<p>In this work, we investigate the regularity of bilinear Fourier integral operators when acting on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1278_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2\times L^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. Specifically, we analyze these operators within the framework of Besov spaces, establishing precise regularity conditions under which they are well-defined and bounded. Our main contribution extends and refines previous results, particularly the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1278_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2\times L^2\rightarrow B^{0}_{1,q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo stretchy="false">→</mo> <msubsup> <mi>B</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> boundedness of such operators with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1278_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le q\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, by considering symbols belonging to the bilinear Hörmander symbols set.</p>

On the Hörmander classes of bilinear Fourier integral operators

期刊论文
发表日期：

2025年7月14日

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Chaimaa Bouchama,
Omar Farouk Aid,
Abderrahmane Senoussaoui

摘要

In this work, we investigate the regularity of bilinear Fourier integral operators when acting on $L^2\times L^2$ $L^{2} \times L^{2}$ . Specifically, we analyze these operators within the framework of Besov spaces, establishing precise regularity conditions under which they are well-defined and bounded. Our main contribution extends and refines previous results, particularly the $L^2\times L^2\rightarrow B^{0}_{1,q}$ $L^{2} \times L^{2} \to B_{1, q}^{0}$ boundedness of such operators with $1\le q\le \infty $ $1 \leq q \leq \infty$ , by considering symbols belonging to the bilinear Hörmander symbols set.