<p>We study the Fourier transform <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> as a bounded but non-compact operator and quantify its degree of non-compactness via strict singularity and finite strict singularity (Bernstein numbers). For <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we prove a dichotomy for <Equation ID="Equ15"> <EquationSource Format="TEX">$ \mathcal {F}:L^p(\mathbb {R}^n)\rightarrow L^{p',r}(\mathbb {R}^n), \qquad \tfrac{1}{p}+\tfrac{1}{p'}=1, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">F</mi> <mo>:</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>r</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mo>′</mo> </msup> </mfrac> </mstyle> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>namely: non-compact and not strictly singular at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$r=p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> (so <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^{p',p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is optimal), and finitely strictly singular for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$r>p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$2<p\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> we show that <Equation ID="Equ16"> <EquationSource Format="TEX">$ \mathcal {F}:L^p(\mathbb {R}^n)\rightarrow B_p^{\,s}(\mathbb {R}^n) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">F</mi> <mo>:</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>B</mi> <mi>p</mi> <mrow> <mspace width="0.166667em" /> <mi>s</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>is not strictly singular at <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$s= d_p^n{:}{=}2n(\frac{1}{p}-\frac{1}{2})<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <msubsup> <mi>d</mi> <mi>p</mi> <mi>n</mi> </msubsup> <mo>:</mo> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and is finitely strictly singular for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$s<d_p^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo><</mo> <msubsup> <mi>d</mi> <mi>p</mi> <mi>n</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>; the dual results for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {F}:B_p^{\,s}\rightarrow L^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>:</mo> <msubsup> <mi>B</mi> <mi>p</mi> <mrow> <mspace width="0.166667em" /> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq14"> <EquationSource Format="TEX">$1<p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) is also studied.</p>

Notes on Non-Compactness of Fourier Transforms on $L^p$ Spaces

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2026年7月7日

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Jan Lang,
David E. Edmunds

摘要

We study the Fourier transform $\mathcal {F}$ $F$ on $\mathbb {R}^n$ $R^{n}$ as a bounded but non-compact operator and quantify its degree of non-compactness via strict singularity and finite strict singularity (Bernstein numbers). For $1<p<2$ $1 < p < 2$ we prove a dichotomy for $ \mathcal {F}:L^p(\mathbb {R}^n)\rightarrow L^{p',r}(\mathbb {R}^n), \qquad \tfrac{1}{p}+\tfrac{1}{p'}=1, $ $F : L^{p} (R^{n}) \to L^{p^{'}, r} (R^{n}), \frac{1}{p} + \frac{1}{p^{'}} = 1,$ namely: non-compact and not strictly singular at $r=p$ $r = p$ (so $L^{p',p}$ $L^{p^{'}, p}$ is optimal), and finitely strictly singular for $r>p$ $r > p$ . For $2<p\le \infty $ $2 < p \leq \infty$ we show that $ \mathcal {F}:L^p(\mathbb {R}^n)\rightarrow B_p^{\,s}(\mathbb {R}^n) $ $F : L^{p} (R^{n}) \to B_{p}^{s} (R^{n})$ is not strictly singular at $s= d_p^n{:}{=}2n(\frac{1}{p}-\frac{1}{2})<0$ $s = d_{p}^{n} : = 2 n (\frac{1}{p} - \frac{1}{2}) < 0$ , and is finitely strictly singular for $s<d_p^n$ $s < d_{p}^{n}$ ; the dual results for $\mathcal {F}:B_p^{\,s}\rightarrow L^p$ $F : B_{p}^{s} \to L^{p}$ ( $1<p<2$ $1 < p < 2$ ) is also studied.

Notes on Non-Compactness of Fourier Transforms on \(L^p\) Spaces

摘要