Then \(\widehat{\psi _k}(\xi )={\hat{\varphi }}(2^{-k\vec {a}}\xi )={\hat{\phi }}(2^{-k\vec {a}}\xi )-{\hat{\phi }}(2^{(1-k)\vec {a}}\xi ),\;k\in {\mathbb {N}}\). We denote by \(\psi _0=\phi \) and we have \(\mathrm{{supp}\, }\widehat{\psi _0}\subset \{|\xi |\le 2\}=:T_0\) and
Also for every \(\xi \in {\mathbb {R}}^n\), \(\lim \limits _{k\rightarrow \infty } {\hat{\phi }}(2^{-k\vec {a}}\xi )=1\), so
We have \(|{\hat{\Phi }}(\xi )|\ge 1/2\) when \(|\xi |\le 2\) (or we modify properly \(\Phi \) and keep the same notation), then by (6.8)
where \({\hat{\eta }}^{(k)}(\xi )=\widehat{\psi _k}(\xi )\Big ({\hat{\Phi }}(2^{-k\vec {a}}\xi )\Big )^{-1}{\hat{\Psi }}(\xi ),\) which gives (4.3).
By (6.7) it is \(\mathrm{{supp}\, }{\hat{\eta }}^{(k)}\subset T_k,\;k\ge 0\). Let \(N>\nu \) and \(|\beta |< \frac{N-\nu }{a_M}\), then
Let \(k\in {\mathbb {N}}\). For every \(\xi \in T_k\) it is \(\xi =2^{k\vec {a}}\zeta \), for some \(\zeta \) with \(2^{-a_M}\le |\zeta |\le 2.\) By (2.8) and (2.3)
which is also true for \(\xi \in T_0\), so
On the other hand
where for every \(\gamma \in ({{\mathbb {N}}}\cup \{0\})^n,\;x^{\gamma _+}=x_1^{{\gamma _1}_+}\cdots x_n^{{\gamma _n}_+}\), and \({\gamma _j}_+=\max (\gamma _j,0).\)
Then by (3.5), (2.11), and since \(N>|\beta |a_M+\nu \)
Furthermore \(|\partial ^\alpha {\hat{\psi }}_k(\xi )|,|\partial ^\alpha ({\hat{\Phi }}(2^{-k\vec {a}}\xi ))^{-1}(\xi )|\le c_\alpha 2^{-k\vec {a}\cdot \alpha }.\) The last estimate and (6.12) give
for \(|\alpha |\le N_1\).
Finally (2.6) leads to
since \(|T_k|=\int _{T_k} \text {d}y\le 2^{k\nu } \int _{\{|\xi |\le 2\}} \text {d}z =c2^{k\nu }\), where we set \(y=2^{k\vec {a}}z\). \(\square \)
1.2 Proof of Lemma 4.4.
Proof
Let \(\varphi \in {\mathcal {S}}_*\). We assume without loss of generality that \({\hat{\varphi }}(0)=1.\) Then \(\lim \limits _{k\rightarrow \infty } {\hat{\varphi }}(2^{-k\vec {a}}\xi )=1,\;\xi \in {\mathbb {R}}^n\). We set \(N:=M+2\) and we observe that
In addition
which implies that
since \(\lim \limits _{k\rightarrow \infty }{\hat{\varphi }}(2^{-k\vec {a}}\xi )=1\). By (6.13) and (6.14) we obtain
It follows that
where
and
It is easy to see that
so \(\Psi \in {\mathcal {S}}_M\). We write (6.15) with \(\xi \) be replaced by \(2^{-\vec {a}}\xi \)
and subtract (6.16) from (6.15) to get
Let \(m>j\) and \(f\in {\mathcal {S}}'\). Then by (6.17)
Now (4.13) follows simply by inverting the Fourier transform. \(\square \)
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Cleanthous, G., Georgiadis, A.G. & Nielsen, M. Anisotropic Mixed-Norm Hardy Spaces. J Geom Anal 27, 2758–2787 (2017). https://doi.org/10.1007/s12220-017-9781-8
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DOI: https://doi.org/10.1007/s12220-017-9781-8