<p>For a tempered distribution <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ g $</EquationSource> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0< p, q, r < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq3.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we show that the operator norm of a Fourier paraproduct <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Pi _g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation>, of the form <Equation ID="Equ77"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_Equ77.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </MediaObject> <EquationSource Format="TEX">$ \Pi _{g}(f) := \sum _{j \in {\mathbb {Z}}} (\varphi _{2^{-j}} * f) \Delta _jg, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mi>g</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mi>j</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </munder> <mrow> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <msup> <mn>2</mn> <mrow> <mo>-</mo> <mi>j</mi> </mrow> </msup> </msub> <mrow /> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>j</mi> </msub> <mi>g</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$ H^p({\mathbb {R}}^n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$ {\dot{H}}^q({\mathbb {R}}^n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> </mrow> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is comparable to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3173_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Vert g\Vert _{{\dot{H}}^r({\mathbb {R}}^n)} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>g</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mrow> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> </mrow> <mi>r</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation>. We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.</p>

The operator norm of paraproducts on Hardy spaces

期刊论文
发表日期：

2025年5月19日

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Shahaboddin Shaabani

摘要

For a tempered distribution $ g $ $g$ , and $ 0< p, q, r < \infty $ $0 < p, q, r < \infty$ with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$ $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$ , we show that the operator norm of a Fourier paraproduct $\Pi _g$ $Π_{g}$ , of the form $ \Pi _{g}(f) := \sum _{j \in {\mathbb {Z}}} (\varphi _{2^{-j}} * f) \Delta _jg, $ $Π_{g} (f) : = \sum_{j \in Z} (φ_{2^{- j}} * f) Δ_{j} g,$ from $ H^p({\mathbb {R}}^n) $ $H^{p} (R^{n})$ to $ {\dot{H}}^q({\mathbb {R}}^n) $ ${\dot{H}}^{q} (R^{n})$ is comparable to $ \Vert g\Vert _{{\dot{H}}^r({\mathbb {R}}^n)} $ ${‖ g ‖}_{{\dot{H}}^{r} (R^{n})}$ . We also establish a similar result for dyadic paraproducts acting on dyadic Hardy spaces.