We consider the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {H}_L ^{s,r} (O)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mi>L</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> consisting of all local Sobolev distributions of order s on an open set O whose Sobolev wave front set of order r is contained in the closed conic set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L\subseteq O\times (\mathbb {R}^m\backslash \{0\})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>⊆</mo> <mi>O</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>m</mi> </msup> <mo stretchy="true">\</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We introduce a locally convex topology on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {H}_L ^{s,r} (O)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mi>L</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and show that the ordinary product of smooth functions uniquely extends to a continuous bilinear mapping <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {H}_{L_1} ^{r_1,r'} (O) \times \mathscr {H}_{L_2} ^{r_2,r''} (O) \rightarrow \mathscr {H}_{L} ^{s,r} (O)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msubsup> <mi mathvariant="script">H</mi> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>,</mo> <msup> <mi>r</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>L</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for appropriate s and r when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are in a favorable position. The key ingredient in our proof is to employ Hörmander’s idea of considering the pullback by the diagonal map <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x\mapsto (x,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the tensor product of two distributions.

On the continuity of the product of distributions in local Sobolev spaces

期刊论文
发表日期：

2026年3月23日

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Stefan Tutić

摘要

We consider the space $\mathscr {H}_L ^{s,r} (O)$ $H_{L}^{s, r} (O)$ consisting of all local Sobolev distributions of order s on an open set O whose Sobolev wave front set of order r is contained in the closed conic set $L\subseteq O\times (\mathbb {R}^m\backslash \{0\})$ $L \subseteq O \times (R^{m} \ {0})$ . We introduce a locally convex topology on $\mathscr {H}_L ^{s,r} (O)$ $H_{L}^{s, r} (O)$ and show that the ordinary product of smooth functions uniquely extends to a continuous bilinear mapping $\mathscr {H}_{L_1} ^{r_1,r'} (O) \times \mathscr {H}_{L_2} ^{r_2,r''} (O) \rightarrow \mathscr {H}_{L} ^{s,r} (O)$ $H_{L_{1}}^{r_{1}, r^{'}} (O) \times H_{L_{2}}^{r_{2}, r^{''}} (O) \to H_{L}^{s, r} (O)$ , for appropriate s and r when $L_1$ $L_{1}$ and $L_2$ $L_{2}$ are in a favorable position. The key ingredient in our proof is to employ Hörmander’s idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ $x \mapsto (x, x)$ of the tensor product of two distributions.