<p>The aim of this work is to provide a characterization for the Besov spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B^{r,q}_p(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mi>p</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>q</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> </InlineEquation> via the continuous shearlet transform in higher dimensions with isotropic dilations to show the regularity of weak solutions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u \in W^{m,p}(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>p</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> </mrow> </math></EquationSource> </InlineEquation> under the equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Qu =f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>, so that for a given <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f \in B^{r,q}_p(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msubsup> <mi>B</mi> <mi>p</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>q</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> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u \in B^{r+m,q}_p(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>B</mi> <mi>p</mi> <mrow> <mi>r</mi> <mo>+</mo> <mi>m</mi> <mo>,</mo> <mi>q</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> </InlineEquation>. In this case, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1< p,q < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0< r < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Q = \sum _{\beta \le m}c_{\beta } \partial ^{\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>β</mi> <mo>≤</mo> <mi>m</mi> </mrow> </msub> <msub> <mi>c</mi> <mi>β</mi> </msub> <msup> <mi>∂</mi> <mi>β</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a partial differential operator of pure order <i>m</i> with positive constant coefficients <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$c_{\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>β</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Regularity of Weak Solutions on Besov Spaces via the Continuous Shearlet Transform in Higher Dimensions

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2026年5月21日

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Jaime Navarro,
Victor A. Cruz-Barriguete

摘要

The aim of this work is to provide a characterization for the Besov spaces $B^{r,q}_p(\mathbb {R}^n)$ $B_{p}^{r, q} (R^{n})$ via the continuous shearlet transform in higher dimensions with isotropic dilations to show the regularity of weak solutions $u \in W^{m,p}(\mathbb {R}^n)$ $u \in W^{m, p} (R^{n})$ under the equation $Qu =f$ $Q u = f$ , so that for a given $f \in B^{r,q}_p(\mathbb {R}^n)$ $f \in B_{p}^{r, q} (R^{n})$ , $u \in B^{r+m,q}_p(\mathbb {R}^n)$ $u \in B_{p}^{r + m, q} (R^{n})$ . In this case, $1< p,q < \infty $ $1 < p, q < \infty$ , $0< r < 1$ $0 < r < 1$ , and where $Q = \sum _{\beta \le m}c_{\beta } \partial ^{\beta }$ $Q = \sum_{β \leq m} c_{β} \partial^{β}$ is a partial differential operator of pure order m with positive constant coefficients $c_{\beta }$ $c_{β}$ .